Korea Institute for Advanced Study

Volume 1 (2021)

The KIAS EXPOSITIONSNam-Gyu Kang, Editor-in-Chief

About the Journal

An official journal of the Korea Institute for Advanced Study, the KIASEXPOSITIONS is designed for high quality expository articles in all areasof mathematics. The authors are invited to submit “articles with overview”,presenting problems of current research with its history and developments,the current states and possible future directions. The article should bepresented to a wider audience than just a few specialists.

i

Organization

Editor-in-ChiefNam-Gyu Kang Korea Institute for Advanced Study

Editorial BoardJaigyoung Choe Korea Institute for Advanced StudyKyeongsu Choi Korea Institute for Advanced StudySang-hyun Kim Korea Institute for Advanced Study

Submission Guidelines

Required ItemsA brief abstract of about 200 words or less and a bibliography must be

included in a submission. The abstract should be self-contained and notmake any reference to the bibliography.

FormatWe ask the authors to submit all source material by email to here. Au-

thors are encouraged to use LaTeX. The class file for the journal is availablefrom here.

Copyright AgreementAuthors of papers which have been accepted for publication will be asked

to sign a copyright agreement.(pdf)

ProofsPage proofs will be made available to authors (or to the designated

corresponding author) in PDF format. If requested by the author, we willalso send her/him a paper copy of the proof.

iii

Contents

About the Journal i

Organization iii

Submission Guidelines iii

Enumerative geometry, before and after string theory 1Young-Hoon Kiem

v

Enumerative geometry, before and after stringtheory

Young-Hoon Kiem

Department of Mathematics and Research Institute of MathematicsSeoul National University, Seoul 08826, Korea

kiem@snu.ac.kr

Abstract Throughout its long history, algebraic geometry has beenenriched by the influences of invariant theory, abstract algebra, num-ber theory, topology, complex analysis, symplectic geometry, categorytheory and so on. One of the most recent influences came from stringtheory circa 1990 when a group of string theorists applied the mirrorsymmetry in string theory to enumerative geometry problems in alge-braic geometry. During the past three decades, we have witnessed oneof the most fruitful interactions between mathematicians and physi-cists in enumerative geometry. In this informal expository article, Iwould like to offer a glimpse into this story about enumerative geom-etry and string theory.

1. IntroductionAccording to Hermann Schubert in 1874, enumerative geometry is the

study about questions like

Question 1.1. How many geometric figures of fixed type satisfy certaingiven conditions?

It is one of the oldest branches of mathematics and dates back to at least200 BCE when Apollonius raised the question:

Question 1.2. How many circles in plane are tangent to three givencircles?

To simplify, let us assume that the disks bounded by the three circlesare disjoint from one another. If a circle X is tangent to the given three

Received: June 11, 2020Kiem is partially supported by Samsung Science and Technology Foundation SSTF-

BA1601-01

1

Young-Hoon Kiem

circles A1, A2, A3, then each of the three circles may lie inside or outside ofX, and hence there are 8 possibilities. Obviously X is determined by threenumbers, namely the coordinates pa, bq of the center and the radius r. Thetangency conditions for the three circles give us three quadratic equations,(0.1) pa´ aiq

2 ` pb´ biq2 “ pr ˘ riq

2, i “ 1, 2, 3

where pai, biq is the center and ri is the radius of Ai. The sign in the righthand side is determined by the choice among eight possibilities of lying insideor outside. It is easy to solve (0.1) and see that the answer to Apolloniusproblem is 8. Of course, one may solve this problem by plane geometrywithout the help of algebra, starting with the fact that the locus1 of centersof circles tangent to two given circles is a hyperbola and Apollonius asksyou to look for the intersection of three hyperbolas.

A circle is the zero locus in plane of a quadratic polynomial of the formx2 ` y2 ` ¨ ¨ ¨ where ¨ ¨ ¨ means lower order terms. More generally, the zerolocus in plane of any quadratic polynomial is called a conic (curve) becausethey are obtained by intersecting a circular cone with planes.2 A moment’sthought may convince the reader that it requires 5 numbers to determine aconic. For instance, to determine an ellipse, we need to know its center, thelengths of long and short axes and the slope of the long (or short) axis. So weneed five equations to determine these numbers (because we may delete onevariable by using one equation) and an equation for the defining parametersusually comes from a geometric constraint like tangency. In 1848, J. Steinerthought about a seemingly naive generalization of Apollonius’ question.

Question 1.3. How many conics in plane are tangent to five givenconics?

Steiner correctly argued that the set of conics tangent to a given conicis the zero locus of degree 6 (sextic) polynomial in 5 variables3 and thus weare looking for the number of solutions of a system of 5 sextic equations in5 variables. So Steiner’s problem is related to the algebraic question:

Question 1.4. Given a system of 5 general4 sextic polynomial equationsin 5 variables, how many solutions are there? More generally, given a system

1The meaning of locus is the same as set, often with a touch of geometric fla-vor (sometimes subvarieties, subschemes or substacks). The zero locus of polynomialsf1, ¨ ¨ ¨ , fr means the set of points p whose coordinates satisfy all the polynomial equa-tions f1ppq “ 0, ¨ ¨ ¨ , frppq “ 0.

2A more appropriate term for a conic from the perspective of modern algebraic ge-ometry is a quadratic curve.

3It suffices to consider a pencil tXtutPP1 of conics and find the number of conics inthis pencil which are tangent to a given smooth conic C. Here a curve is smooth if andonly if the tangent space at each point is 1-dimensional. In particular, a plane curve Cdefined by a polynomial f is smooth if and only if the gradient vector of f at every pointin C is nonzero. The intersection Xt XC is a divisor of degree 4 in C – P1 and hence givesus a a degree 4 map C Ñ P1. There are precisely 6 ramification points by the Hurwitzformula (6 “ 2 ¨ 4 ´ 2). Hence the pencil intersects with the locus of conics tangent to Cat 6 points.

4We want to avoid extreme cases which behave badly, like the case where there areinfinitely many solutions. The word general means that we are excluding such unfortunatepossibilities. Mathematically, it means that we are choosing the parameters (coefficientsof the equations in this case) in an open dense subset (in the Zariski topology).

2

Enumerative geometry, before and after string theory

of r polynomial equations of degrees d1, ¨ ¨ ¨ , dr respectively, in r variables,how many solutions are there?

For an insight, it is always a good idea to consider a simplified versionof a difficult problem.

Question 1.5. If you have 2 general polynomial equations of degreesm,n in 2 variables, how many common solutions do they have?

Geometrically speaking, two general curves of degrees m,n in plane meetat mn points.5 Why? Since the coordinates of intersection points are thesolutions of polynomial equations which vary continuously as we vary thecoefficients of the polynomials, the number of intersection points remainsconstant as we vary the coefficients continuously and hence we can deformthe equations continuously until we reach

(0.2)ź

0ďiăm

px´ iq,ź

0ďjăn

py ´ jq

without changing the number of intersection points. Obviously the numberof intersection points of the curves defined by (0.2) is mn. This methodof deforming or degenerating into simple cases is one of the most impor-tant techniques in mathematics and in fact, a huge part of geometry andtopology is based on this single techinque of stability under degeneration ordeformation.6

The above geometric argument looks simple but actually requires manypages of justification to make it precise. Can you prove it algebraically? Thestarting point is the simple observation that a polynomial factorizes into aproduct of irreducible polynomials.7 If two polynomials(0.3) fpxq “ a0x

m ` a1xm´1 ` ¨ ¨ ¨ ` am´1x` am,

gpxq “ b0xn ` b1x

n´1 ` ¨ ¨ ¨ ` bn´1x` bn

in one variable of degrees m,n share a root, then they have a common factorand hence their least common multiple has degree less than mn. So we canfind polynomials h, k with deg h ă n, deg k ă m such that hf ` kg “ 0.If we think of the coefficients of h, k as variables and write the equationhf ` kg “ 0 as a system of linear equations in the coefficients, we obtain anpm` nq ˆ pm` nq matrix

(0.4)

¨

˚

˚

˚

˚

˚

˚

˚

˚

˚

˚

˝

a0 a1 a2 ¨ ¨ ¨ am 0 0 00 a0 a1 ¨ ¨ ¨ am´1 am 0 0

¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨

0 0 ¨ ¨ ¨ 0 a0 ¨ ¨ ¨ am´1 amb0 b1 b2 ¨ ¨ ¨ bn 0 0 00 b0 b1 ¨ ¨ ¨ bn´1 bn 0 0

¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨

0 0 ¨ ¨ ¨ 0 b0 ¨ ¨ ¨ bn´1 bn

˛

‹

‹

‹

‹

‹

‹

‹

‹

‹

‹

‚

5The degree of a plane curve is the degree of the defining polynomial.6The idea is the same but their names may vary in different branches of mathematics.7For a field F , a nonzero polynomial with coefficients in F is the product of irreducible

polynomials in a unique way up to the order of multiplication and units.

3

Young-Hoon Kiem

whose determinant is called the resultant of f, g and denoted by Rpf, gq.The two polynomials f, g have a common root if and only if hf ` kg “ 0for some nonzero polynomials h, k with deg h ă n, deg k ă m, which isequivalent to saying that their resultant Rpf, gq is zero.8

Now if fpx, yq and gpx, yq are polynomials in two variables, we maywrite them as (0.3) with ai, bj being polynomials in y of degrees ď i and ď jrespectively. It is not hard to see that the resultant Rpf, gq is a polynomialin y of degree mn for general f, g. In the general case where two curvesdefined by f and g intersect only at finitely many points, we may pick thecoordinate system so that the y coordinates of the intersection points are alldistinct, which means that the system fpx, yq “ 0 “ gpx, yq has at most onesolution if y is fixed. For the real numbers, the only conclusion we can drawis that there are at most mn solutions to the system fpx, yq “ 0 “ gpx, yq

because Rpf, gq “ 0 has at most mn solutions.We can do much better with complex numbers. As there is a solution for

any polynomial equation in one variable, the system fpx, yq “ 0 “ gpx, yq

has exactly one solution of the form px, y0q, for each (simple) root y0 ofRpf, gq “ 0. Thus if Rpf, gq “ 0 has k solutions, then so does the systemfpx, yq “ 0 “ gpx, yq. Moreover, if we use complex numbers, for general f, g,we can see that Rpf, gq “ 0 has exactly mn distinct solutions and hence thesystem fpx, yq “ 0 “ gpx, yq has exactly mn solutions. Therefore the answerto Question 1.5 is mn over C.

Remark. As you can see, for a consistent theory, it is much better touse the complex number field C (or an algebraically closed field) instead ofreal numbers. So from now on, we will always use complex numbers9 as ourscalars.

Motivated by our answer to Question 1.5, you may expect that generalpolynomial equations f1 “ ¨ ¨ ¨ “ fr “ 0 of degrees d1, ¨ ¨ ¨ , dr in r variablesshould have

śri“1 di solutions over C. Indeed this is true although it requires

a more sophisticated proof.10 In particular, 5 general sextic polynomialequations in 5 variables should have 65 “ 7776 solutions and this number isSteiner’s answer to Question 1.3.

8In fact, Rpf, gq is a constant multiple ofś

i,jpαi ´ βjq where tαiu1ďiďm (resp.tβju1ďjďn) are the roots of f “ 0 (resp. g “ 0).

9For real numbers, one can first look for complex solutions and see which of them areinvariant under conjugation.

10By Hilbert’s Nullstellensatz, the number of solutions is the same as the dimension

dimCrx1, ¨ ¨ ¨ , xrsxf1, ¨ ¨ ¨ , fry.

To compute this dimension, by adding an extra variable x0, homogenize f1, ¨ ¨ ¨ , fr intohomogeneous polynomials F1, ¨ ¨ ¨ , Fr of degrees d1, ¨ ¨ ¨ , dr. For ℓ large enough, we havean isomorphism

Crx1, ¨ ¨ ¨ , xrsxf1, ¨ ¨ ¨ , fry – Crx0, ¨ ¨ ¨ , xrsℓxF1, ¨ ¨ ¨ , Fryℓ

where the subscript ℓ stands for the degree ℓ homogeneous part. The dimension of theright hand side can be computed by the Koszul complex

0 ÐÝ Crx0, ¨ ¨ ¨ , xrsℓxF1, ¨ ¨ ¨ , Fryℓ ÐÝ Crx0, ¨ ¨ ¨ , xrsℓpF1,¨¨¨ ,Frq

ÐÝ ‘1ďiďr Crx0, ¨ ¨ ¨ , xrsℓ´di

ÐÝ ‘1ďiăjďrCrx0, ¨ ¨ ¨ , xrsℓ´di´dj ÐÝ ¨ ¨ ¨ ÐÝ Crx0, ¨ ¨ ¨ , xrsℓ´d1´¨¨¨´dr ÐÝ 0

4

Enumerative geometry, before and after string theory

It turned out that 7776 is the correct answer to Question 1.4 but notto Question 1.3. Why? The problem is that the 5 sextic polynomials forthe tangency of 5 given conics are never general. The correct answer toQuestion 1.3 was found to be 3264 by Chasles in 1864 but a mathematicallyrigorous proof was provided by Fulton and MacPherson, only in 1978! Ittook more than a century to settle such a naive looking problem.

Why 3264 instead of 7776? In the space of all conics, namely in thespace of all quadratic polynomials in two variables up to constant, there isa subspace of double lines, namely the set of squares of linear polynomials.Double lines are tangent to any conic although we are not interested in them.So in the space of conics, the subset Z of conics tangent to the 5 given conicsconsists of 3264 honest smooth conics and also infinitely many double lines.If we deform the defining equations of the 5 sextic polynomials defining thesubset Z, extra 4512 intersection points arise from the locus of double lineswhich we do not want to consider.11

Enumerative geometry is about natural questions on curves and surfacesetc but solving such problems often requires an immense amount of technicaltraining and ingenuities.

In his book What is mathematics?, Richard Courant wrote:Understanding of mathematics cannot be transmitted by painless entertain-ment any more than education in music can be brought by the most brilliantjournalism to those who never have listened intensively. Actual contact withthe content of living mathematics is necessary.

I agree that the best way to appreciate mathematics is walking throughcarefully selected problems. In this note, I will try to convey the ideas inenumerative geometry through explicit examples as much as possible. If youenjoyed so far, you will find more interesting stories below.

In the subsequent sections, we will see more of classical enumerativegeometry problems, its connection with string theory and recent develop-ments. Everything in this note was taken from books, papers and memories.No part of this note is original.

together with the identityr

ź

i“1

di “

rÿ

i“0

p´1qi

ÿ

1ďj1㨨¨ăjiďr

˜

r ` ℓ ´ dj1 ´ ¨ ¨ ¨ ´ djir

¸

.

11To remove the contribution from the locus of double lines, we blow up the space ofconics along the locus of double lines (Veronese surface) and then compute the intersectionnumber p6H ´ 2Eq5 “ 3264 where H is the hyperplane class and E is the exceptionaldivisor. The space of conics P5 is the moduli space of ideal sheaves (Hilbert scheme) andhence Steiner’s answer 7776 is a Donaldson-Thomas (DT) invariant in algebraic geometryor BPS state counting in physics terms. The blowup of P5 along the Veronese surfaceis the moduli space of stable maps to P2 of degree 2 and hence Chasles’s answer 3264 isthe Gromov-Witten (GW) invariant in modern terms. The GW/DT correspondence, orthe MNOP conjecture, claims that the GW invariants have the same amount of informa-tion on enumerative geometry of curves as the DT invariants. It is remarkable that thefirst nontrivial case of the GW/DT correspondence or the gauge/string duality alreadyappeared in mathematics, 150 years before string theory.

5

Young-Hoon Kiem

2. Classical enumerative geometryIt is great fun to play with enumerative geometry problems.

2.1. Schubert calculus. Schubert calculus is the enumerative geom-etry of lines, planes and more generally geometric figures defined by linearequations and sometimes quadratic polynomial equations. For instance, asingle linear equation in 3 variables defines a plane while two linear equa-tions usually define a line. Let us consider the following question which youmay find in a standard undergraduate algebraic geometry textbook.

Question 2.1. How many lines in 3 dimensional space meet 4 givengeneral lines?

A line in space tpx, y, zq |x, y, z scalarsu is the intersection of two planesax` by ` cz ` d “ 0, ex` fy ` gz ` h “ 0,

pa, b, cq ∦ pe, f, gq.

Since the equations are determined by the coefficients, we may just recordthem in the matrix form

(0.5) A “

ˆ

a b c de f g h

˙

P M2ˆ42

where M2ˆ4 denotes the set of 2 ˆ 4 matrices and the subscript 2 denotesthe open subset of matrices of rank 2. It is easy to see that two such 2 ˆ 4matrices A and A1 represent the same line if and only if A1 “ gA for aninvertible 2 ˆ 2 matrix g P GL2.

We denote by GLn the set of invertible n ˆ n matrices. The identitymatrix I lies in GLn and if ξ, ξ1 P GLn, then ξ´1, ξξ1 P GLn, i.e. GLnis a group. The closed subset of n ˆ n matrices ξ with determinant 1 isdenoted by SLn which is a subgroup of GLn because the determinant ismultiplicative. Constant multiples of I form another subgroup Z of GLnwhich is complementary to SLn.12

For A P M2ˆ42 , the set tξA | ξ P GL2u is called the orbit of A and the set

of orbits in M2ˆ42 is denoted by

GL2zM2ˆ42 “: Grp2, 4q.

Therefore the set X of lines in space is an open subset of Grp2, 4q, namelythe set of orbits of matrices whose left 2 ˆ 3 submatrix has rank 2. Whatis the geometry of X and Grp2, 4q, the orbit spaces by the action of GL2?Let us first consider the orbits by SL2 and later by Z. By the multiplicativeproperty detpξξ1q “ detpξq detpξ1q, we have a map

Φ : SL2zM2ˆ42 ÝÑ C6, A ÞÑ pΦ12,Φ13,Φ14,Φ23,Φ24,Φ34q

whose coordinates are the 2 ˆ 2 minor determinants13

(0.6) Φ12pAq “ af ´ be, Φ13pAq “ ag ´ ce, Φ14pAq “ ah´ de,

12To be precise, GLn “ SLn ˆµn Z “ SLn ˆ Zµn, where µn “ tζ | ζn “ 1u.13An affine variety (a subset of a vector space defined as the zero locus of a finitely

many polynomials) is entirely determined by its coordinate ring of polynomial functions(functions given by polynomials) up to isomorphism. The coordinate ring for the quotientSL2zM2ˆ4 is the ring of polynomial functions on the vector space M2ˆ4 which are in-variant under the action of SL2. In the 19th century, one of the most intensively studied

6

Enumerative geometry, before and after string theory

Φ23pAq “ bg ´ cf, Φ24pAq “ bh´ df, Φ34pAq “ ch´ gd

of the 2 ˆ 4 matrix (0.5). The six polynomials satisfy only one algebraicrelation

(0.7) Φ12Φ34 ´ Φ13Φ24 ` Φ14Φ23 “ 0.

It is easy to see that the map Φ is injective and hence the set of SL2 orbitsin M2ˆ4

2 is a quadric hypersurface

Q “ tpz1, ¨ ¨ ¨ , z6q | z1z6 ´ z2z5 ` z3z4 “ 0, not all zi are 0u Ă C6 ´ t0u

and the set X of lines in C3 is identified with the open subset of the quadrichypersurface Q, defined by pz1, z2, z4q ‰ p0, 0, 0q. Here a hypersurface meansthe zero set of a single polynomial. A hyperplane will mean a hypersurfacedefined by a linear polynomial.

Let us now consider the action of the center Z “ ttI | t P C˚u. We seethat ΦptAq “ t2ΦpAq and hence we have to consider the orbit space

C˚zpC6 ´ t0uq “: P5

where the constants t P C˚ act on C6 ´ t0u by scalar multiplication. Thisorbit space is called the 5 dimensional projective space and let us denoteby rz1, ¨ ¨ ¨ , z6s the orbit of pz1, ¨ ¨ ¨ , z6q P C6 ´ t0u.14 We call z1, ¨ ¨ ¨ , z6homogeneous coordinates of P5.

Combining the discussions above, we find that the space X of all linesin space is the open subset in the quadric hypersurface

Grp2, 4q “ trz1, ¨ ¨ ¨ , z6s P P5 | z1z6 ´ z2z5 ` z3z4 “ 0u Ă P5,

branches of mathematics is Invariant Theory whose goal is to find the ring of polynomi-als invariant under a group action. Many branches in current mathematics arose frominvariant theory, such as linear algebra, algebraic group theory, representation theory,geometric invariant theory and more. Also the influence of invariant theory on quantumphysics is pervasive and evident. The fundamental theorems in invariant theory for SLn

tell us that the ring of SL2-invariant polynomial functions on M2ˆ4 is generated by the2 ˆ 2 minor determinants (0.6) with the Plucker relation (0.7).

14The projective space P5 is a smooth manifold of (complex) dimension 5 with thequotient topology by the map

C6´ t0u Ñ C˚

zpC6´ t0uq “ P5, pz1, ¨ ¨ ¨ , z6q ÞÑ rz1, ¨ ¨ ¨ , z6s.

P5 is compact as there is a surjective continuous map S5 Ă C6 ´ t0u Ñ P5 from the unitsphere in C6 which is a closed and bounded subset. Recall that given a surjective mapf : X Ñ Y from a topological space X to a set Y , the quotient topology of Y is obtainedby declaring that a subset U of Y is open if and only if f´1pUq is open. All orbits of pointspz1, ¨ ¨ ¨ , z6q with z1 ‰ 0 in C6 ´ t0u meet the hyperplane z1 “ 1 at a unique point andwe may identify this open subset U1 “ pz1 ‰ 0q of P5 with C5 “ tp1, z2, ¨ ¨ ¨ , z6q | zi P Cu.Ignoring z1, we find that its complement pz1 “ 0q is

P5´ U1 “ C˚

zpC5´ t0uq “: P4.

By induction, we thus find that

P5“ C5

\ P4“ C5

\ C4\ P3

“ ¨ ¨ ¨ “ C5\ C4

\ ¨ ¨ ¨ \ C \ pt.

Hence the projective space P5 is a compactification of C5, meaning that it is a compactspace containing C5 as a dense open subset. The open subsets Ui “ pzi ‰ 0q are charts ofP5 which make P5 a smooth manifold.

7

Young-Hoon Kiem

defined by pz1, z2, z4q ‰ p0, 0, 0q.15 It is a 4 dimensional space as you canexpect.16 This wraps up our description of the space X of all lines in space.

Given a line

(0.8) x´ p

ℓ“y ´ q

m“z ´ r

n

in space, a line determined by (0.5) meets (0.8) if and only if

(0.9) detA

¨

˚

˚

˝

ℓ pm qn r0 1

˛

‹

‹

‚

“ 0.

It is easy to see that (0.9) equalspℓq ´ pmqΦ12 ` pℓr ´ pnqΦ13 ` ℓΦ14 ` pmr ´ qnqΦ23 `mΦ24 ` nΦ34 “ 0,

a linear combination of tΦiju. So we see that the set of lines meeting (0.8)is a hyperplane in P5 defined by(0.10) pℓq ´ pmqz1 ` pℓr ´ pnqz2 ` ℓz3 ` pmr ´ qnqz4 `mz5 ` nz6 “ 0.

For instance, consider the 4 lines given by

(0.11)

¨

˚

˚

˝

ℓ pm qn r0 1

˛

‹

‹

‚

“

¨

˚

˚

˝

0 00 01 00 1

˛

‹

‹

‚

,

¨

˚

˚

˝

1 00 00 00 1

˛

‹

‹

‚

,

¨

˚

˚

˝

0 01 0

´1 10 1

˛

‹

‹

‚

,

¨

˚

˚

˝

1 1´1 00 00 1

˛

‹

‹

‚

.

Then (0.10) gives usz6 “ z3 “ z4 ` z5 ´ z6 “ z1 ` z3 ´ z5 “ 0.

Together with the quadratic equation z1z6 ´ z2z5 ` z3z4 “ 0, we find thatthere are exactly two solutions

rz1, ¨ ¨ ¨ , z6s “ r1, 0, 0,´1, 1, 0s, r0, 1, 0, 0, 0, 0s P X Ă Grp2, 4q Ă P5.

So the set of lines meeting all 4 given lines is the intersection of the 4hyperplanes (of lines meeting each of the given 4 lines) with X in P5. For ageneral choice of 4 hyperplanes H1, ¨ ¨ ¨ ,H4 of P5, the intersection(0.12) Grp2, 4q XH1 X ¨ ¨ ¨ XH4

is disjoint from the lower dimensional subvarieties Grp2, 4q ´ X and pz1 “

0q “ P5 ´ C5. Hence finding the set (0.12) amounts to solving a system of4 linear equations and 1 quadratic equation in 5 variables. By eliminatingone variable by using one equation at a time (or by Question 1.4), we findthat (0.12) consists of precisely 2 points and equals X XH1 X ¨ ¨ ¨ XH4.

As we’ve seen an example where there are exactly 2 lines meeting all 4given lines,17 we conclude that the answer to Question 2.1 is 2.

15The set of lines in a 3-dimensional vector space is thus a rank 2 vector bundle overP2.

16To specify a line L, we only need 2 parameters for the direction (namely a point inP2) of L and 2 parameters for the intersection point of L with the plane orthogonal to Land passing through the origin (rank 2 bundle).

17The Grassmannian is irreducible and the condition that (0.12) avoid Grp2, 4q ´ Xand pz1 “ 0q is open.

8

Enumerative geometry, before and after string theory

Is there an easier way to see it? Once again, try to convince yourself thatif you vary your 4 given lines L1, ¨ ¨ ¨ , L4 continuously, the lines L meeting allof them will vary continuously. So we may consider a special configuration oflines. For instance, suppose L1 XL2 “ tp12u ‰ H and L3 XL4 “ tp34u ‰ H.Let P12 (resp. P34) denote the unique plane containing L1 and L2 (resp. L3

and L4). Then it is easy to see that the line P12 X P34 and the line joiningp12 and p34 are the only two lines meeting all 4 lines tLiu. In fact, (0.11) isan example of such a configuration.

2.2. Enumerating rational curves in plane. As we saw above, Pn isa compactification of the vector space Cn and a curve in Cn is compactified inPn by taking the closure. If one is interested in finding curves in plane C2, itmakes sense to find compact curves in P2. In this subsection, we enumeratecurves in the projective plane P2 which are the images of polynomial mapsf : P1 Ñ P2.

An irreducible rational curve in Pn of degree d ě 1 is the image fpP1q ofa polynomial map18

(0.13) f : P1 Ñ Pn, fprt, ssq “ rf0pt, sq, ¨ ¨ ¨ , fnpt, sqs

for homogeneous polynomials19 f0, ¨ ¨ ¨ , fn P Crt, ssd of degree d (without acommon factor), which is generically injective, i.e. there exists a finite setA Ĺ fpP1q such that f is injective on P1 ´ f´1pAq.

A curve in P2 of degree 1 is a line P1 and hence an irreducible rationalcurve. A conic is defined by a homogeneous quadratic polynomial in z0, z1, z2and hence a symmetric 3 ˆ 3 matrix. By linear algebra, we can find acoordinate system such that the conic is isomorphic to

(1) z20 “ 0 (rank 1, double line), or(2) z0z1 “ 0 (rank 2, two lines), or(3) z0z2 “ z21 (rank 3, smooth).

The last case is the image of the polynomial mapP1 ÝÑ P2, rt, ss ÞÑ rt2, ts, s2s.

Therefore any smooth curve in P2 of degree ď 2 is irreducible rational.By definition, a polynomial map f : P1 Ñ Pn of degree d is determined

by a choice of n` 1 homogeneous polynomials f0, ¨ ¨ ¨ , fn P Crt, ssd. Writing

(0.14) fjpt, sq “ÿ

0ďiďd

aijtd´isi,

we obtain a pd` 1q ˆ pn` 1q matrixA “ paijq P M pd`1qˆpn`1q “ Cpd`1qpn`1q

whose C˚-orbit rAs P Ppd`1qpn`1q´1 determines f . The image fpP1q doesnot change even if f is composed with a fractional linear transformation

18A polynomial map is more often called a regular map or a morphism in algebraicgeometry.

19A homogeneous polynomial is a polynomial whose nonzero monomials have thesame degree. A polynomial f P Crx1, ¨ ¨ ¨ , xrs is homogeneous of degree d if and onlyif fptx1, ¨ ¨ ¨ , txrq “ tdfpx1, ¨ ¨ ¨ , xrq for t P C˚. The vector space of homogeneous poly-nomials of degree d in the polynomial ring Crx1, ¨ ¨ ¨ , xrs together with 0 is denoted byCrx1, ¨ ¨ ¨ , xrsd.

9

Young-Hoon Kiem

η : P1 Ñ P1.20 Hence the set RdpPnq of all irreducible rational curves in Pnof degree d is an open subset in the orbit space(0.15) SL2zPM pd`1qˆpn`1q “ SL2zPpd`1qpn`1q´1

whose dimension is pd ` 1qpn ` 1q ´ 4. In particular, when n “ 2, thedimension of the space of all irreducible rational plane curves of degree d is3d´ 1.

Given a collection of objects and an equivalence relation, their modulispace is the set of equivalence classes of objects in the collection. For exam-ple, the moduli space of circles in R2 is R2 ˆ Rą0 where R2 parameterizesthe center and Rą0 parameterizes the radius. The moduli space of rectan-gles in R2 up to translation and rotation is tpa, bq P R2

ą0 | a ě bu. Oftenmoduli spaces come with natural topologies and even algebraic structures ifthe collection consists of algebraic objects.

Now a curve in P2 of degree d is defined as the zero locus(0.16) trz0, z1, z2s P P2 | fpz0, z1, z2q “ 0u

of a nonzero homogeneous polynomial fpz0, z1, z2q P Crz0, z1, z2sd which isuniquely determined by the curve up to constant. Hence the moduli spaceof all plane curves of degree d is

C˚zpCrz0, z1, z2sd ´ 0q “ Pdpd`3q2.

For a nonzero homogeneous polynomial fpz0, z1, z2q, we can detect its degreeby restricting it to a general line in P2 and then finding the number of zeros.The irreducible rational curve in P2 of degree d defined by a polynomial map(0.13) for n “ 2 intersects with a line at d points obviously and hence themoduli space RdpP2q of all irreducible rational curves is a subset of the spacePdpd`3q2 of all plane curves of degree d. In this section, a rational curve inP2 of degree d is defined as a curve in P2 of degree d in the closure RdpP2q inPdpd`3q2 of RdpP2q. In other words, a rational plane curve of degree d is alimit of irreducible rational curves of degree d in P2. For instance, all planecurves of degree d ď 2 are rational curves and we will see that a plane curveof degree 3 is rational if and only if it is singular.

A plane curve (0.16) passes through a point rz0, z1, z2s P P2 if and onlyif fpz0, z1, z2q “ 0, which is a linear equation in the coefficients aijk of

fpz0, z1, z2q “ÿ

i`j`k“d

aijkzi0zj1zk2 .

Hence, the set of plane curves of degree d passing through a given point isa hyperplane. As the moduli space of rational curves of degree d in plane is3d´1 dimensional, we may expect that there are only finitely many rationalcurves in P2 of degree d passing through 3d´ 1 points.21 So we may ask thefollowing.

Question 2.2. How many rational curves in plane of degree d passthrough given 3d´ 1 general points?

20The automorphism group PGL2 “ SL2t˘1u of P1 acts on PM pd`1qˆpn`1q asCrt, ssd is an irreducible representation of SL2 of dimenson d ` 1.

21In algebraic terms, we are asking for the degree of the projective variety RdpP2q inPdpd`3q2.

10

Enumerative geometry, before and after string theory

Let us denote the answer by Nd for d ě 1. For d “ 1, we are lookingfor lines through 2 given distinct points. Of course the nubmer is N1 “ 1.For d “ 2, we are looking for conics through 5 general points. The space ofconics in P2 is the space of coefficients

P5 “ tra0, a1, ¨ ¨ ¨ , a5su

of quadratic polynomials

(0.17) a0z20 ` a1z0z1 ` a2z

21 ` a3z0z2 ` a4z1z2 ` a5z

22

in the homogeneous coordinates z0, z1, z2 of P2, up to constant. The locusof conics through a given point is given by a linear polynomial in a0, ¨ ¨ ¨ , a5.Hence the locus of conics through 5 general points is the intersection of 5hyperplanes in P5. For example, we find that there is precisely one conicpz20 ` z21 “ z22q passing through

r1, 0, 1s, r0, 1, 1s, r1, 0,´1s, r0, 1,´1s, r1,?

´1, 0s

by plugging in these coordinates to (0.17). By Question 1.4, we find thatN2 “ 1.

For d “ 3, it is an elementary exercise22 to show that any irreduciblehomogeneous cubic polynomial in z0 “ x, z1 “ y, z2 “ z is transformed to

(0.18) y2z “ xpx´ µzqpx´ λzq, λ, µ P C

by a linear change of coordinates, i.e z1i “

ř

j aijzj for an A “ paijq P GL3.It is easy to see that the cubic curve defined by (0.18) is smooth if and onlyif λ, µ, 0 are all distinct, in which case the cubic curve is not rational.23

When singular, a plane cubic defined by (0.18) is transformed to eithery2z “ x3 (cusp) or y2z “ x2px´zq (node) by a linear change of coordinates.In both cases, the singular cubic is a rational curve because it is the imageof the polynomial map

P1 Ñ P2, rt, ss ÞÑ rt2s, t3, s3s (cusp), or

P1 Ñ P2, rt, ss ÞÑ rspt2 ` s2q, tpt2 ` s2q, s3s (node).Moreover, all reducible cubic polynomials define rational curves. Thereforean irreducible plane cubic curve is rational if and only if it is singular. Hence,Question 2.2 for d “ 3 is the same as the following.

Question 2.3. How many singular cubic curves in P2 pass through 8given general points?

22An irreducible cubic curve C “ pfpx, y, zq “ 0q has at most one singular pointbecause if there are two, the line joining them meets the cubic curve at two points withmultiplicity ě 2, which is impossible. A point rx0, y0, z0s P C is an inflection point of Cif the Hessian determinant of f at px0, y0, z0q is zero. As the Hessian is a polynomial ofdegree 3, there are 9 inflection points and we may pick a smooth inflection point. By acoordinate change, we may assume the inflection point is r0, 1, 0s. By simple algebra, wecan choose coordinates such that the irreducible cubic polynomial f is of the form (0.18).

23The smooth cubic plane curve defined by (0.18) is a double cover of P1 branchedat 4 points. Hence it is topologically a torus S1 ˆ S1 which does not allow a ramifiedcovering by P1, homeomorphic to S2, by the Hurwitz formula.

11

Young-Hoon Kiem

The locus of cubics passing through a point is a hyperplane in the modulispace P9 of all plane cubic curves. Hence the locus of cubics passing through8 general points is a line P1 “ trt, ssu, which means that we have two cubicpolynomials F,G P Crz0, z1, z2s3 such that a cubic curve passing through8 general points is the zero locus Crt,ss of tF ` sG for some rt, ss P P1. Asingular point of Crt,ss is characterized by the vanishing of the three partialderivatives

(0.19) tBF

Bz0` s

BG

Bz0, t

BF

Bz1` s

BG

Bz1, t

BF

Bz2` s

BG

Bz2,

which are homogeneous of degree 1 in t, s and of degree 2 in z0, z1, z2. Sucha system of three equations has 12 common solutions.24 For example, let

F px, y, zq “ y2z ´ x3 ` x2z, Gpx, y, zq “ x3 ` y3 ` z3,

with x “ z0, y “ z1, z “ z2, whose zero loci are denoted by C and C 1

respectively. Then one can check by hand that C XC 1 consists of 9 distinctpoints and a cubic curve passing through 8 points among C XC 1 is the zerolocus Crt,ss of tF ` sG for some rt, ss P P1. It is straightforward to checkfrom the vanishing of (0.19) that there are exactly 12 singular curves Crt,ss

in this family, each of which contains exactly one singular point. Hence theanswer to Question 2.3 is N3 “ 12, as was computed by Steiner in 1848.

How about Nd for d ą 3? The cubic case was considerably more difficultthan d ď 2 and you may expect that the problem will become much moredifficult as d increases. In fact, between 1848 and 1993 CE, only N4 “ 620and N5 “ 87304 were computed. So it was taken as a complete surprisewhen Kontsevich proved the formula

(0.20) Nd “ÿ

d1`d2“d

Nd1Nd2d21d2

ˆ

d2

ˆ

3d´ 4

3d1 ´ 2

˙

´ d1

ˆ

3d´ 4

3d1 ´ 1

˙˙

in 1994. With the input N1 “ N2 “ 1, (0.20) enables us to compute all Nd

inductively.In 1990s, string theory, especially the mirror symmetry, challenged math-

ematicians with conjectures on enumeration of curves and Kontsevich for-mulated the notion of stable maps for a mathematical theory of the Gromov-Witten invariant. Actually, once equipped with the notion of stable maps,Question 2.2 is not so hard any more. The formula (0.20) is the first majorsuccess story in enumerative geometry motivated by string theory, and wewill see a sketchy proof of (0.20) in §4.1.

24Each of the three partial derivatives is a section of the line bundle OP1ˆP2p1, 2q overP1 ˆ P2. Their common zero locus has 12 points byż

P1ˆP2c1pOP1ˆP2p1, 2qq

3“ pp ` 2ℓq3 “ p3 ` 6p2ℓ ` 12pℓ2 ` ℓ3 “ 12pℓ2 “ 12 P H0pP1

ˆ P2q

where ℓ is the class of a line in P2 and p is the class of a point in P1 since p2 “ 0, ℓ3 “ 0.As there are 12 singular points in

tprt, ss, rz0, z1, z2sq P P1ˆ P2

| tF pz0, z1, z2q ` sGpz0, z1, z2q “ 0u,

there are 12 singular curves for general choices of 8 points, as one can check with explicitexamples.

12

Enumerative geometry, before and after string theory

2.3. Enumerating rational curves in hypersurfaces. Another clas-sical enumerative geometry problem is about rational curves in hypersur-faces. Let Y Ă Pn be a smooth hypersurface25 defined by the vanishing of ahomogeneous polynomial F pz0, ¨ ¨ ¨ , znq of degree k.

A polynomial map (0.13) is a map to Y if and only if(0.21) F pf0pt, sq, f1pt, sq, ¨ ¨ ¨ , fnpt, sqq “ 0 P Crt, ssdk.

By (0.14), we find that (0.21) is equivalent to the vanishing of dk ` 1 ho-mogeneous polynomials in the coefficients aij . Let W be the subvariety ofM pd`1qˆpn`1q defined by the vanishing of these dk` 1 homogeneous polyno-mials, whose expected dimension is pd`1qpn`1q ´dk´1. Then the modulispace RdpY q of irreducible rational curves of degree d in Y is an open subsetin the orbit space

SL2zPW Ă SL2zPM pd`1qˆpn`1q

whose expected dimension is(0.22) pd` 1qpn` 1q ´ dk ´ 5.

It is easy to see that the actual dimension is not smaller than the expecteddimension because an equation can drop the dimension by at most 1.

Let us enumerate lines in Y first. As d “ 1, the expected dimension is2n ´ 3 ´ k and hence if k ă 2n ´ 3, we find that there are infinitely manylines in the hypersurface Y of degree k. For k ą 2n´3, no lines are expectedin Y because the expected dimension is negative. So we ask the following.

Question 2.4. How many lines lie in a smooth hypersurface Y of degree2n´ 3 in Pn for n ě 2?

For n “ 2, the answer is 1 obviously. For n “ 3, 2n ´ 3 “ 3 and Y is acubic surface in P3. The following is a beautiful result which you can findin an undergraduate textbook on algebraic geometry.

Theorem 2.5. Every smooth cubic surface in P3 has exactly 27 lines.

There are many ways to prove this theorem as follows:(1) Check that the number of lines is independent of the cubic surface

and enumerate lines in a special cubic surface like the Fermat cubicz30 ` z31 ` z32 ` z33 “ 0

explicitly. For instance, the line defined by z0 ` e2π

?´1

3 z1 “ 0 “

z2 ` z3 is contained in the Fermat cubic surface. Did you get 27?(2) Show that every smooth cubic surface is the blowup of P2 at 6

points q1, ¨ ¨ ¨ , q6.26 The lines in Y are obtained from 15 lines in25A hypersurface Y defined by a polynomial F P Crz0, ¨ ¨ ¨ , zns is smooth if and only

if the differentialdF “

BF

Bz0dz0 ` ¨ ¨ ¨ `

BF

Bzndzn

is nonzero at every point of Y .26The linear system 3H ´E1 ´ ¨ ¨ ¨ ´E6 gives an embedding of the blowup of P2 at 6

points into P3 as a cubic surface, where Ei are the exceptional divisors. The moduli of cubicsurfaces is 4 “

`

63

˘

´ 42 dimensional while the choice of 6 points modulo automorphism ofP2 has 4 “ 6 ¨ 2 ´ p32 ´ 1q moduli. One can further compare the deformation theories.

13

Young-Hoon Kiem

P2 through 2 points among tqiu, 6 conics in P2 through 5 pointsamong tqiu and 6 exceptional divisors. Hence 27 “ 15`6`6 lines.

(3) Show that the lines in Y are all rigid, i.e. they do not deform inY . As (0.21) is a section of the vector bundle E “ π˚OPU p3q ofrank 4 over the Grassmannian Grp2, 4q of lines in P3, the numberof points in its vanishing locus is computed by the Euler class27

of E, where U is the universal rank 2 bundle over Grp2, 4q andπ : PU Ñ Grp2, 4q is the bundle projection. By Poincaré-Hopf andRiemann-Roch, we find the number of lines in Y to be

ż

Grp2,4q

c4pπ˚OPU p3qq “ 27.

More details of the proofs and related geometry can be found in most text-books on algebraic geometry.

The next case for Question 2.4 is when n “ 4 and hence k “ 2n´ 3 “ 5.This case is quite special in that the expected dimension (0.22) vanishesfor all d. So we expect a finite number of rational curves of degree d in asmooth quintic hypersurface in P4 for any d ą 0. Therefore a complementaryquestion to Question 2.4 is the following.

Question 2.6. How many rational curves of degree d ě 1 lie in a smoothquintic 3-fold Y in P4?

To answer this question, one first proves that rational curves in Y arerigid.28 As in the case of cubic surfaces, for d “ 1, by Poincaré-Hopf, theanswer to Question 2.6 is the integral

(0.23)ż

Grp2,5q

c6pπ˚OPU p5qq

27A complex vector bundle E of rank r over a topological space X is a continuousmap π : E Ñ X of topological spaces such that there is an open cover X “ YαXα with ahomeomorphism φα : π´1pXαq Ñ Xα ˆ Cr for each α such that

φβ ˝ φ´1α : pXα X Xβq ˆ Cr

Ñ pXα X Xβq ˆ Cr, px, vq ÞÑ px, gpxqpvqq

for a continuous g : Xα X Xβ Ñ GLrpCq. A section of the vector bundle π : E Ñ X is acontinuous map s : X Ñ E satisfying π ˝ s “ idX . For all x P X, π´1pxq is a vector spaceand the assignment x ÞÑ 0 P π´1pxq is a continuous map 0 : X Ñ E satisfying π ˝0 “ idX ,called the zero section. It is obvious that the zero section is a homeomorphism onto itsimage and hence we can think of X as a subspace of E by the inclusion 0 : X Ñ E. Underreasonable assumptions like X being a CW complex, it is not hard to see that for anycycle ξ in X we have a (perturbed zero) section which is transversal to ξ. The Euler classof E is a homomorphism

crpEq : H˚pXq ÝÑ H˚´2rpXq, ξ ÞÑ crpEq X ξ “

ż

ξ

crpEq

sending ξ to the intersection of ξ with a section of E, transveral to ξ. If X is a complexmanifold, crpEq is represented by a differential form of degree 2r, supported in a smallneighborhood of the zero section.

28More precisely, we need H0pNCY q “ 0 for any irreducible rational curve C in Y .The rigidity was proved by Sheldon Katz in 1986 for d ď 7. It is not so hard to seedirectly for d “ 1. For higher d, Katz first proves that the incidence variety of pairspC, Y q where Y is a quintic hypersurface and C is a rational curve in Y is irreducible byregularity, for d ď 7. Next he proves that there is at least one smooth rational curve inY of degree d which is rigid by Mori’s argument. Then one can easily conclude that theproper subvariety of rational curves C with H0pNCY q ‰ 0 is empty for general Y .

14

Enumerative geometry, before and after string theory

of the Euler class of the rank 6 vector bundle π˚OPU p5q over the Grass-mannian Grp2, 5q of lines in P4. Here the notation is similar to the cubicsurface case above. Once again it is straightforward to compute (0.23) byRiemann-Roch. The answer for d “ 1 is 2875.

Similar computations with the moduli spaces of conics and twisted cubicstell us that the answer to Question 2.6 is 609250 for d “ 2, 317206375 ford “ 3. The integral like (0.23) for d ě 3 is much more tricky because we needto compactify the moduli space RdpP4q of irreducible rational curves and anatural compactification such as the Hilbert scheme is often not smooth. Sothe computations were very difficult and algebraic geometers couldn’t makemuch progress on Question 2.6.

This was more or less the state of knowledge at around 1990 when agroup of string theorists found a (conjectural) answer to Question 2.6 forall d. Certainly, algebraic geometers were completely taken by surprise andthis challenge by string theory has nourished algebraic geometry enormouslyduring the past three decades.

2.4. Moduli space and intersection theory. Methods employed in§2.1 work for many problems about lines, planes and hyperplanes. Answer-ing enumerative geometry problems on them requires finding intersectionnumbers of subsets in the corresponding moduli spaces of linear objects,called the Grassmannians.

Schubert calculated various intersection numbers on Grassmannians, in-geniously but sometimes not so rigorously. In 1900, Hilbert included in hisfamous list of 23 problems for the 20th century mathematicians the problemof providing a rigorous foundation for Schubert’s calculus. It seems fair tosay that such a foundation was provided in 1978 by Fulton and MacPherson.

The keys for Fulton and MacPherson’s intersection theory are the de-generation to normal cone29 and rational equivalence.30 When intersectinga subset Y in a variety X with another subset, we may replace X by thenormal cone CY X up to rational equivalence by a degeneration, i.e. a con-tinuous family Xt for t P C with X1 “ X and X0 “ CY X . If Y is a smoothsubvariety so that the normal cone is in fact the normal bundle

NY X “ TX |Y TY ,

the quotient of the tangent bundle of X by the tangent bundle of Y , thenthe intersection of Z with Y is the result of intersecting the cone CY XZZ Ă

NY X with the zero section of NY X .Intersection with the zero section of a vector bundle is taken care of

by the Euler class of the bundle, and is obtained by either (1) perturbingthe zero section until it becomes transversal to the cone CY XZZ or (2)perturbing the cone to make it transversal to the zero section. Usually

29The normal cone is obtained by considering the lowest order terms of the definingpolynomials. For instance, the normal cone of the singular cubic curve y2 “ x2px ` 1q

(resp. y2 “ x3) at the origin is the union y2 “ x2 of the two tangent lines (resp. thedouble line y2 “ 0).

30Two irreducible subvarieties X,Y of dimension d in a variety V are rationally equiv-alent if there is a d`1 dimensional subvariety W of V and a rational function on W whosezero is X and pole is Y . This relation gives us the rational equivalence on the free groupgenerated by irreducible subvarieties.

15

Young-Hoon Kiem

the first perturbation is preferred in topology but in algebraic geometrythe second perturbation has to be used because often the zero section isimpossible to perturb.

Even when the normal cone CY X is not a vector bundle, if we canembed it into a vector bundle E, a similar argument works but then theintersection theory depends on the embedding ı : CY X Ñ E. So if you canfind E and ı in a natural fashion from geometric considerations, we can stillperform intersections of subvarieties. Modern enumerative geometry since1995 is based on this observation, under the name of virtual intersectiontheory that we will see below.

We found the answer to Question 2.1 in two ways and in fact these arethe main roads in enumerative geometry. Let us recapitulate them. Thefirst way to answer Question 1.1 proceeds as follows:

(1) Construct the moduli space31 of all geometric figures of fixed type(like Grp2, 4q);

(2) Find the loci of geometric figures satisfying the given constraints(like the hyperplanes);

(3) Find the intersection of the loci.It is usually impossible to find the intersection points of subsets explicitly,but in case there are only finitely many intersection points, their numberis often expressed in terms of cohomology pairings and hence integrals ofdifferential forms (by Poincaré duality). Therefore, enumerative geometryis mostly about constructing moduli spaces and intersection theory ofsubvarieties.32

The second way is to deform the problem continuously till you reacha more tractable situation. But you need to prove ahead that the answerremains invariant (unchanged) during the deformation. So it proceeds asfollows:

(1) Prove that the answer is deformation invariant;(2) Solve the special case.

The technique of deformation (or degeneration in case we allow singularities)is very important in classical and modern enumerative geometry.

3. Enter string theorySo far, we’ve seen a few snapshots of classical enumerative geometry.

Throughout history, mathematics has been enriched by interactions withother disciplines like philosophy, astronomy, navigation, gambling, artillery,

31As we saw above, moduli spaces are often constructed as orbit spaces under groupactions. The art of finding nice orbit spaces or closest geometric objects belonged to therealm of geometric invariant theory (GIT) but now the method of algebraic stacks is asuseful as GIT. The moduli spaces often depend on choices and they undergo suitable trans-formations as we vary our choices. The changes in the answers to enumerative problemsas we vary the choices are called wall crossing terms.

32Intersection theory is the study of finidng the equivalence class of the intersectionof suitable subsets. Depending on the objects and the equivalence relation, there aredifferent layers of intersection theories, like Chow theory, K-theory, Borel-Moore homology,algebraic cobordism, motivic cohomology etc, and Riemann-Roch compares them.

16

Enumerative geometry, before and after string theory

physics and engineering. Quite recently, enumerative geometry has beenconsiderably expanded by interacting with string theory since 1990s. Inthis section, I want to discuss why the interaction was possible and how ithappened. But I have to keep the discussion minimal because my knowledgeis quite limited.

Newton’s mechanics was founded on Euclidean geometry. By calculus,he proved that when dealing with a solid body in mechanics, we may assumethat the object is a point located at its center of mass. So Newton’s physicsis about points and their trajectories. Even today, the standard physicsexplains elementary particles as points. To a geometer like me, string theorylooks like an attempt to make physics more interesting by adding moregeometric structures to physics. Thanks to string theory, we can now thinkof elementary particles as curves, surfaces and so on. The price we pay isthat to make the theory consistent with our universe, we have to accept theassumption that the universe has extra dimensions, on top of the usual 4dimensional space-time.

The use of extra dimension dates back to 1919 when Kaluza observedthat if we increase the dimension of the space-time R4 by adding a tiny circleS1, then Einstein’s general relativity and Maxwell’s electromagnetism areamalgamated into a concise and united theory. Around 1985, string theoristsincluding Witten realized that the correct way to enlarge the space-time instring theory is to add a Calabi-Yau 3-fold.33 Perhaps the simplest exampleof a compact Calabi-Yau 3-fold is a smooth quintic hypersurface in P4 likethe Fermat quintic

Y “ pz50 ` z51 ` ¨ ¨ ¨ ` z54 “ 0q Ă P4.

So now all the physics takes place in R4ˆY for a Calabi-Yau 3-fold Y . Whena particle is a point, its trajectory is a curve. But when a particle is a stringlike S1, its trajectory (world sheet) is a map from a surface S1ˆR to R4ˆY .By projecting to Y , we obtain a map S1 ˆR Ñ Y . The length of the stringmay change from 0 to R and then to 0. Also strings may ramify into twostrings or more. Hence we think of a world sheet as a map f : Σ Ñ Y from aRiemann surface Σ. If we choose a Lagrangian suitably, the Euler-Lagrangeequation is satisfied at f if and only if f is holomorphic, i.e. fpΣq is analgebraic curve in case Y is contained in a projective space. So enumeratingcurves in the Calabi-Yau 3-fold Y may be interpreted as an integral on thecritical set of the action functional, a physics quantity in string theory. Thisobservation connects string theory with enumerative geometry.

There are two ways to obtain Lagrangians relevant to enumerative ge-ometry, A-model and B-model. An interesting proposition in string theorysays that for a Calabi-Yau 3-fold Y , there is a mirror Calabi-Yau 3-fold Ysuch that the A-model physics of Y (resp. Y ) is equivalent to the B-modelphysics of Y (resp. Y ). This is called the mirror symmetry.

Around 1990, it occurred to a group of string theorists that the mir-ror symmetry may be used to solve enumerative geometry problems. The

33A 3-fold is a Kahler manifold of dimension 3. A complex manifold Y of dimension ris Calabi-Yau if there is a nowhere vanishing holomorphic r-form on Y , i.e. the canonicalline bundle KY “ detT˚

Y – OY is trivial.

17

Young-Hoon Kiem

A-model physics of a quintic 3-fold Y in P4 enumerates rational curves inY while the B-model of the mirror partner Y gives us an explicit (hyper-geometric) series. So the physics of mirror symmetry predicted the num-ber of rational curves of degree d in Y for all d. It was a huge blow toalgebraic geometers because string theory provided all the numbers whilealgebraic geometry was not making progress towards Question 2.6. Sincethen, many conjectural formulas on enumerative geometry problems havebeen produced by applying string theory and interesting new developmentsfollowed through attempts to rigorously prove or disprove the conjectures inmathematics.

4. Modern enumerative geometryIn this section, let us take a look at new aspects of enumerative geometry

since the interaction with string theory.As we saw in §2, a standard approach to enumerative geometry problems

consists of(1) construction of a moduli space,(2) intersection theory of subvarieties and(3) checking that the numbers are actually enumerating the objects we

want.Let us not worry about the issue (3) and focus on getting a number by (1)and (2). When the moduli space is smooth, by Poincaré duality,34 intersec-tion theory of subvarieties is taken care of by integrals of closed differentialforms η. We call an integral

ş

X η of a differential form η (or a cohomologyclass) against the fundamental class35 of a moduli space X, an enumerativeinvariant, if the number is fixed under deformation of the enumerative prob-lem. As noted in §2, deformation to a simpler case is an extremely valuabletechnique.

In reality, moduli spaces we will use are highly singular and an integralof the form

ş

X η will never be deformation invariant, even if we could makesense of it. The way out is to find a replacement of the fundamental class,namely a homology class of expected dimension which is invariant underdeformation. The replacement is now called a virtual fundamental class anddenoted by rXsvir. So the issue (2) should be split into two steps of findinga virtual fundamental class rXsvir and then integrating cohomology classesagainst rXsvir. In summary, designing an enumerative invariant requiresthree steps:Step 1. Construct a nice moduli space X;Step 2. Construct a virtual fundamental class rXsvir;Step 3. Integrate cohomology classes against rXsvir.

34The Poincaré duality identifies homology classes (submanifolds without boundarymodulo homological equivalence) with cohomology classes (closed differential forms mod-ulo exact forms).

35The fundamental class of an oriented manifold is the whole space with the cho-sen orientation. To make sense of an integral, we need an orientation and a change oforientation may change the sign componentwisely.

18

Enumerative geometry, before and after string theory

Let us see how each step works through the example of Gromov-Witten in-variant. Unfortunately, the discussions from now on will get more technical,sometimes beyond the level of undergraduate mathematics.

4.1. Step 1: Moduli space. As we saw in §3, the motivation formodern enumerative invariants came from the mirror symmetry. The firsttask was to make clear the meaning of the numbers calcuated by the mirrorsymmetry. Mathematicians realized very quickly that the numbers enumer-ate symplectic surfaces in the symplectic manifold underlying the quintic3-fold Q but not as embedded symplectic manifolds but as maps.36 In alge-braic setting, the numbers from the mirror symmetry enumerate polynomial(regular) maps f : C Ñ Q, up to automorphism, where C is a projectivealgebraic curve, with at worst nodal singularities.37 Such a map f is calledstable if the automorphism group

Autpfq “ tφ : C–

ÝÑC | f ˝ φ “ fu

of invertible polynomial maps of C preserving f is finite.How do we construct the moduli space of stable maps to a smooth pro-

jective variety Q? There are several approaches: One may use geometricinvariant theory starting from a big Hilbert scheme, or one may show thatthe moduli functor is an algebraic stack and then prove its expected prop-erties like projectivity. A more down-to-earch construction due to Fultonand Pandharipande goes as follows: Let us denote by Mg,npQ, dq the mod-uli space of stable maps f : C Ñ Q Ă Pr from curves C of genus g with nsmooth distinct marked points such that the direct image f˚rCs of the fun-damental class of C meets a general hyperplane at d points. A polynomialmap f : C Ñ Pr, fppq “ rf0ppq, ¨ ¨ ¨ , frppqs is determined by its componentfunctions fi. Each polynomial function fi is determined by its d roots upto constant. For a stable map f , we can choose a coordinate system for Prsuch that all these roots are distinct and away from the marked and singularpoints of C. Hence the roots determine a pointed curve for which we alreadyhave a moduli space Mg,mpSdqr`1 for m “ n ` dpr ` 1q whose dimension

36A mathematical count of embedded symplectic surfaces Σ in a symplectic manifoldQ is called the Gromov invariant, which equals the Seiberg-Witten invariant when Qis a symplectic 4-manifold. When the 4-manifold is a smooth projective surface, theGromov/Seiberg-Witten invariant is the virtual integral (called the Poincaré invariant)on the Hilbert scheme of curves and is equivalent to the Donaldson invariant (after wallcrossing) by Mochizuki’s formula.

37Here are the definitions: A projective algebraic curve is a projective variety ofdimension 1. A projective variety Q is a subset of a projective space Pn defined by thevanishing of some homogeneous polynomials f1, ¨ ¨ ¨ , fm in the homogeneous coordinatesx0, ¨ ¨ ¨ , xn. The dimension of Q is 1 if n minus the rank of the Jacobian of pf1, ¨ ¨ ¨ , fmq

is 1 at all points of Q except finitely many points. A projective algebraic curve C has atworst nodal singularity if in an analytic neighborhood of each point p P C, the curve issmooth or looks like the union of two lines pxy “ 0q. Topologically a projective algebraiccurve C with at worst nodal singularities is the union of finitely many doughnuts possiblywith any number of holes, glued at points. The genus of C is the total number of holesplus the number of loops in the dual graph. When the genus g is zero, we say C is rational.

19

Young-Hoon Kiem

is m ` 3g ´ 3.38 The choice of f from the roots requires r parameters39

and a fiber bundle over an open subspace in Mg,mpSdqr`1 gives us a chartof Mg,npPr, dq. Gluing these charts constructs Mg,npPr, dq.40 If you under-stand this construction, it is not hard to see that for any projective varietyQ Ă Pr, the moduli space of stable maps to Q is a closed subspace

Mg,npQ, dq Ă Mg,npPr, dq

which is also projective.41

As an application of the construction of the moduli space of stable maps,let us prove Kontsevich’s formula (0.20).

Sketchy proof of Kontsevich’s formula (0.20). We will prove theformula for d “ 3 (Question 2.3) and leave the general case to the readerwhich is no more difficult. Pick 7 general points q3, q4, ¨ ¨ ¨ , q9 and two linesL1 and L2 in general position. In the moduli space M0,9pP2, 3q, let Y bethe locus of stable maps whose first marked point p1 maps to a point in L1,the second marked point p2 to a point in L2, and the jth marked point pjto qj for j ě 3. The dimension of M0,9pP2, 3q is 17 and the requirements forp1 and p2 cut the dimension by 1 each while the requirement for pj , j ě 3drops the dimension by 2 each. Hence you find that Y is a projective curve.

Consider the composition

φ : Y Ă M0,9pP2, 3q ÝÑ M0,9 ÝÑ M0,4 – P1

where the first arrow forgets the map to P2 and the second arrow forgetsthe marked points pj for j ą 4. By the Cauchy integration theorem or itsalgebraic counterpart, a meromorphic function on a projective curve has thesame number of zeros and poles. Hence, φ´1p0q and φ´1p8q have the samenumber of points. By change of coordinates if necessary, we may say 0 P P1

(resp. 8 P P1) represents a nodal rational curve

C “ C1 Y C2 P M0,4 – P1

with two components, with p1, p2 P C1 (resp. p1, p3 P C1) and p3, p4 P C2

(resp. p2, p4 P C2).Now let us count the number of stable maps f P φ´1p0q.(1) If the degree of f restricted to the component C1 is 0, then this

component should map to L1XL2 and the image of f passes through

38The moduli space Mg,n “ Mg,nppt, 0q of stable curves was constructed by Deligneand Mumford in 1968. Here a stable curve means a stable map to a point. Any subgroupG of the symmetric group Sn acts on Mg,n by permuting the marked points.

39We have to find isomorphisms of the line bundles Li defined by the Cartier divisorzeropfiq. Since HompLi, Liq “ H0pOCq is 1 dimensional, the identification of line bundlesL0, ¨ ¨ ¨ , Lr requires r dimensional parameter space. The cohomology H1pOCq “ Cg is theobstruction space to gluing of line bundles and hence the expected dimension is m` 3g ´

3 ` rp1 ´ gq “ n ` p3 ´ rqpg ´ 1q ` dpr ` 1q.40The projectivity follows from Kollar’s criterion.41To be precise, the Fulton-Pandharipande construction gives us the Deligne-

Mumford stack which represents the natural moduli functor of stable maps whose coarsemoduli space is a projective scheme. Coarse moduli means that we are forgetting thestabilizer groups of points.

20

Enumerative geometry, before and after string theory

all the eight points L1 X L2, q3, ¨ ¨ ¨ , q9. Hence there are N3 suchstable maps.

(2) If the degree of f restricted to C1 is 1, then the other component C2

has degree 2 and hence fpC2q may contain at most 3 points out ofq5, ¨ ¨ ¨ , q9 because fpC2q already passes through q3 and q4. HencefpC1q should pass through exactly 2 points out of q5, ¨ ¨ ¨ , q9. Wehave

`

52

˘

“ 10 choices of the two points and the gluing point of C1

and C2 may be mapped to one of the two points of the intersectionof the line by C1 and the conic by C2. Hence there are 10 ˆ 2 suchstable maps.

(3) If the degree of f restricted to C1 is 2, the line fpC2q has to bethe line joining q3 and q4. Thus fpC1q is the unique conic throughq5, ¨ ¨ ¨ , q9. As fpC1q X L1 and fpC1q X L2 have two points each,there are 2ˆ2 choices for fpp1q and fpp2q. As the conic fpC1q meetsthe line fpC2q at two points, we have two choices for fpC1 X C2q.Hence there are 2 ˆ 2 ˆ 2 such stable maps.

In summary, there are N3 ` 20 ` 8 points in φ´1p0q.Let us count the number of stable maps f P φ´1p8q.(1) The degree of f restricted to the component C1 cannot be 0 or 3

because q3 R L1 and q4 R L2.(2) If the degree of f restricted to C1 is 1, fpC1q has to be one of

the five lines joining q3 with one of q5, ¨ ¨ ¨ , q9 while fpC2q is theunique conic passing through q4 and the remaining 4 points amongq5, ¨ ¨ ¨ , q9. As fpC2q meets L2 at 2 points, there are 2 choices forfpp2q. As the line fpC1q meets the conic fpC2q at 2 points, thereare two choice for fpC1 XC2q. Hence there are 5ˆ2ˆ2 such stablemaps.

(3) If the degree of f restricted to C1 is 2, the same argument tells usthat there are 20 such stable maps.

In summary there are 20 ` 20 “ 40 points in φ´1p8q.We thus find that N3`28 “ 40 and hence N3 “ 12 as we saw in §2.2.

4.2. Step 2: Virtual fundamental class. To discuss anything mean-inful, we absolutely need the notion of a vector bundle. Let us begin withthe tangent bundle and define a vector bundle as a generalization.

For an open U Ă Rn, tangent vectors are expressions like

(0.24) a “ a1B

Bx1` ¨ ¨ ¨ ` an

B

Bxn, ai P R

and if f is a smooth fuction on U , its derivative in the direction of a is

dfpaq “ÿ

i

aiBf

Bxi.

By the assignment a ÞÑ pa1, ¨ ¨ ¨ , anq P Rn, you may think of U ˆ Rn as theset of all tangent vectors on U and let pr : U ˆ Rn Ñ U be the projectionto the first component. A vector field on U is then a map a : U Ñ U ˆ Rnwhose composition with pr is the identity map on U . For a map π : X Ñ Y ,a map s : Y Ñ X with π˝s “ idY is often called a section of π. For instance,

21

Young-Hoon Kiem

0 : U Ñ U ˆ Rn, 0ppq “ pp, 0q is a section of pr : U ˆ Rn Ñ U , called thezero section.

However this description of tangent vectors and vector fields depends onthe choice of a coordinate system. If we employ a new coordinate systemty1, ¨ ¨ ¨ , ynu instead of tx1, ¨ ¨ ¨ , xnu, then the expression (0.24) is trans-formed to

(0.25) a “

nÿ

i“1

a1i

B

Byi, where a1

i “

nÿ

j“1

ajByiBxj

.

Hence the identification of the set of all tangent vectors on U with U ˆ Rnrequires the transformation

U ˆ Rn ÝÑ U ˆ Rn, pp, aq ÞÑ pp, Jaq

where J “ pByiBxj

q is the Jacobian, which is a smooth map J : U Ñ GLpn,Rq.Let M be a smooth manifold of real dimension n. Then we have an

open cover M “ YαMα together with an open embedding φα : Mα Ñ

Rn. By the above paragraph, the set of all tangent vectors on Mα can beidentified with Mα ˆ Rn. On the intersection Mαβ :“ Mα X Mβ, we havetwo coordinate systems coming from φα and φβ. By the above paragraph,the identifications of the set of tangent vectors on Mαβ are related by theJacobian Jαβ : Mαβ Ñ GLpn,Rq of φβ ˝ φ´1

α . It is easy to see that theidentity, called the 2-cocycle condition,

Jγα ˝ Jβγ ˝ Jαβ “ id

holds on Mα X Mβ X Mγ for all indices α, β, γ, by the chain rule. Let usdefine the tangent bundle of M to be the set of equivalence classes

TM “ \αpMα ˆ Rnq „

where pp, vq „ pq, wq for p P Mα and q P Mβ if and only if p “ q andw “ Jαβppqv. This set comes with the projection π : TM Ñ M which equalsthe projection pr : Mα ˆ Rn Ñ Mα over the open set Mα. A vector fieldon M is just a section of the tangent bundle π : TM Ñ M . By declaringthat π´1pMαq is open in TM and the obvious map MαˆRn Ñ π´1pMαq is ahomeomorphism for every α, the tangent bundle TM becomes a topologicalspace and a smooth manifold.

More generally, a real vector bundle of rank r over a topological spaceM is a topological space E and a continous map π : E Ñ M such that thereare an open cover M “ YαMα, a commutative diagram

π´1pMαqψα //

π$$J

JJJJ

JJJJ

Mα ˆ Rr

przzttt

tttttt

t

Mα,

for each α, and a continuous map ξαβ :Mαβ Ñ GLpr,Rq for every pair α, βof indices, such that

ψβ ˝ ψ´1α pp, vq “ pp, ξαβppqvq, @p P Mαβ, v P Rr.

22

Enumerative geometry, before and after string theory

If we replace continuous (resp. real) by smooth (resp. complex), we obtaina smooth (resp. complex) vector bundle. When M is a complex manifold,42

a complex vector bundle π : E Ñ M defined by ξαβ : Mαβ Ñ GLpr,Cq

is a holomorphic bundle if ξαβ is a holomorphic map. For a holomorphicbundle π : E Ñ M , E is a complex manifold. A section s : U Ñ E on anopen U Ă M of a holomorphic bundle π : E Ñ M is holomorphic if it is aholomorphic map of complex manifolds. For a vector bundle π : E Ñ M anda subset N of M , π´1pNq is denoted by E|N and the restriction E|N Ñ Nof π is a vector bundle over N .

Given a vector bundle E of rank r over M and a homomorphism ρ :GLpr,Rq Ñ GLpk,Rq, we obtain a new vector bundle of rank k on M bythe recipe

ρpEq :“ \αpMα ˆ Rkq „

where pp, vq „ pq, wq for p P Mα, q P Mβ if and only if p “ q and w “

ρpξαβppqqv. For instance, the determinant det : GLpr,Rq Ñ GLp1,Rq definesthe determinant line bundle detpEq for a vector bundle E of rank r on M .Here a line bundle means a vector bundle of rank 1. Moreover, given vectorbundles E and F , applying linear algebra operations to ξαβ, we can generatenew vector bundles E ‘F , E bF , E˚, ^iE, symiE and so on. With a helpof functional analysis, one can extend the theory of vector bundles to infiniterank vector bundles over infinite dimensional Banach manifolds.

Many moduli spaces in geometry are defined as the zero locus of a smoothsection s of an infinite dimensional vector bundle E over an infinite dimen-sional (Banach) manifold Y, at least locally. For instance, if we are lookingfor the moduli space of holomorphic maps f : C Ñ Q from a compact Rie-mann surface43 C to a smooth projective variety Q, we consider the space Yof smooth maps f : C Ñ Q with fixed f˚rCs P H2pQq. By the existence anduniqueness of solutions of ordinary differential equations, for each f P Y,the tangent space to Y at f is the infinite dimensional vector space

TfY “ ΓpC, f˚TQq

of smooth sections of the pullback bundle

f˚TQ “ tpp, qq P C ˆ TQ |πpqq “ fppqu ÝÑ C, pp, qq ÞÑ p,

and there is an open neighborhood of f in Y diffeomorphic to an openneighborhood of 0 in TfY. In particular, Y is an infinite dimensional smoothmanifold.

The condition that the smooth map f should be holomorphic is thevanishing of the section s “ df ˝ j ´ J ˝ df of the infinite dimensional vectorbundle E whose fiber44 over f is the vector space ΓpC,HompTC , f

˚TQqq of

42A complex manifold of complex dimension n is a smooth manifold M with an opencover M “ YαMα, a homeomorphism ϕα : Mα Ñ Uα Ă Cn for an open set Uα in Cn suchthat the transitions ϕβ ˝ ϕ´1

α are holomorphic on ϕαpMαβq.43A Riemann surface is a complex manifold of dimension 1.44The fiber of a vector bundle π : E Ñ M over p P M is the vector space π´1ppq.

23

Young-Hoon Kiem

smooth sections of the bundle HompTC , f˚TQq, where j (resp. J) is the

complex structure on TC (resp. TQ).45

Given an infinite dimensional manifold Y and an infinite dimensionalvector bundle E , a section s of E is called Fredholm if the kernel and coker-nel of the differential ds : TY Ñ E at each p P s´1p0q are finite dimensional.In many moduli problems in geometry, the differential ds is an elliptic dif-ferential operator and hence Fredholm.

We are interested in the geometry of M “ s´1p0q where 0 Ă E denotesthe zero section. For any compact subset K of M , we can find an openneighborhood U in Y of K and a subbundle rF of E |U of finite rank such that

rF ` impdsq “ E |U .

Then s gives a section s of the quotient E “ E |U rF of E |U by rF whosedifferential ds is surjective by our choice of rF . By the implicit functiontheorem, Y “ s´1p0q is a finite dimensional manifold and s becomes asection of F :“ rF |Y whose zero locus is the open subset U :“ U XM of M .We thus obtain a finite dimensional local model(0.26) F

U “ s´1p0q // Y

s

VV

for the zero locus M of a Fredholm section, where Y is a finite dimensionalmanifold and F Ñ Y is a vector bundle of finite rank. The differential dsgives us a homomorphism(0.27) TY |U

dsÝÑF |U .

By the implicit function theorem again, if ds is surjective over U , U issmooth of dimension(0.28) vdpUq “ dimY ´ rankF “ rank kerpds|U q ´ rank cokerpds|U q

and the tangent bundle TU of U is the kernel of (0.27). As the vanishing ofthe cokernel of (0.27) implies the smoothness of U , we call the cokernel of(0.27) the obstruction sheaf 46 of the local model (0.26) and (0.27) is calleda tangent-obstruction complex. We call (0.28) the expected dimension or thevirtual dimension of U defined by the local model (0.26).

Most moduli spaces in algebraic geometry are very singular and for afinite dimensional local model (0.26), U usually has a bigger dimension thanthe expected (0.28). The singularity of U “ s´1p0q arises if the section s isnot transversal to the zero section 0 of F . If we perturb s slightly to obtaina section s1 transversal to the zero section 0, we obtain a smooth manifoldU 1 “ s1´1p0q of expected dimension (0.28). By construction, U 1 is very close

45If M is a complex manifold, the tangent bundle TM is a holomorphic vector bundleobviously. The complex structure on TM is the multiplication

?´1 : TM Ñ TM by

?´1.

46If one tries to extend a map Cm Ñ U in the mth jet of U to the m ` 1st jet, thereis an element in the obstruction sheaf whose vanishing is equivalent to the existence of anextension.

24

Enumerative geometry, before and after string theory

to U with a proper47 map U 1 Ñ U . We wish to use U 1 instead of U todefine enumerative invariants but U 1 depends on the choice of perturbations1. To remove the dependency, we use a homology theory48 and push downU 1 to U homologically. Algebraic topology neatly summarizes this methodof perturbation of the section in terms of (refined) Euler class

s! “ epF, sq : H˚pY q ÝÑ H˚´2rpUq

for a complex vector bundle F of rank r and a section s of F . In case Yis a complex manifold, we obtain the virtual fundamental class by applyings! “ epF, sq to the fundamental class49 rY s of Y , i.e.

rU svir :“ epF, sq X rY s “ s!rY s P HvdRpUqpUq.

In this way, moduli spaces M we are interested in usually admit an opencover tMα Ñ Mu together with finite dimensional local models

(0.29) Fα

Mα “ s´1

α p0q // Yα

sα

UU

where Yα is a complex manifold and Fα is a holomorphic vector bundle whilesα is a holomorphic section. So we have local virtual fundamental classes

rMαsvir “ s!αrYαs P H˚pMαq.

The question is then:Can we glue rMαsvir P H˚pMαq to a homology class rM svir P H˚pMq?

In other words, we want a homology class rM svir whose restriction to Mα

is rMαsvir. Of course, we cannot expect a positive answer for free. Forinstance, the virtual dimensions should be independent of α.

Let Iα be the image of the dual s_α : F_

α Ñ OYα of the section sα.50

Holomorphic functions belonging to Iα determine Mα as their common zerolocus. Hence the dual of the tangent-obstruction complex(0.30) TYα |Mα

dsαÝÑFα|Mα

fits into the commutative diagram

(0.31) F_α |Mα

ds_α //

s_α

T_Yα

|Mα

Iα|Mα

d // T_Yα

|Mα .

47A proper map is a continuous map such that the inverse image of a compact set iscompact.

48In a sense, a homology theory is a way to identify submanifolds which does notchange an integral of a closed differential form, by Stokes’ theorem.

49The fundamental class of a complex manifold is the whole space together with thecanonical orientation

?´1

ndz1 ^ dz1 ^ ¨ ¨ ¨ ^ dzn ^ dzn in local coordinates.

50A holomorphic section s : M Ñ E of a holomorphic vector bundle π : E Ñ M canbe thought of as a map OM Ñ E of the trivial bundle OM “ M ˆ C into E which mapspx, λq to λspxq for x P M .

25

Young-Hoon Kiem

For all α, the lower arrows are canonical (up to quasi-isomorphism) andalways glue to a global object, called the truncated cotangent complex LMof M . The correct condition discovered in 1995 by Li-Tian and Behrend-Fantechi for gluing the local virtual fundamental classes rMαsvir to a globalvirtual fundamental class rM svir is that the tangent-obstruction complexes(0.30) glue to a complex E “ rE0 Ñ E1s of vector bundles on M51 and(0.31) also glue to a morphism ϕ : E_ Ñ LM . This complex E togetherwith the morphism ϕ is called a perfect obstruction theory on M .

Given a perfect obstruction theory E “ rE0 Ñ E1s, the normal conesCMαYα induce cones

CMαYα ˆ E0|MαTYα |Mα Ă E1|Mα

which glue to a cone C Ă E1 over M . Now the virtual fundamental classof M is defined to be the intersection of C with a perturbation of the zerosection of E1 by

(0.32) rM svir “ 0!E1rCs “ epE1|C , τq X rCs

where τ is the tautological section c ÞÑ pc, cq of

E1|C “ tpv, cq P E1 ˆ C |πpvq “ πpcqu Ñ C, pv, cq ÞÑ c

and π : E1 Ñ M is the bundle projection.52

Since 1995, lots of enumerative invariants have been defined by integralsof cohomology classes against virtual fundamental classes. Enumerative in-variants defined in this way are sometimes called virtual invariants and havebeen the focus of intensive research during the past 25 years in enumerativegeometry because they automatically satisfy nice properties like deformationinvariance under reasonable assumptions.

Unfortunately, it is not easy to handle the virtual fundamental classesand usually computation of virtual invariants is extremely difficult. As faras I know, there are only three techniques to handle virtual fundamentalclasses as follows.

(1) Lefschetz hyperplane principle: If ı :M ãÑ N is a closed immersionand the normal cone CMN embeds into V |M for some holomorphic

51More precisely, we only need a derived category object E P DbpCohMq whoserestriction to Mα is isomorphic to (0.30). Li-Tian’s condition is weaker than this, nowcalled a semi-perfect obstruction theory.

52Nowadays, the construction of the virtual fundamental class rM svir requires muchless assumptions: We don’t need a global resolution rE0 Ñ E1s by vector bundles andwe only need a derived category object which locally is isomorphic to (0.30). In fact, wedon’t even need a global derived category object, just local objects (0.30) together with acompatibility condition on the obstruction sheaf and obstruction assignments. However,for convience of explanation, I will assume that we have a global resolution rE0 Ñ E1s ofthe perfect obstruction theory.

26

Enumerative geometry, before and after string theory

vector bundle V on N ,53 we can compare the virtual fundamentalclasses of M and N by

(0.33) rM svir “ ı!rN svir, ı˚rM svir “ epV q X rN svir

where epV q denotes the Euler class of V .54

(2) Torus localization: Let M be equipped with an action of the multi-plicative group C˚ and a C˚-equivariant perfect obstruction theoryE “ rE0 Ñ E1s. Then the virtual fundamental class is localized tothe fixed point locus MC˚ by the formula

(0.34) rM svir “rMC˚

svir

epNvirq, where epNvirq “

epE0|mvMC˚ q

epE1|mvMC˚ q

and Ei|mvMC˚ (i “ 0, 1) denotes the subbundle of Ei|MC˚ where C˚

acts with nontrivial weights.(3) Cosection localization: Let M be equipped with a perfect obstruc-

tion theory rE0 Ñ E1s whose cokernel ObM admits a homomor-phism σ : ObM Ñ OM to the trivial bundle. We call σ a cosectionof ObM . Then rM svir is localized to a class rM svirloc supported in thezero locus of σ. Often σ´1p0q is much simpler than M , sometimesjust the empty set or a smooth point, and hence the cosection lo-calization gives us a vanishing result or simplifies the computationof a virtual invariant.

Often a combination of these techniques turns out to be quite powerful. Forinstance, one can compute the genus 0 GW invariants of a hypersurfacein a projective space by first pushing the computation to the projectivespace by the Lefschetz hyperplane principle and then localizing to the fixedpoint locus by the torus localization. One way to use the torus or cosectionlocalization is to define a virtual invariant even when the moduli space Mis not compact. In general, we cannot integrate a cohomology class over anoncompact space. If there is a C˚ action on M and the fixed point setMC˚ is compact, we can define rM svir by (0.34). If there is a cosection σ ofObM and σ´1p0q is compact, the cosection localized virtual cycle rM svirloc iscompactly supported and hence we can integrate cohomology classes againstrM svirloc.

4.3. Step 3: Virtual invariants. As we mentioned before, virtualinvariants are integrals of cohomology classes against virtual fundamental

53We further need that the perfect obstruction theories EM and EN for M and N fitinto a commutative diagram of exact triangles

V _|M // E_N |M //

ϕN

E_M

//

ϕM

V _|M // LN |M // LM

// .

54There is a more general form of the (quantum) Lefschetz hyperplane principle,namely the functoriality of virtual pullbacks.

27

Young-Hoon Kiem

classes. For the moduli space Mg,npQ, dq of stable maps to a smooth pro-jective variety Q in §4.1, there is a natural perfect obstruction theory.55 Bythe recipe of §4.2, we thus obtain the virtual fundamental class

rMg,npQ, dqsvir P H˚pMg,npQ, dqq

and the Gromov-Witten (GW for short) invariants of Q are defined as in-tegrals of cohomology classes against rMg,npQ, dqsvir. In particular, whenQ is a Calabi-Yau 3-fold, the expected dimension of Mg,0pQ, dq is zero forany d P H2pQ,Zq and thus rMg,0pQ, dqsvir P H0pMg,0pQ, dqq, a linear com-bination of points. By taking the sum of the coefficients, we obtain the GWinvariant(0.35) GWQ

g,d “ deg rMg,0pQ, dqsvir P Q.

When Q is a general quintic hypersurface in P4, there are only finitelymany rational curves of degree d ď 7 and it is conjectured that the sameshould be true for all d. Hence the count of rational curves of degree d in Q(Question 2.6) can be taken care of by the GW invariant of Q if we removecontributions from multiple covers.

Let us sketch how Givental and Lian-Liu-Yau computed the genus 0Gromov-Witten invariant of a quintic 3-fold Q in P4. By composing anystable map to Q with the inclusion Q Ă P4, we have an inclusion

X :“ M0,0pQ, dq Ă M0,0pP4, dq “: Y

into the smooth projective variety Y of dimension 5d ` 1. The universalfamily

Cf //

π

P4

Y

of stable maps gives us a vector bundle N “ π˚f˚OP4p5q of rank 5d` 1 and

the defining quintic polynomial of Q pulls back to a section s of N by thehomomorphism

H0pP4,OP4p5qqf˚

ÝÑH0pC, f˚OP4p5qq “ H0pY,Nq

so that we have a diagram

N “ π˚f˚OP4p5q

X “ s´1p0q

ı // Y

s

EE.

By the Lefschetz hyperplane principle, we have

rXsvir “ epN, sq X rY s P H0pXq.

55The deformation theory of a holomorphic map f : C Ñ Q was well studied since1960s. The perfect obstruction theory at a point pf : C Ñ Qq P Mg,npQ, dq comes fromthe hyperext groups Ext˚prf˚ΩQ Ñ Ωlog

C s,OCq where ΩQ “ T_Q denotes the cotangent

bundle of Q.

28

Enumerative geometry, before and after string theory

The genus 0 GW invariant of Q is thus

(0.36)ż

rXsvir1 “

ż

YepNq P Q.

So the computation of the GW invariant of Q was pushed to Y but still Yis not easy for an explicit computation. We further push the computationto the projective space Wd “ P5d`4 by the diagram

M0,0pP1 ˆ P4, p1, dqq

// PpH0pP1,Opdqq b C5q “ Wd

X “ M0,0pQ, dq // Y “ M0,0pP4, dq

where the vertical arrow is the composition of stable maps f : C Ñ P1 ˆ P4

with the projection P1 ˆ P4 Ñ P4 and the top horizontal arrow is obtainedby allowing the linear system to have base points and contracting compo-nents of the domain C whose degrees are 0 in the P1 direction. Exceptfor X, the action of pC˚q5 by multiplication on homogeneous coordinatesof P4 makes the diagram pC˚q5-equivariant. By establishing a functorialtorus localization, one can push the computation of (0.36) to the projectivespace Wd where integrals of cohomology classes are straightforward. In thisway, the generating function of genus 0 GW invariants (0.35) is given by ahypergeometric series.

A general element of Wd represents a base point free linear system on P1

of degree d and thus defines a map P1 Ñ P4. A quasi-map from a curve C toPr is defined as a line bundle L on C together with r` 1 sections s0, ¨ ¨ ¨ , srof L. In particular, Wd is the moduli space of nontrivial quasi-maps fromP1 to P4 of degree d.

4.4. More virtual invariants. We’ve seen the three steps to define avirtual invariant. So if you can find a compact56 moduli space M equippedwith a perfect obstruction theory, you have a virtual invariant. Nowadays,there are many virtual invariants defined in this way. Let us take a look ata few of them related to curve counting.

So we want a compactified moduli space of curves in a smooth projectivevariety Q which admits a perfect obstruction theory. The moduli spaceMg,npQ, dq is such an example. Are there any other? Let’s think about theunit circle S1 in plane centered at the origin. Probably the first definitionof the unit circle S1 you learned says that it is the set of points in planewith distance 1 from the origin. A little later, you probably learned thatthe unit circle is the same thing as the locus of points in the coordinateplane whose coordinates satisfy x2 ` y2 “ 1. A projective variety Z Ă Pnis uniquely determined by the homogeneous ideal I “ IpZq of polynomialsvanishing on Z. Recall that an ideal I of a polynomial ring Crz0, ¨ ¨ ¨ , zns

is homogeneous if it is generated by homogeneous polynomials. We maywrite I “ ‘dě0Id where Id “ I X Crz0, ¨ ¨ ¨ , znsd and Crz0, ¨ ¨ ¨ , znsd is the

56If the moduli space is not compact, you can still define a virtual invariant by findinga torus action whose fixed locus is compact or by finding a cosection of the obstructionsheaf whose zero locus is compact.

29

Young-Hoon Kiem

vector space spanned by homogeneous polynomials in z0, ¨ ¨ ¨ , zn of degree d.Two homogeneous (radical) ideals I and I 1 in Crz0, ¨ ¨ ¨ , zns define the sameprojective subvariety in Pn if and only if Id “ I 1

d for d ąą 0 and we writeI „ I 1. So you may think of a projective curve in Pn as a homogeneous idealI up to the equivalence „. For any homogeneous ideal I of Crz0, ¨ ¨ ¨ , zns,dimCrz0, ¨ ¨ ¨ , znsdId is a polynomial function of d for d ąą 0 whose definingpolynomial is called the Hilbert polynomial. Grothendieck proved that theset of homogeneous ideals I Ă Crz0, ¨ ¨ ¨ , zns up to „ with a fixed Hilbertpolynomial forms a projective scheme,57 called the Hilbert scheme. From thisit is straightforward to conclude that the set of all projective subschemeswith a fixed Hilbert polynomial in a projective variety Q is a projectivescheme M in a natural way.58

Let M “ MP pQq be the Hilbert scheme of curves (more generally pro-jective subschemes) with fixed Hilbert polynomial P in a Calabi-Yau 3-foldQ. Then M admits a perfect obstruction theory59 with expected dimension0. We thus obtain the virtual fundamental class rM svir P H0pMq whosedegree60

(0.37) DTQP “ deg rMP pQqsvir

is defined to be the Donaldson-Thomas (DT for short) invariant of Q. Usu-ally DT is more difficult to compute than GW. However there are manyadvantages of DT over GW:

(1) DT can enumerate not just curves but also projective subschemesor vector bundles.

(2) The perfect obstruction theory of DT is symmetric and the tangentspace is dual to the obstruction space at each point.

(3) The DT moduli space is locally the critical locus of a holomorphicfunction on a complex manifold.

(4) DT is an integer while GW is a rational number.As GW and DT both can enumerate curves in a smooth projective vari-ety Q, they should be related somehow. The Maulik-Nekrasov-Okounkov-Pandharipande conjecture states a precise formula for the equivalence ofGW and DT after a suitable change of coordinates and removing contri-butions by points floating away from curves etc. It was proved for quintic3-folds and some other cases.

There are still many more ways to think of a curve C in a smoothprojective variety Q. We can think of the coordinate ring Crz0, ¨ ¨ ¨ , znsIC

57A projective scheme is a generalization of a projective variety, in that projectivevarieties in Pn are in bijection with radical homogeneous ideals in Crz0, ¨ ¨ ¨ , zns up to „

while projective schemes are in bijection with all homogeneous ideals strictly contained inthe maximal ideal pz0, z1, ¨ ¨ ¨ , znq.

58A homogeneous ideal IC Ă Crz0, ¨ ¨ ¨ , zns contains the ideal IQ of Q if and only ifthe composition IQ ãÑ Crz0, ¨ ¨ ¨ , zns Ñ Crz0, ¨ ¨ ¨ , znsIC is zero.

59The tangent and obstruction spaces at I P M are the traceless ext-groupsExtiQpI, Iq0 for i “ 1, 2.

60deg : H0pMq Ñ H0pptq – Q is the pushforward by the proper map M Ñ pt. Thismap is counting points with multiplicity.

30

Enumerative geometry, before and after string theory

of C as a homogeneous module over Crz0, ¨ ¨ ¨ , znsIQ.61 This perspectivegives us another way to define a virtual invariant enumerating curves inQ, also called the DT invariant. We can think of a curve C in Q as ahomomorphism

OQ “ Crz0, ¨ ¨ ¨ , znsIQ Ñ Crz0, ¨ ¨ ¨ , znsIC “ OC

and this perspective gives us the virtual invariant of stable pairs, called thePandharipande-Thomas (PT for short) invariant. We can also think of acurve as a quasi-map and hence construct a virtual invariant.

Any reasonably defined curve counting virtual invariant should be re-lated with known invariants like GW, DT and PT.

5. Recent developmentsTo end this note, let us review some of recent developments in enumer-

ative geometry. Inevitably, I have to be very sketchy and cannot explainall the terms and ideas, because each of the topics deserves an independentsurvey article.

5.1. Cohomological, K-theoretic and motivic refinements andwall crossing. For a smooth projective variety Q, the collection of all vec-tor bundles over Q with fixed topological type (Chern classes) is usually notcompact and we need to add coherent sheaves to compactify it. Roughlyspeaking, a coherent sheaf is like a vector bundle on a subvariety with round-ing off along the boundaries. The collection of all coherent sheaves is bigenough for compactification but it is not Hausdorff. So we need to takeout some undesired coherent sheaves and a criterion to determine whichsheaves to delete is called a stability condition. A most popular choice -Gieseker stability - comes from geometric invariant theory and it is givenby an inequality of Hilbert polynomials. After fixing a stability condition,under favorable circumstances, the collection of all stable coherent sheaveson Q forms a compact Hausdorff space which admits a virtual fundamentalclass when Q is a Calabi-Yau 3-fold. So, the DT invariants of Calabi-Yau3-folds depend on the choice of a stability condition. A wall crossing formularecords the differences of DT invariants as we vary the stability condition.

If we choose the stability condition in a clever way, the DT invariantsare easy to compute and hence the DT invariants for arbitrary stabilityconditions can be computed if we know all the wall crossings. Of course,wall crossing is really useful and has been a key tool for at least three decadesin algebraic geometry.

The DT invariants are known to be motivic which means that we cancut the moduli space M into locally closed pieces and add the (weighted)DT invariants of the pieces to get the DT invariant associated to M . This

61So we can think of a curve C in Q as a coherent sheaf OC on Q. By GIT, weknow that we have to confine ourselves to the study of semistable sheaves and thereis a projective stack M parameterizing semistable sheaves on Q after fixing an ampleline bundle. The open set of stable sheaves admits a perfect obstruction theory (whendimQ “ 3 and KQ ď 0) and hence we obtain a DT invariant when all semistable sheavesare stable. If there are semistable sheaves which is not stable, we need to use the Hallalgebra machinery or the Kirwan blowup, to obtain a generalized DT invariant.

31

Young-Hoon Kiem

motivic property enables us to employ the Hall algebra machinery and thewall crossing can be handled in a completely systematic way. Furthermore,this Hall algebra technique enables us to define DT invariants even whenthe circumstances are not favorable.62

When Q is a Calabi-Yau 3-fold, the moduli space of stable sheaves on Qis locally the critical locus of a regular function on a smooth variety. Usingthe motivic vanishing cycles, Kontsevich and Soibelman generalized the DTinvariant to a motivic invariant which takes values in a motivic ring andstudied their wall crossing.

Another refinement of DT is by K-theory. The virtual fundamentalclass may be defined as a class in the K-group of coherent sheaves by takingthe K-theoretic intersection of the cone C with the zero section of E1 inthe notation of (0.32). By taking the holomorphic Euler characteristic ofa vector bundle twisted by the K-theoretic virtual fundamental class, weobtain the K-theoretic DT invariant. As K-theory is more refined thanhomology by (virtual) Riemann-Roch, the K-theoretic DT invariants containmore information about the geometry of Q. Recently, interesting links ofK-theoretic DT with represenation theory have been studied by Okounkovand others.

Finally, there is a cohomological refinement of DT invariants which mayproduce a correct mathematical theory of the Gopakumar-Vafa invariant orBPS invariant. The critical locus of a regular function on a smooth varietycomes with a perverse sheaf of vanishing cycles. One can glue the locallydefined perverse sheaves of vanishing cycles to a globally defined perversesheaf P when the moduli space M is oriented.63 So we can consider thehypercohomology H˚pM,P q of P whose Euler characteristic is the usualDT invariant. Now if the perverse sheaf P is semisimple, as it underlies amixed Hodge module, the morphism to Chow scheme defines an action ofsl2 ˆ sl2 on H˚pM,P q by the hard Lefschetz property. The first sl2 actioncomes from the hard Lefschetz on the cohomology of fibers and the secondsl2 action comes from the base.

Imagine a coherent sheaf is a line bundle on a subvariety and M is thespace of pairs pC,Lq where C Ă Q is a smooth curve and L is a line bundleon C. The morphism to Chow is the forgetful map pC,Lq ÞÑ C whose fiberover C is the Jacobian of C, an abelian variety, which is topologically aproduct Jg of g copies ellipses where g is the genus of C. Gopakumar-Vafa’srecipe to enumerate curves of genus g in Q is to write

H˚pM,P q –à

g

H˚pJgq bRg

by using the sl2ˆsl2 action and then calculate the Euler characteristic of Rg.The conjecture is that the invariants defined in this way contain equivalentinformation as GW invariants. GV’s invariants are integers and supposedto be free from multiple cover contributions - more purified curve countinginvariants.

62Like when there are semistable sheaves which are not stable.63A moduli space of stable coherent sheaves is always orientable when there is a

universal family.

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Enumerative geometry, before and after string theory

5.2. Quantum singularity theory. Quantum singularity theory is anattempt to enumerate curves in a smooth projective variety by the singu-larity of the associated affine cone. Let me try to explain the ideas for thecase of Fermat quintic 3-fold

Q “ p

5ÿ

i“1

x5i “ 0q Ă P4.

The general case is not more difficult.Consider the quadruple pC,L, x, pq of(1) a projective curve C with at worst nodal singularities,(2) an algebraic line bundle L on C,(3) five sections x “ px1, ¨ ¨ ¨ , x5q P H0pLq‘5 of L, and(4) a section p P H0pL´5ωCq where ωC is the dualizing line bundle (the

sheaf of 1-forms).The collection of all such quadruples pC,L, x, pq forms an algebraic stack,as it is locally the quotient of an affine scheme by a group action. Let usdenote the algebraic stack by X.

If x : O‘5C Ñ L is surjective, it induces a morphism f : C Ñ P4. Let us

denote the open substack of quadruples with x : O‘5C Ñ L surjective by X`.

The moduli spaceMg,npP4, dqp “ tpC,L, x, pq | pC,L, xq P Mg,npP4, dq, p P H0pL´5ωCqu –

of stable maps to P4 with p-fields is an open substack in X`.If p : L5 Ñ ωC is surjective, it has to be an isomorphism and we say

pC,Lq is a 5-spin curve. Let us denote the open substack of quadruples withp : L5 Ñ ωC surjective by X´. The moduli space

Sg,n “ tpC,L, pq | p : L5 – ωC , C P Mg,nu –

of 5-spin curves is a compact space and hence integrals of cohomology classesare well defined. Obviously Sg,n is closed in X´ and the space

Wg,n “ tpC,L, x, pq | pC,L, pq P Sg,n, x P H0pLq‘5u –

is open in X´.Here are the key points of the quantum singularity theory for

ř5i“1 x

5i :

(1) Any open separated Deligne-Mumford substack U of X admits aperfect obstruction theory whose obstruction sheaf ObU admits acosection σ.

(2) For Mg,npP4, dqp, the cosection localized virtual fundamental classhas support in the moduli space Mg,npQ, dq of stable maps to Qand satisfies

(0.38) rMg,npQ, dqsvir “ ˘rMg,npP4, dqpsvirloc.

(3) For Wg,n, the cosection localized virtual fundamental class has sup-port in the moduli space Sg,n of 5-spin curves and hence integratingagainst rWg,nsvirloc enables us to define virtual invariants, called theFan-Jarvis-Ruan-Witten (FJRW for short) invariants.

(4) There are ways to compare the virtual invariants by rMg,npP4, dqpsvirloc(i.e. the GW invariants of Q) with the FJRW invariants.

33

Young-Hoon Kiem

To be precise, we should include orbifold structures on curves for the FJRWinvariants and the complete set of FJRW invariants shares many nice prop-erties with GW invariants, some of which are codified in the axioms of aCohomological Field Theory.

The comparison of GW and FJRW turned out to be a powerful newtechnique to investigate the GW invariants of Calabi-Yau 3-folds and someold conjectures were proved by this method.

5.3. Donaldon-Thomas theory for Calabi-Yau 4-folds. An alge-braic version of the Donaldson-Floer theory is as follows:

(1) For each smooth projective surface S, we have the Donaldson in-variant DpSq which enumerates stable coherent sheaves on S.

(2) For each smooth projective curve C, we should have a finite dimen-sional vector space HpCq with a pairing ˚ : HpCq bHpCq Ñ Q andHpHq “ Q.

(3) For each smooth projective surface S with a smooth divisor C,we should have a vector DF pS,Cq P HpCq satisfying DF pS,Hq “

DpSq.(4) If S degenerates into the gluing of two smooth projective surfaces

S1 and S2 along the smooth divisor C, we have the gluing formula

DpSq “ DF pS1, Cq ˚DF pS2, Cq.

Intuitively, we are gluing a vector bundle F 1 on S1 with a vector bundle F 2

on S2 by an isomorphim F 1|C – F 2|C , to get a vector bundle on S1 YC S2

and deforming it to a vector bundle on S. So far, no one knows how to makethis fantastic dream rigorous.64

More generally, one may dream about a theory which assigns a vectorspace HpSq to each d dimensional smooth projective variety S and a vectorDpY, Sq P HpSq to each d`1 dimensional smooth projective variety Y whichcontains S as a divisor. When S “ H, HpHq “ Q and DpY q “ DpY,Hq P Qis an invariant of Y . Moreover we expect a gluing formula

(0.39) DpY q “ DpY 1, Sq ˚DpY 2, Sq,

when Y degenerates into the union of smooth projective varieties Y 1 andY 2 glued along a smooth divisor S. When S is a Calabi-Yau 3-fold, wedo have a vector space HpSq “ H˚pM,P q by the perverse sheaf P overthe moduli space of stable coherent sheaves, mentioned above. So one maydream about an invariant of 4-folds which may be obtained from HpSq likein the algebraic Donaldson-Floer theory.

Such an invariant for a Calabi-Yau 4-fold (DT4 for short) was conceivedby Donaldson and Thomas about 20 years ago but rigorous mathematicalapproaches were developed only recently. There are two ways: differen-tial geometric method by Cao-Leung, and the Kuranishi atlas method byBorisov-Joyce.65 The key point is that the tangent-obstruction theory for

64The main difficulty is that we don’t have an effective intersection theory of Artinstacks.

65There was an announcement of a third (algebraic) approach using half-Euler classby Oh-Thomas.

34

Enumerative geometry, before and after string theory

the moduli space of stable sheaves on a Calabi-Yau 4-fold is a 3-term com-plex of vector bundles but we can locally write it as the sum E ‘ E_r´2s

where E is a (real analogue of) perfect obstruction theory. After a choiceof orientation on the moduli space, the local (Kuranishi) models give us thevirtual fundamental class.

Here are some big open questions about DT4:(1) Does the cosection localization work for DT4? If so, we will get

some vanishing results for hyperkahler 4-folds. Also it will help usin computation of DT4.

(2) What is the wall crossing formula for DT4? For the usual DTinvariant of a Calabi-Yau 3-fold, there is a beautiful wall crossingformula. A wall crossing for DT4 will be extremely important forcomputation and theory at the same time.

(3) Will DT4 invariants satisfy a degeneration formula like (0.39) withHpSq “ H˚pM,P q?

I expect that a big progress will follow from an algebraic construction ofDT4. Differential geometry is intuitive and easier to grasp. However whenproving a statement, algebraic geometry is often much more reliable.

5.4. Vafa-Witten invariant. In mid 1990s, Vafa and Witten com-puted the Euler numbers of moduli spaces of semistable sheaves with fixeddeterminant on smooth projective surfaces, in a physical manner. In fa-vorable circumstances, their Euler numbers coincide with the usual Eulernumbers (up to sign). But until very recently, no one could find a mathe-matical theory of VW’s Euler numbers in general. When the surface S isK3, VW’s Euler numbers gave generating series which are modular forms!

What is the Euler number of a compact smooth manifold? By Poincare-Hopf, ifM is a compact complex manifold of dimension r, then the cotangentbundle ΩM “ T_

M is a vector bundle of rank r whose Euler class epΩM q givesus the Euler characteristic

χpMq “ p´1qrż

MepΩM q.

In particular, the number of points in the zero locus of a 1-form, if finite, isthe Euler characteristic of M up to sign. How can we generalize this formulato the case where M is not smooth?

Let me try to convey VW’s idea from the perspective of virtual inter-section theory. As we saw in §4.2, the moduli spaces M we are interested inoften admit perfect obstruction theories, which means that we have the localdescription (0.29). Think of the open set Mα of M as the smooth manifoldYα with constraints sα “ 0. Do you remember the Lagrange multipliersmethod in calculus? If we want to deal with constraints, we should not tryto cut the manifold by the constraints but instead increase the manifoldby adding more variables. Geometrically, the Lagrange multipliers methodsays that we should embed Yα into the dual bundle F_

α of Fα as the zerosection. This is exactly what we do in the Lagrange multipliers method:we add as many variables as the constraints and pair them by the naturalpairing of Fα with F_

α . The section sα of Fα gives us a holomorphic functionsα : F_

α ÝÑ C, ξ ÞÑ ξ ¨ sαpπαpξqq

35

Young-Hoon Kiem

where πα : F_α Ñ Yα is the bundle projection.

Suppose that dsα : TYα Ñ Fα is surjective so that Mα is smooth. Thena holomorphic function φ on Yα which has discrete critical locus gives us arepresentative

Critpφ|Mαq “ zeropdφ|Mαq

of the Euler class of the cotangent bundle ΩMα . On the other hand, π˚αφ`

sα is a holomorphic function on F_α whose critical locus is a perturbation

of Critpφ|Mαq by the Lagrange multipliers method. So the critical locusCritpπ˚

αφ ` sαq, a perturbation of Critpsαq, on F_α gives us the Euler class

of Mα up to sign.Now the interesting point is that if M admits a perfect obstruction the-

ory with local models (0.29), the critical loci tCritpsαqu glue to an algebraicstack N “ Ob_

M , called the dual obstruction cone of M . It comes with localmodels

ΩF_α

Nα “ pdsαq´1p0q // F_

α

dsα

TT

and hence a symmetric (semi-perfect) obstruction theory. Therefore, we canthink of the virtual invariant

p´1qvdpN q

ż

rN svir1

as the Euler number of M .Now back to VW’s Euler numbers. The moduli stack M of semistable

sheaves with fixed determinant on a smooth projective surface S has thetangent and obstruction spaces

TM |E “ Ext1SpE,Eq0, ObM |E “ Ext2SpE,Eq0

at E P M , where the subscript 0 denotes the traceless part. For the dualobstruction cone N , we have to add the dual space

HompE,E bKSq0 – Ext2SpE,Eq_0

of the obstruction space by Serre duality where KS “ ^2ΩS denotes thecanonical line bundle of S. A homomorphism ϕ : E Ñ E b KS is calleda Higgs field and the pair pE, ϕq is called a Higgs pair.66 Hence we shouldconsider the moduli space N of Higgs pairs pE, ϕq on S with detE fixedand trpϕq “ 0. Tanaka and Thomas proved that N admits a symmetricobstruction theory and defined the Vafa-Witten invariant as

ż

rN svir1 “

ż

rNC˚s

1

epNvirq

by using the torus localization, where t P C˚ acts on pE, ϕq by pE, tϕq. Theircalculations match with the conjectured formulas by Vafa-Witten. Thomasalso constructed a K-theoretic refinement of VW’s Euler characteristic andcomputed the refined invariant in some examples.

66A Higgs pair on S is the same as a coherent sheaf on the Calabi-Yau 3-fold KS

with compact support. If we don’t fixed detE and trpϕq, the VW invariant becomes theDT invariant of KS which does not match VW’s Euler number.

36

Enumerative geometry, before and after string theory

For the case where there are strictly semistable sheaves, Tanaka-Thomasused the Mochizuki/Joyce-Song technique of stable pairs. In a sense, addinga section to a moduli problem is like blowing up the moduli space andhence this M/J-S technique should be related to the singularity at strictlysemistable points. It will be interesting to find a direct way of definingthe VW invariant by investigating the singularity and of establishing themodularity.

After all these years, we still do not understand the Euler characteristicso well!

The references below are never meant to be anything close to a completelist, which may easily take a dozen or more pages. The list below justindicates a few starting points for further reading.

37

Bibliography

[1] K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997),no. 1, 45-88.

[2] W. Fulton, Intersection theory. Second edition. Ergebnisse der Mathematik und ihrerGrenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 2. Springer-Verlag, Berlin, 1998.

[3] W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology,Algebraic geometry, Santa Cruz 1995, 45–96, Proc. Sympos. Pure Math., 62, Part 2,Amer. Math. Soc., Providence, RI, 1997.

[4] T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135(1999), no. 2, 487-518.

[5] Y.-H. Kiem and J. Li, Localizing virtual cycles by cosections, J. Amer. Math. Soc. 26(2013), no. 4, 1025-1050.

[6] F. C. Kirwan, Complex algebraic curves. London Mathematical Society Student Texts,23. Cambridge University Press, Cambridge, 1992.

[7] S. L. Kleiman, Intersection theory and enumerative geometry: a decade in re-view. Proc. Sympos. Pure Math., 46, Part 2, Algebraic geometry, Bowdoin, 1985(Brunswick, Maine, 1985), 321–370, Amer. Math. Soc., Providence, RI, 1987.

[8] J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraicvarieties, J. Amer. Math. Soc. 11 (1998), no. 1, 119–174.

[9] D. Mumford, Algebraic geometry. I. Complex projective varieties. Reprint of the 1976edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.

[10] D. Mumford, J. Fogarty and F. Kirwan, Geometric Innvariant Theory. Third edition.Ergebnisse der Mathematik und ihrer Grenzgebiete (2), 34. Springer-Verlag, Berlin,1994.

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