Matthew Baker's Project Group 3: The Poincaré-Lelong...

IntroductionPoincare-Lelong for the Berkovich Projective Line

Poincare-Lelong for General Algebraic Curves

Matthew Baker’s Project Group 3:The Poincare-Lelong Formula

Xander Faber, Andrew Obus, Jorge Pineiro, Jerome Poineau,Christian Wahle

March 20, 2007

X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula

Poincare-Lelong for General Algebraic CurvesThe Berkovich Projective Line

Throughout, K will denote an algebraically closed field completewith respect to a nontrivial nonarchimedean norm.

I P1Berk is a tree, infinitely branched at a dense set of points.

The points of P1Berk can be classified into 4 types, of which we

will primarily focus on 3:I Type I points are the classical points of P1(K ).I The set of type II and III points will be denoted by HR

I There is a canonical metric on HRBerk. For example, if

ζ0,a, ζ0,b ∈ HRBerk correspond to balls in K centered at zero

with radii a < b, respectively, then

ρ(ζ0,a, ζ0,b) = logv b − logv a.

I An extended real-valued function on P1Berk is continuous

piecewise affine (CPA) roughly if, on small paths [x , y ], it is ofthe form t 7→ Aρ(x , t) + B for some A,B ∈ R.

will primarily focus on 3:

I Type I points are the classical points of P1(K ).I The set of type II and III points will be denoted by HR

will primarily focus on 3:I Type I points are the classical points of P1(K ).

I The set of type II and III points will be denoted by HRBerk.

The Laplacian

View HRBerk as the direct limit of a direct system of metrized

graphs Γ (with metric induced by ρ).

We build a Laplacian of a CPA function f by restricting to finitemetrized subgraphs Γ of P1

I For p ∈ Γ, ∆(f |Γ)(p) = −∑

~v∈TΓ,pd~v (f |Γ)(p).

I ∆(f |Γ) =∑

p∈Γ ∆(f |Γ)(p) δp

I Define ∆(f ) on P1Berk to be the weak limit of the Laplacians

of f |Γ on finite subgraphs Γ (pushed forward to P1Berk).

The Laplacian

I For p ∈ Γ, ∆(f |Γ)(p) = −∑

I ∆(f |Γ) =∑

The Laplacian

I For p ∈ Γ, ∆(f |Γ)(p) = −∑

I ∆(f |Γ) =∑

The Laplacian

I For p ∈ Γ, ∆(f |Γ)(p) = −∑

I ∆(f |Γ) =∑

The Laplacian

I For p ∈ Γ, ∆(f |Γ)(p) = −∑

I ∆(f |Γ) =∑

The Poincare-Lelong Formula

I If ϕ ∈ K (T ) is a rational function on P1K , then the function

z 7→ − logv |ϕ(z)| extends to a CPA function on P1Berk.

I Theorem[Poincare-Lelong] For any rational functionϕ ∈ K (T ), we have the identity

∆(− logv |ϕ|) = δdiv(ϕ).

I Key Point: The left side depends on the canonical metric,while the right is independent of it.

I Question: If one replaces the canonical metric by anothermetric ρ′ on HR

Berk, can Poincare-Lelong still hold?

A visual representation of P1Berk r∞

EEDDCCBBBBrAA( ��

��#��

��

(( ��

��

#��

��

hh``` XXX PPPP

SSSSSSSr b

...��EELLLSS

HHQQZZcEEEE

hh``` XXX PPPP

QQZZcEEDDCCBBAA ��(( ��

��

#s ζGauss

EEEEEEEEEEEr...b

(((( ��

��

��r

��

EE��BBB��

��r XX...b

aaaaaaaaaa

HHHHHHHH

cccccccccccccr

(( ��

��

��#��

��

��EEEE

...b ��r A

AAr��r@r

(( ��

��

��#��

��

EEDDCCBBAA

( ��#��

��

Poincare-Lelong for General Algebraic CurvesThe case of the Berkovich Projective Line

Conditions on the Metric

I Say ρ′ is another metric on P1Berk for which Poincare-Lelong

holds, and let ∆′ be the associated Laplacian.

Thus,∆′(− logv |ϕ|) = δdiv(ϕ) for all ϕ ∈ K (T ).

I Consider the (unique) path in P1Berk from 0 to ∞, and let

ϕ = T .

Then ∆′(− logv |T |) = δ0 − δ∞.

I Note that − logv |T |ζ0,r = − logv r , and is locally constant offthe path from 0 to ∞.

I Consider two discs of radii a < b, centered around the origin,and the interval Γ connecting the corresponding points ζ0,a

and ζ0,b.

Since Γ is a finite metrized graph, Poincare-Lelongsays that ∆′(− logv |T |) on Γ is δζ0,a − δζ0,b

I So − logv |T | is CPA in the interior of Γ, and is thus clearlyaffine here.

Since a and b can be chosen arbitrarily, − logv |T |is affine with respect to ρ′ in (0,∞).

holds, and let ∆′ be the associated Laplacian. Thus,∆′(− logv |ϕ|) = δdiv(ϕ) for all ϕ ∈ K (T ).

ϕ = T .

Then ∆′(− logv |T |) = δ0 − δ∞.

and ζ0,b.

ϕ = T .

Then ∆′(− logv |T |) = δ0 − δ∞.I Note that − logv |T |ζ0,r = − logv r , and is locally constant off

the path from 0 to ∞.I Consider two discs of radii a < b, centered around the origin,

and the interval Γ connecting the corresponding points ζ0,a

and ζ0,b.

ϕ = T . Then ∆′(− logv |T |) = δ0 − δ∞.

and ζ0,b.

ϕ = T . Then ∆′(− logv |T |) = δ0 − δ∞.I Note that − logv |T |ζ0,r = − logv r , and is locally constant off

the path from 0 to ∞.

and ζ0,b.

.I So − logv |T | is CPA in the interior of Γ, and is thus clearly

affine here.

and ζ0,b. Since Γ is a finite metrized graph, Poincare-Lelongsays that ∆′(− logv |T |) on Γ is δζ0,a − δζ0,b

affine here.

affine here. Since a and b can be chosen arbitrarily, − logv |T |is affine with respect to ρ′ in (0,∞).

Conditions on the Metric (cont’d)

I Since − logv |T | is affine on (0,∞), we have that ∃c ∈ Rsuch that ∀0 < a ≤ b, − logv a− (− logv b) = cρ′(ζ0,a, ζ0,b).

I So ρ′(ζ0,a, ζ0,b) = logv (b/a)c =

ρ(ζ0,a,ζ0,b)c .

I Since multiplying distances by 1/c multiplies slopes by c , wehave ∆′(− logv |T |) = cδ0 − cδ∞.

Thus c = 1, and ρ = ρ′ on(0,∞)!

I We can repeat this argument for the path between any twoK -rational points y and z by using the function T−y

T−z in placeof T (where, by convention, we take T −∞ to be 1).

Thus,ρ = ρ′ on all type II and type III points.

I By continuity, ρ = ρ′ on type IV points, and thus on all ofHBerk.

ρ(ζ0,a,ζ0,b)c .

Thus c = 1, and ρ = ρ′ on(0,∞)!

ρ(ζ0,a,ζ0,b)c .

Thus c = 1, and ρ = ρ′ on(0,∞)!

ρ(ζ0,a,ζ0,b)c .

I Since multiplying distances by 1/c multiplies slopes by c , wehave ∆′(− logv |T |) = cδ0 − cδ∞. Thus c = 1, and ρ = ρ′ on(0,∞)!

ρ(ζ0,a,ζ0,b)c .

T−z in placeof T (where, by convention, we take T −∞ to be 1). Thus,ρ = ρ′ on all type II and type III points.

ρ(ζ0,a,ζ0,b)c .

T−z in placeof T (where, by convention, we take T −∞ to be 1). Thus,ρ = ρ′ on all type II and type III points.

Poincare-Lelong for General Algebraic CurvesThe Case of a General Curve

Main Result

Let X denote a connected smooth proper algebraic curve over K .

I There is a canonical metric ρ on the associated analytic spaceXBerk and a notion of Laplacian.

I Theorem.[Thuillier] For any rational function ϕ ∈ K (X ), thefunction z 7→ − logv |ϕ| lies in CPA(XBerk), and

I Theorem.[F., O., P., P., W.] Under the assumption that theresidue field of K is countable, the canonical metric is theunique metric on XBerk such that Thuillier’s result holds.

Main Result

Idea of the Proof

The proof has essentially two steps:

I Use the fact that − logv |ϕ| ∈ CPA(XBerk) to show that thenew metric must be a constant scaling of the canonical metric.

I Choose a clever rational function ϕ and compute∆(− logv |ϕ|) locally to see that the two metrics must agree.

The proof is trickier than for P1Berk because we cannot immediately

construct rational functions with prescribed divisors and poles.To get around this, we apply the following result:

Theorem.[Rumely] Suppose K has countable residue field, and letX be a connected smooth proper curve over K . Let U ⊂ XBerk bean open neighborhood and let ζ ∈ X (K ) \ U(K ) be any classicalpoint. Then there exists a rational function ϕ ∈ K (X ) such thatall of the zeros of ϕ lie in U, and the only pole of ϕ is at ζ.

Idea of the Proof

construct rational functions with prescribed divisors and poles.

To get around this, we apply the following result:

Idea of the Proof

Matthew Baker's Project Group 3: The Poincaré-Lelong...

Documents

Transcript of Matthew Baker's Project Group 3: The Poincaré-Lelong...