Monads and instanton bundles on smooth hyperquadrics

Math. Nachr. 282, No. 2, 169 – 179 (2009) / DOI 10.1002/mana.200610730

Monads and instanton bundles on smooth hyperquadrics

Laura Costa∗1 and Rosa Maria Miro–Roig∗∗1

1 Facultat de Matematiques, Universitat de Barcelona, Gran Via, 585, 08007 Barcelona, Spain

Received 23 January 2006, revised 23 February 2007, accepted 15 March 2007Published online 19 January 2009

Key words Instantons, monads, hyperquadricsMSC (2000) 14F05

In this paper we study instanton bundles on quadric hypersurfaces from the algebraic and geometric point ofview. We discuss examples and we give a cohomological characterization.

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1 Introduction

A mathematical instanton bundle on P2l+1 with quantum number k is by definition a rank 2l vector bundle Ewith Chern polynomial ct(E) = 1

(1−t2)k , natural cohomology in the range −2l− 1 ≤ j ≤ 0 and trivial splitting

type. Via the Penrose transformation a certain subset of the mathematical instanton bundles on P3 corresponds toself-dual solutions of the SU(2) Yang–Mills equations on S4. This correspondence was generalized by Salamonand this generalization was the main motivation in [16] to introduce the notion of mathematical instanton bundleon P2l+1, and to give a characterization of such bundles by means of cohomology bundles of special monads.Among other reasons, the importance of instanton bundles on P

2l+1 relies on the fact that they provide examplesof rank 2l vector bundles on P2l+1. Indeed, there are only few examples of indecomposable vector bundles ofsmall rank on Pn. For instance, the only known indecomposable rank two vector bundles on P4 are the Horrocks–Mumford bundles and its pullbacks under a finite morphism. There is the rank 3 Horrocks bundle on P5 and noindecomposable rank 2 vector bundle is known on P5.

The main goal of this paper is to introduce and characterize mathematical instanton bundles on the quadrichypersurface Q2l+1 ⊂ P2l+2. We believe that the study of instanton bundles on hyperquadrics may give someinsight in the study of rank 2l vector bundles on P

2l+2. As in the case of instanton bundles on P2l+1, instanton

bundles on Q2l+1 will be defined by means of special monads (see Definition 3.2). Monads appear in a widevariety of contexts within algebraic geometry as in, for instance, the construction of locally free sheaves on Pn.Once we will have introduced the instanton bundles on Q2l+1 as the cohomology bundles of some special mon-ads, using the generalization of the Beilinson spectral sequence obtained in [7], we will give a cohomologicalcharacterization of such bundles (see Theorem 3.5). Beilinson’s type spectral sequences have been extensivelyused in many works on vector bundles and their precise statement needs the language of derived categories. Thegeneralizations given in [7] have provided important tools for describing quite explicitly derived categories of co-herent sheaves. In this work, we will link the abstract and general context of the generalization of the Beilinson’stheorem to concrete examples and applications, mainly to introduce and describe the family of indecomposablerank 2l vector bundles on Q2l+1 provided by the instanton bundles on Q2l+1 (see also [1] and [7] for otherapplications of Beilinson type spectral sequences).

Next we outline the structure of the paper. In Section 2, after recalling the notion of m-block collectionσ = (E0, E1, . . . , Em), Ei =

(Ei

1, . . . , Eiαi

)of coherent sheaves on X which generates D, we introduce the

notion of special monads and special bundles on X with respect to σ. The definitions are somewhat mysteriousand they are made natural when we illustrate them by means of precise examples and we recover mathematical

∗ Corresponding author: e-mail: [email protected]∗∗ e-mail: [email protected]

c© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

170 Costa and Miro–Roig: Monads and instanton bundles on smooth hyperquadrics

instanton bundles on P2n+1 as the cohomology bundle of a certain special monad. We end the section giving acohomological characterization of special sheaves on Pn of arbitrary rank. In Section 3, we introduce the notionof instanton bundle on Q2l+1 and we give a cohomological characterization of these bundles. We also determineunder which conditions there exist special monads on Qn whose maps are given by linear forms. We end thepaper with some question/problems which naturally arise in our context.

Notation Throughout this paper X will be a smooth projective variety defined over the complex numbersC and we denote by D = Db(OX -mod) the derived category of bounded complexes of coherent sheaves ofOX -modules. Notice that D is an abelian linear triangulated category. We identify, as usual, any coherent sheafF on X to the object (0 → F → 0) ∈ D concentrated in degree zero, we will not distinguish between a vectorbundle and its locally free sheaf of sections and we will use the definition of stability and semistability due toMumford–Takemoto ([15]).

We refer to [7] for the basic definitions and properties on full strongly exceptional collections of sheaves andm-block collections of sheaves on a smooth projective variety needed in the sequel as well as for the generaliza-tion of Beilinson’s spectral sequence from Pn to any smooth projective variety of dimension n with an n-blockcollection which generates D.

2 Special monads and special bundles

Let X be a smooth projective variety of dimension n with an m-block collection σ = (E0, E1, . . . , Em), Ei =(Ei

1, . . . , Eiαi

)of coherent sheaves on X which generates D. The goal of this section is to introduce the notions

of special monads and special bundles on X with respect to σ. We will discuss their existence and give acohomological characterization of certain special bundles in the case X = P

2n+1 and X = Q2n+1 ⊂ P2n+2.

Let us start recalling the precise definition of m-block collection and gathering basic facts about monads.

Definition 2.1 Let X be a smooth projective variety.(i) An object F ∈ D is exceptional if Hom•

D(F, F ) is a 1-dimensional algebra generated by the identity.(ii) An ordered collection (F0, F1, . . . , Fm) of objects of D is an exceptional collection if each object Fi is

exceptional and Ext•D(Fk, Fj

)= 0 for j < k.

(iii) An exceptional collection (F0, F1, . . . , Fm) of objects of D is a block if ExtiD(Fj , Fk

)= 0 for any i

and j �= k.(iv) An m-block collection of type (α0, α1, . . . , αm) of objects of D is an exceptional collection

(E0, E1, . . . , Em) =(E0

1 , . . . , E0α0, E1

1 , . . . , E1α1, . . . , Em

1 , . . . , Emαm

)such that all the subcollections Ei =

(Ei

1, Ei2, . . . , E

iαi

)are blocks.

Definition 2.2 Let X be a smooth projective variety. A monad on X is a complex of vector bundles:

M• : 0 −→ Aα−→ B

β−→ C −→ 0

which is exact at A and at C. The sheaf E := Ker(β)/Im(α) is called the cohomology sheaf of the monadM•.

Remark 2.3 Clearly, the cohomology sheaf E of a monad M• is always a coherent sheaf, but more can besaid in particular cases. In fact, E is torsion free if and only if the localized maps αx are injective away from asubset Y ⊂ X of codimension 2, E is reflexive if and only if the localized maps αx are injective away from asubset Y ⊂ X of codimension 3, and E is locally free if and only if the localized maps αx are injective for allx ∈ X .

Monads were first introduced by Horrocks who showed that all vector bundlesE on P3 can be obtained as thecohomology bundle of a monad of the following kind:

0 −→⊕

i

OP3(ai) −→⊕

j

OP3(bj) −→⊕

n

OP3(cn) −→ 0.

Monads appeared in a wide variety of contexts within algebraic-geometry, like the construction of locallyfree sheaves on Pn, the classification of space curves in P3 and surfaces in P4. In this paper, we will focus ourattention on the so-called special monads defined as follows:

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Math. Nachr. 282, No. 2 (2009) 171

Definition 2.4 Let X be a smooth projective variety of dimension n with an m-block collectionσ = (E0, E1, . . . , Em), Ei =

(Ei

1, . . . , Eiαi

)of coherent sheaves on X which generates D. A monad on X is

called special with respect to σ if it has the form

S• : 0 −→αi−1⊕j=1

(Ei−1

j ⊗ V i−1j

) −→αi⊕

j=1

(Ei

j ⊗ V ij

) −→αi+1⊕j=1

(Ei+1

j ⊗ V i+1j

) −→ 0

where V ij are C-vector spaces of dimension ri

j ≥ 0 and 1 ≤ i ≤ m− 1.We will simply call special monad when there is no confusion about σ.


(Ei

1, . . . , Eiαi

)of coherent sheaves on X which generates D. Let F be a coher-

ent (locally free) sheaf on X . We say that F is a special sheaf (resp. special bundle) with respect to σ if there isa line bundle L on X such that F ⊗ L is the cohomology sheaf (resp. cohomology bundle) of a special monad

S• : 0 −→αi−1⊕j=1

(Ei−1

j ⊗ V i−1j

) −→αi⊕

j=1

(Ei

j ⊗ V ij

) −→αi+1⊕j=1

(Ei+1

j ⊗ V i+1j

) −→ 0.

This somewhat mysterious definitions are made natural once we recall that, using Penrose–Ward correspon-dence, in [3] Atiyah–Drinfeld–Hitchin–Manin showed that instanton bundles on P3 satisfying a reality conditioncorrespond to self dual Yang Mills Sp(1)-connections over the 4-dimensional sphere S4. More general, we define

Definition 2.6 An instanton bundle on P2n+1 with quantum number k is a rank 2n vector bundleE on P2n+1

satisfying the following properties(i) ct(E) = 1

(1−t2)k ,

(ii) E has natural cohomology in the range −2n− 1 ≤ j ≤ 0, i.e., for every j in that range at most one ofthe cohomology groupsHq(P2n+1, E(j)) is non-zero,

(iii) E has trivial splitting type, i.e., for a general line l ⊂ P2n+1 we have E|l ∼= Ol2n.

The existence of instantons bundle on P2n+1 was given by Okonek and Spindler in [16] answering a question

posed by Salamon. They proved

Proposition 2.7 Any instanton bundleE on P2n+1 with quantum number k is a special bundle; more precisely,E is the cohomology bundle of a special monad

0 −→ V ⊗OP2n+1(−1) −→W ⊗OP2n+1 −→ U ⊗OP2n+1(1) −→ 0

with dimV = dim = k and dimW = 2k + 2n.Conversely, a special bundle arising as the cohomology bundle of a special monad

0 −→ V ⊗OP2n+1(−1) −→W ⊗OP2n+1 −→ U ⊗OP2n+1(1) −→ 0

with dimV = dimU = k and dimW = 2k + 2n is an instanton bundle provided it has trivial splitting type.

In [2, Proposition 2.11], Ancona and Ottaviani proved that any instanton bundle E on P2n+1 with quantumnumber k is simple. Moreover, if k = 1 or n = 1 then E is stable and the stability is left as an open problemwhen k ≥ 2 and n ≥ 2. The existence of the moduli space MIP2n+1(k) of instantons bundle on P2n+1 withquantum number k was established by Okonek and Spindler in [16, Theorem 2.6]. Determining the irreducibilityand smoothness of MIP2n+1(k) is a long standing question far from being solved, see [5] and [11] for a recentsurvey on the topic.

To end this section and as an application of Beilinson’s spectral sequence, we will give a cohomologicalcharacterization of special sheaves on Pn constructed and classified by Fløystead in [8]. Indeed, let n ≥ 1 andfix integers a, b and c such that

(1) b ≥ 2c+ n− 1 and b ≥ a+ c, or(2) b ≥ a+ c+ n.

Using the degeneration of the Beilinson’s spectral sequence on one hand and, on the other hand, the displayassociated to a monad, we are able to prove:

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Proposition 2.8 Let E be a rank b − a − c torsion free sheaf on Pn with Chern polynomial ct(E) =1

(1−t)a(1+t)c . It holds:

(1) If b < c(n+1) and E has natural cohomology in the range −n ≤ j ≤ 0, then E is the cohomology sheafof a special monad

0 −→ OPn(−1)a α−→ ObPn

β−→ OPn(1)c −→ 0.

(2) If E is the cohomology sheaf of a special monad

0 −→ OPn(−1)a α−→ ObPn

β−→ OPn(1)c −→ 0

and H0(Pn, E) = 0, then E has natural cohomology in the range −n ≤ j ≤ 0.

3 Instanton bundles on hyperquadrics

There are very few examples of indecomposable vector bundles of small rank r, say r ≤ n − 2, on Pn, n ≥ 4.The only known indecomposable rank r, r ≤ n − 2, vector bundles on P

n are the rank 2 Horrocks–Mumfordbundle on P4 (and its pullbacks) and the rank 3 Horrocks bundle in P5. We hope that the study of rank n − 2vector bundles on hypersurfaces X ⊂ Pn may give some insight to the study of rank n − 2 vector bundles onPn. It is the aim of this section to study special bundles on quadric hypersurfaces Qn ⊂ Pn+1. So, let us startrecalling the results on coherent sheaves on Qn needed in the sequel.

Let Qn ⊂ Pn+1, n > 2, be a smooth quadric hypersurface. It is well-known that Pic(Qn) ∼= Z and that thecanonical line bundle ωQn

∼= OQn(−n). If n = 2l+ 1, the cohomology ring of Qn is given by

H∗(Qn; Z) = Ze1 + Ze2 + · · · + Zen

with ei ∈ H2i(Qn; Z) and the following relations

ei · en−i = en for all i,

er1 =

{er if r ≤ l,

2er if r > l.

If n = 2l, the cohomology ring of Qn is given by

H∗(Qn; Z) = Ze1 + Ze2 + · · · + (Ze′l + Ze′′l ) + Zel+1 + · · · + Zen

where ei ∈ H2i(Qn; Z) for i �= l, e′l, e′′l ∈ H2l(Qn; Z) and the following relations

ei · en−i = en for i < l,

e′l · e′l = e′′l · e′′l =

{0 if l is odd,

en if l is even,

e′l · e′′l =

{en if l is odd,

0 if l is even,

er1 =

⎧⎪⎨⎪⎩er if r < l,

e′l + e′′l if r = l,

2er if r > l.

Let F be a coherent sheaf on Qn. The Chern polynomial of F is

ct(F ) = 1 + c1(F )e1t+ c2(F )e2t2 + · · · + cn(F )entn


Math. Nachr. 282, No. 2 (2009) 173

if n = 2l + 1 and we simply call ci(F ) = ci ∈ Z the Chern classes of F ; and

ct(F ) = 1 + c1(F )e1t+ c2(F )e2t2 + · · · + (c′l(F )e′l + c′′l (F )e′′l ) tl + · · · + cn(F )entn

if n = 2l and we simply call ci(F ) = ci ∈ Z, for i �= l, and (c′l(F ), c′′l (F )) = cl ∈ Z2 the Chern classes of F .In [13], M. M. Kapranov defined the locally free sheaves ψi, i ≥ 0, on Qn and the Spinor bundles Σ on

Qn to construct a resolution of the diagonal Δ ⊂ Qn × Qn and to describe the bounded derived categoryDb(OQn −mod). In particular, he obtained the results that if n is even and Σ1, Σ2 are the Spinor bundles onQn, then

(Σ1(−n),Σ2(−n),OQn(−n+ 1), . . . ,OQn(−1),OQn)

is a full strongly exceptional collection of locally free sheaves on Qn; and if n is odd and Σ is the Spinor bundleon Qn, then

(Σ(−n),OQn(−n+ 1), . . . ,OQn(−1),OQn)

is a full strongly exceptional collection of locally free sheaves on Qn ([13, Proposition 4.9]). Set Ωj := ΩjPn+1

and we define inductively ψj :

ψ0 := OQn , ψ1 := Ω1(1)∣∣Qn

and, for all j ≥ 2, we define the locally free sheaf ψj as the unique non-splitting extension(

note that

Ext1(ψj−2,Ωj(j)

∣∣Qn

)= C

):

0 −→ Ωj(j)∣∣Qn

−→ ψj −→ ψj−2 −→ 0.

Our next goal is to prove the existence of special bundles on Q2n+1 ⊂ P2n+2.

Construction. Let Q2l+1 ⊂ P2l+2 be a smooth quadric hypersurface. After a suitable linear change ofvariables the equation f ∈ C[x0, . . . , x2l+2] defining Q2l+1 can be brought into the form f = x2

0 + x21 + · · · +

x22l+2. Consider the k × (l + k) matrices with linear entries

A1 =

⎛⎜⎜⎜⎜⎝

x0 x1 . . . . . . xl 0 0 . . . . . . 00 x0 x1 . . . . . . xl 0 0 . . . 00 0 x0 x1 . . . . . . xl 0 . . . 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0 0 . . . . . . . . . x0 x1 . . . . . . xl

⎞⎟⎟⎟⎟⎠ ,

A2 =

⎛⎜⎜⎜⎜⎝

xl+1 xl+2 . . . . . . x2l+1 0 0 . . . . . . 00 xl+1 xl+2 . . . . . . x2l+1 0 0 . . . 00 0 xl+1 xl+2 . . . . . . x2l+1 0 . . . 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0 0 . . . . . . . . . xl+1 xl+2 . . . . . . x2l+1

⎞⎟⎟⎟⎟⎠ .

Let α : OQ2l+1(−1)k → O2l+2kQ2l+1

be the morphism associated to the k × (2k + 2l) matrix with linear entries

A = (A1 A2) and let β : O2k+2lQ2l+1

→ OQ2l+1(1)k be the morphism associated to the (2k + 2l) × k matrix with

linear entries B = At(

transpose with respect to the standard symplectic form G :=(

0 −1k+11k+1 0

)). Since the

localized maps αx are injective for all x ∈ Q2l+1, the cohomology sheaf of the special monad

S• : 0 −→ OQ2l+1(−1)k α−→ O2k+2lQ2l+1

β−→ OQ2l+1(1)k −→ 0

is a rank 2l special bundle with Chern polynomial ct(E) = 1(1−e1t)k(1+e1t)k .



Remark 3.1 We want to point out that in the literature one can find other occasions where special symmetricmatrices also have been used to construct monads (see for example [2], [16]–[17]).

The analogy between Q2l+1 ⊂ P2l+2 and P2l+1 leads us to the following definition.

Definition 3.2 A mathematical instanton bundle on Q2l+1 with quantum number k is a rank 2l vector bundleE on Q2l+1 with trivial splitting type

(i.e., for a general line L ⊂ Q2l+1 we have E|L ∼= OL

2l)

and defined asthe cohomology bundle of a monad

S• : 0 −→ OQ2l+1(−1)k A−→ O2k+2lQ2l+1

Bt−→ OQ2l+1(1)k −→ 0

where A and B are k × (2l + 2k) matrices with linear entries; the fact that S• is a monad is equivalent to thefollowing conditions on A, B

(i) A, B have rank k at every point of Q2l+1,(ii) ABt = 0.

Proposition 3.3 Let E be any mathematical instanton bundle on Q2l+1. Then, E is simple andH0(Q2l+1, E) = 0. In particular, any mathematical instanton bundle E on Q3 is stable.

P r o o f. Let

S• : 0 −→ OQ2l+1(−1)k A−→ O2k+2lQ2l+1

Bt−→ OQ2l+1(1)k −→ 0

be the monad associated to E. Applying [2, Theorem 2.8], changing the role of Pm by Q2l+1, to the exactsequences

0 −→ K = ker(Bt

) −→ O2k+2lQ2l+1

−→ OQ2l+1(1)k −→ 0 and (3.1)

0 −→ OQ2l+1(−1)k −→ K −→ E −→ 0, (3.2)

we get that K is stable and H0(Q2l+1, E ⊗ E∗) = C or, equivalently, E is simple. Since K is a stable vectorbundle and c1(K) = −k < 0, there exists an integer λ ≥ 0 such that H0(Q2l+1,Knorm) = H0(Q2l+1,K ⊗OQ2l+1(λ)) = 0 and we get

H0(Q2l+1, E) = H0(Q2l+1,K) = 0. (3.3)

The last assertion is now obvious because for a rank 2 vector bundle on Q2l+1 being stable is equivalent to beingsimple.

We will now give a cohomological characterization of mathematical instanton bundles on Q2l+1 ⊂ P2l+2

involving only the Spinor bundle. To this end we introduce the following definition.


(Ei

1, . . . , Eiαi

)of coherent sheaves on X which generates D. Let F be a coher-

ent sheaf on X . We say that F has natural cohomology with respect to σ if and only if for all i, 0 ≤ i ≤ n, andall j, 1 ≤ j ≤ αi, at most one of the cohomology groupsHq(X,F ⊗ Ei

j) is non-zero.

Theorem 3.5 Let Q2l+1 ⊂ P2l+2 be a quadric hypersurface and let E be a rank 2l vector bundle on Q2l+1

with Chern polynomial ct(E) = 1(1−e1t)k(1+e1t)k . Then E is the cohomology bundle of a special monad of the

following type

S• : 0 −→ OQ2l+1(−1)k −→ O2k+2lQ2l+1

−→ OQ2l+1(1)k −→ 0

if and only if E has natural cohomology with respect to

σ =(Σ(−2l− 1),OQ2l+1(−2l), . . . ,OQ2l+1(−1),OQ2l+1

)and Hi(Q2l+1, E ⊗ Σ(−2l)) = 0 for all i ≥ 0.


Math. Nachr. 282, No. 2 (2009) 175

P r o o f. Assume that E has natural cohomology with respect to σ and Hi(Q2l+1, E ⊗ Σ(−2l)) = 0 forall i ≥ 0 and let us prove that E is the cohomology bundle of a special monad as in the statement. Sincect(E) = 1

(1−e1t)k(1+e1t)k , using the Riemann–Roch formula we find that

χE(t) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

2l− k(2l + 1) if t = 0,−k if t = −1,0 if − 2l+ 1 ≤ t ≤ −2,k if t = −2l.

(3.4)

By [6, Proposition 5.4],

τ = (E0, . . . , E2l+1) =(OQ2l+1(−2l),Σ(−2l),OQ2l+1(−2l+ 1), . . . ,OQ2l+1(−1),OQ2l+1

)is also a full strongly exceptional collection of locally free sheaves on Q2l+1 and its right dual is(

R(0)E2l+1, R(1)E2l, . . . , R

(2l+1)E0

)=

(OQ2l+1 , ψ∗1 , ψ

∗2 , . . . , ψ

∗2l−1,Σ,O(1)

).

Therefore, by [7, Theorem 3.16], for any coherent sheaf E on Q2l+1 there is a spectral sequence with E1-term

IIEpq1 = Extq

(E∗

p+2l+1, E) ⊗ (

R(−p)Ep+2l+1

)∗

=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

Extq(OQ2l+1(−p), E

) ⊗ ψ−p if p �= −2l− 1,−2l, 0,

Extq(OQ2l+1 , E

) ⊗OQ2l+1 if p = 0,

Extq(Σ(−2l)∗, E

) ⊗ Σ∗ if p = −2l,

Extq(OQ2l+1(2l), E

) ⊗OQ2l+1(−1) if p = −2l− 1,

(3.5)

situated in the square −2l− 1 ≤ p ≤ 0, 0 ≤ q ≤ 2l+ 1 which converges to

IIEi∞ =

{E for i = 0,

0 for i �= 0.

Using the degeneration of this spectral sequence, the values of the Euler characteristic χE(t) given in (3.4)together with the fact that E has natural cohomology with respect to σ, we deduce that E is the cohomologybundle of the monad

0 −→ OQ2l+1(−1) ⊗ U −→ ψ1 ⊗ V −→ OQ2l+1 ⊗W −→ 0 (3.6)

where U = H2l(Q2l+1, E(−2l)), V = H1(Q2l+1, E(−1)) and W = H1(Q2l+1, E) are C-vector spaces ofdimension χE(−2l) = k, −χE(−1) = k and −χE = k(2l+ 1) − 2l, respectively.

By definition, ψ∗1 = R(1)E2l and hence ψ∗

1 is given by the aid of the exact sequence

0 −→ OQ2l+1(−1) −→ Hom∗ (OQ2l+1(−1),OQ2l+1

) ⊗OQ2l+1 −→ ψ∗1 −→ 0.

Tensoring it with V and combining it with the monad (3.6) we get the following commutative diagram(O =

OQ2l+1

):

0 0⏐� ⏐�0 −→ O(−1) ⊗H2lE(−2l) −→ ψ1 ⊗H1E(−1) −→ O ⊗H1E −→ 0⏐� ⏐� ∣∣∣∣0 −→ O ⊗H1(ψ1 ⊗ E(−1)) −→ O ⊗H0O(1) ⊗H1E(−1) −→ O ⊗H1E −→ 0⏐� ⏐�

O(1) ⊗H1E(−1) = O(1) ⊗H1E(−1)⏐� ⏐�0 0

(3.7)



where H1(ψ1 ⊗ E(−1)) is a C-vector space of dimension 2l + 2k. As a consequence of the hypercohomologyspectral sequence of the double complex (3.7), the first row and the first column are monads with the samecohomology. The first row has cohomology E, so we get that E is also the cohomology bundle of a specialmonad of the following type

0 −→ OQ2l+1(−1)k −→ O2k+2lQ2l+1

−→ OQ2l+1(1)k −→ 0.

Conversely, assume that E is the cohomology bundle of a special monad

0 −→ OQ2l+1(−1)k α−→ O2k+2lQ2l+1

β−→ OQ2l+1(1)k −→ 0.

We have the exact sequences

0 −→ K := ker(β) −→ O2k+2lQ2l+1

β−→ OQ2l+1(1)k −→ 0 and (3.8)

0 −→ OQ2l+1(−1)k −→ K −→ E −→ 0, (3.9)

which gives us⎧⎪⎪⎪⎨⎪⎪⎪⎩Hi(Q2l+1, E(t)) = 0 for all t and 2 ≤ i ≤ 2l− 1,Hi(Q2l+1, E(−2l)) = 0 for all i �= 2l,Hi(Q2l+1, E(−1)) = 0 for all i �= 1,Hi(Q2l+1, E) = 0 for all i �= 0, 1,

(3.10)

and by Proposition 3.3,

H0(Q2l+1, E) = 0. (3.11)

Using the exact sequences

0 −→ K ⊗ Σ −→ O2k+2lQ2l+1

⊗ Σ −→ OQ2l+1(1)k ⊗ Σ −→ 0, and (3.12)

0 −→ OQ2l+1(−1)k ⊗ Σ −→ K ⊗ Σ −→ E ⊗ Σ −→ 0 (3.13)

we get that Hi(Q2l+1, E ⊗ Σ(−2l)) = 0 for all i; and finally using the exact sequences (3.12), (3.13) and

0 −→ Σ(−1) ⊗ E −→ O2l+1

Q2l+1⊗ E −→ Σ ⊗ E −→ 0

we get that

Hi(Q2l+1, E ⊗ Σ(−2l− 1)) = 0 for all i �= 2l. (3.14)

The equalities (3.10), (3.11) and (3.14) proves that E has natural cohomology with respect to

σ =(Σ(−2l− 1),OQ2l+1(−2l), . . . ,OQ2l+1(−1),OQ2l+1

)and we are done.

In particular, this result reflects the fact that the notion of instanton bundle on Q2l+1 given in Definition 3.2 isthe one which naturally generalizes the notion of instanton bundle on P2l+1. Indeed, we have:

Corollary 3.6 Any instanton bundle E on Q2l+1 with quantum number k satisfies:(i) ct(E) = 1

(1−e1t)k(1+e1t)k ,

(ii) E has natural cohomology with respect to

σ =(Σ(−2l− 1),OQ2l+1(−2l), . . . ,OQ2l+1(−1),OQ2l+1

)and Hi(Q2l+1, E ⊗ Σ(−2l)) = 0 for all i ≥ 0,


Math. Nachr. 282, No. 2 (2009) 177

(iii) E has trivial splitting type.

Conversely, any rank 2l vector bundleE onQ2l+1 verifying the conditions (i), (ii) and (iii) is an instanton bundleE on Q2l+1.

P r o o f. It follows from the definition of instanton bundle on Q2l+1 and Theorem 3.5.

As we pointed out before, the main motivation of this paper was the generalization of the notion of instantonbundles on P2n+1 to the notion of instanton bundle on Q2l+1. Nevertheless, we want to finish this section withthe following results that can be useful in future works. First of all, following the ideas developed by Fløystead in[8], it is not difficult to determine when there exists a special monad onQn whose maps are linear forms. Indeed,we have

Proposition 3.7 Let n ≥ 3. There exist monads on Qn whose entries are linear maps, i.e. special monads

0 −→ OQn(−1)a α−→ ObQn

β−→ OQn(1)c −→ 0

if and only if at least one of the following conditions holds:

(1) b ≥ 2c+ n− 1 and b ≥ a+ c.

(2) b ≥ a+ c+ n.

If so, there actually exists a special monad with the map α degenerating in expected codimension b− a− c+ 1.

P r o o f. First, let us prove the existence part. Without lost of generality we may assume thatQn is the quadrichypersurface in Pn+1 = Proj(k[x0, x1, . . . , xn+1]) defined by f(x0, . . . , xn+1) = x2

0 + x21 + · · · + x2

n+1. By[8, Main Theorem], if b ≥ 2c+ n and b ≥ a+ c or b ≥ a+ c+ n+ 1 then there exist special monads

0 −→ OPn+1(−1)a α−→ ObPn+1

β−→ OPn+1(1)c −→ 0 (3.15)

with the map α degenerating in the expected codimension b − a − c + 1. So restricting a general monad (3.15)to Qn we get a special monad


β−→ OQn(1)c −→ 0

with the map α degenerating in the expected codimension b − a− c+ 1. So, it is enough to consider the cases

(a) b = 2c+ n− 1 and b ≥ a+ c.

(b) b = a+ c+ n.

(a) Assume b = 2c+ n− 1 and b ≥ a+ c. We distinguish two subcases

(a1) b = 2c+ n− 1 and b = a+ c.

(a2) b = 2c+ n− 1 and b > a+ c.

(a1) b = 2c + n − 1 and b = a + c. Set n1 = n−12 if n is odd and n1 = n−2

2 if n is even. Consider the(n1 + c) × c, (n− 1 − n1 + c) × c, (n− 1 + c) × (n− 1 − n1 + c) and (n− 1 + c) × (n1 + c) matrices

A1 =

⎛⎜⎜⎝

x0 x1 . . . . . . xn1 0 0 . . . . . . 00 x0 x1 . . . . . . xn1 0 0 . . . 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0 0 . . . . . . . . . x0 x1 . . . . . . xn1

⎞⎟⎟⎠ ,

A2 =

⎛⎜⎜⎝

xn1+1 xn1+2 . . . . . . xn 0 0 . . . . . . 00 xn1+1 xn1+2 . . . . . . xn 0 0 . . . 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0 0 . . . . . . . . . xn1+1 xn1+2 . . . . . . xn

⎞⎟⎟⎠ ,

A3 =

⎛⎜⎜⎝

x0 x1 . . . . . . xn1 0 0 . . . . . . 00 x0 x1 . . . . . . xn1 0 0 . . . 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0 0 . . . . . . . . . x0 x1 . . . . . . xn1

⎞⎟⎟⎠ ,



A4 =

⎛⎜⎜⎝

xn1+1 xn1+2 . . . . . . xn 0 0 . . . . . . 00 xn1+1 xn1+2 . . . . . . xn 0 0 . . . 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0 0 . . . . . . . . . xn1+1 xn1+2 . . . . . . xn

⎞⎟⎟⎠ .

Define the complex


β−→ OQn(1)c −→ 0 (3.16)

where β is the map given by the matrix B = (A1 A2) and α is the map given by

A =(

A4

−A3

).

It is not difficult to see that α degenerates in codimension b− c− a+ 1 = 1.

(a2) If b = 2c + n − 1 and b > a + c, we consider a sufficiently general injection φ : OQn(−1)a −→OQn(−1)b−c and the composition αφ : OQn(−1)a −→ O2c+n−1

Qnwhere α is the map appearing in (3.16). Since

for φ general enough, αφ degenerates in codimension b− c− a+ 1 we get the special monad

0 −→ OQn(−1)a αφ−→ Ob=2c+n−1Qn

β−→ OQn(1)c −→ 0

we were looking for.

(b) b = a + c + n. We may assume that b < 2c + n − 1 since otherwise we are in one of the cases alreadycovered. Since b = a+ c+ n and b < 2c+ n− 1 we get a ≤ c− 1. Thus b = a+ c+ n ≥ 2a+ n− 1 and thereexists a special monad

0 −→ OQn(−1)c ρ−→ ObQn

η−→ OQn(1)a −→ 0

with the map ρ degenerating in the expected codimension b− a− c+ 1 ≥ n+ 1 and so ρ does not degenerates.Dualizing we get a special monad

0 −→ OQn(−1)a η∗−→ Ob

Qn

ρ∗−→ OQn(1)c −→ 0

with the map η∗ not degenerating and we are done.

Pursuing the ideas developed by Fløystead in [8] and essentially changing the role of Pn by Qn we get that

the numerical conditions on a, b, c and n are indeed necessary.

In this last result, we use the unified notation Σ∗ meaning that for even n both Spinor bundles Σ1 and Σ2 areconsidered, and for odd n, the Spinor bundle Σ.

Proposition 3.8 Let Qn ⊂ Pn+1 be a quadric hypersurface and let E be a rank b − a− c torsion free sheafon Qn with Chern polynomial ct(E) = 1

(1−e1t)a(1+e1t)b . It holds:(a) If b− c(n+ 2) < 0, E has natural cohomology with respect to

σ = (Σ∗(−n),OQn(−n+ 1), . . . ,OQn(−1),OQn)

and Hi(Qn, E ⊗ Σ∗(−n+ 1)) = 0 for all i ≥ 0, then E is the cohomology bundle of a special monad ofthe following type

S• : 0 −→ OQn(−1)a −→ ObQn

−→ OQn(1)c −→ 0.

(b) If E is the cohomology bundle of a special monad of the following type

S• : 0 −→ OQn(−1)a −→ ObQn

−→ OQn(1)c −→ 0

and H0(Qn, E) = 0, then E has natural cohomology with respect to

σ = (Σ∗(−n),OQn(−n+ 1), . . . ,OQn(−1),OQn)

and Hi(Qn, E ⊗ Σ∗(−n+ 1)) = 0 for all i ≥ 0.


Math. Nachr. 282, No. 2 (2009) 179

4 Final remarks and open problems

The notions introduced in this paper as well as the results proved give rise to a number of quite interestingquestions and possible generalizations that we gather together in this last section.

We have seen in Proposition 3.3 that every instanton bundle onQ3 is stable and we are led to pose the followingattractive question.

Question Is any instanton bundle E on Q2l+1 stable?

We would also like to study the stability of special bundles of higher rank on Qn. So, we ask:

Question Is any special bundle E on Qn stable or at least semistable?

Let X be a smooth projective variety of dimension n with an m-block collection σ = (E0, E1, . . . , Em),Ei =

(Ei

1, . . . , Eiαi

)of coherent sheaves on X which generates D. The construction of higher rank special

bundles and higher rank special sheaves which are not locally free on X is an interesting direction for futureresearch. More precisely, in Proposition 2.8 (resp. Theorem 3.5 and Proposition 3.8 ) we have established a coho-mological characterization of special sheaves on P

n (resp. onQ2l+1). We hope that an analogous cohomologicalcharacterization will work for special sheaves on other smooth projective varieties.

Acknowledgements The authors were partially supported by MTM2007-61104.

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Monads and instanton bundles on smooth hyperquadrics

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