Singular metrics and the Calabi-Yau theorem in non-Archimedean - - PowerPoint PPT Presentation

Singular metrics and the Calabi-Yau theorem in non-Archimedean geometry

Mattias Jonsson

University of Michigan

Tuesday February 14, 2012 Joint work with Charles Favre and S´ ebastien Boucksom

Mattias Jonsson Singular metrics and Calabi-Yau

Plan

◮ The Calabi-Yau theorem over C. ◮ A non-Archimedean analogue. ◮ Idea of proof.

Mattias Jonsson Singular metrics and Calabi-Yau

Metrized line bundles on complex projective manifolds

◮ X = smooth, complex proj. variety =

⇒ X cplx manifold.

◮ L → X line bundle. ◮ · metric on L:

◮ Picture! ◮ A section s ∈ Γ(U, L) gives rise to function s on U. ◮ · is continuous if s ∈ C 0(U) for all s. ◮ · is smooth if s ∈ C ∞(U) for all s. ◮ Say (L, · ) is a metrized line bundle.

◮ Curvature form c1(L, · ) of a smooth metrized line bundle:

◮ c1(L, · ) is a closed (1, 1)-form on X. ◮ c1(L, · )|U = −ddc log s, s ∈ Γ(U, L), s = 0. ◮ ddc =

1 2π∂∂ is a differential operator.

◮ De Rham class of curvature form is c1(L) ∈ H2(X, Z).

◮ Metric is positive if the curvature form is positive. ◮ By Kodaira, L admits positive metric iff L is ample.

Mattias Jonsson Singular metrics and Calabi-Yau

The Calabi-Yau theorem

◮ Define “curvature volume form” c1(L, · )n. This is:

◮ smooth volume form on X. ◮ smooth positive measure of mass c1(L)n if metric is positive.

◮ Calabi-Yau Theorem: if L is ample and µ is a smooth positive

measure on X of mass c1(L)n, then there exists a unique (up to scaling) positive metric · on L such that c1(L, · )n = µ. (1)

◮ Can view (1) as nonlinear PDE, the Monge-Amp`

ere equation.

◮ Ko

lodziej generalized this to the case when µ is a positive, not too singular measure. Metric is then (only) continuous.

◮ Our main result is a non-Arch version of Ko

lodziej’s theorem.

◮ Lots of things to define to make sense of this!

Mattias Jonsson Singular metrics and Calabi-Yau

Berkovich spaces

◮ K = non-Archimedean field. ◮ Will work with K := k((t)), char k = 0. ◮ Norm on K: |t| = e−1 (trivial norm on k). ◮ Valuation ring is k[[t]]. ◮ X = smooth projective variety over K. ◮ X an = analytification of X as Berkovich space: ◮ If X = i Spec Ai, then X an = i(Spec Ai)an, where

(Spec Ai)an = {mult. seminorms on Ai extending norm on K}

◮ Can understand X an through integral models of X.

Mattias Jonsson Singular metrics and Calabi-Yau

Models and Berkovich spaces

◮ A model of X is given by a flat, projective morphism

X → Spec k[[t]] with generic fiber X. Write X0 for special fiber.

◮ Picture! ◮ Work mainly with SNC models:

◮ X is regular; ◮ X0 =

I∈I biEi has simple normal crossing support.

◮ for J ⊂ I, EI :=

i∈J Ei is empty or irreducible.

◮ To SNC model X associate dual complex ∆X . Picture! ◮ Have retraction X an → ∆X and embedding ∆X ֒

→ X an.

◮ Thm: X an ∼

→ lim ← −X ∆X .

◮ Can visualize X an when X = curve. Picture! ◮ As in complex case, will write X = X an. Justified by GAGA.

Mattias Jonsson Singular metrics and Calabi-Yau

Metrics on line bundles

◮ How to metrize a line bundle L → X? ◮ Extend L (somehow) to line bundle L on SNC model X. ◮ Get continuous metric · L on X such that

sL = 1 for every local frame s of L.

◮ Such a metric is called a model metric. ◮ Say that · L is semipositive if L is (relatively) nef, i.e.

(L · C) ≥ 0 for every proper curve C ⊆ X0.

◮ Def [S.-W. Zhang] A continuous metric · is semipositive if

it is the uniform limit of semipositive model metrics.

◮ This implies that L is nef (on X).

Mattias Jonsson Singular metrics and Calabi-Yau

The Chambert-Loir measure

◮ Consider line bundle L with semipos. cont. metric · . ◮ Chambert-Loir defined positive measure c1(L, · )n on

X = X an of mass c1(L)n.

◮ When L is an extension of L to an SNC model X with special

fiber X0 =

i∈I biEi, then

c1(L, · L)n =

• i∈I

bi(L|Ei)nδxi is an atomic measure.

◮ In general, c1(L, · )n is constructed using approximation.

Mattias Jonsson Singular metrics and Calabi-Yau

A non-Archimedean Calabi-Yau theorem

• Thm. Assume X is algebraizable and L is ample. Let µ be a

positive measure on X of mass c1(L)n supported on some dual complex ∆X . Then there exists a semipositive continuous metric · on L such that c1(L, · )n = µ. The metric is unique up to a multiplicative constant.

◮ X algebraizable means X = Y ⊗F K, where:

◮ F = function field of curve having K as a completion ◮ Y = smooth projective F-variety.

◮ Previous results:

◮ Kontsevich-Tschinkel (2001): sketch for µ = Dirac mass. ◮ Thuillier (2005): X = curve. ◮ Yuan-Zhang (2009): uniqueness. ◮ Liu (2010): existence when X = abelian var. Mattias Jonsson Singular metrics and Calabi-Yau

Steps in proof

◮ Write problem “additively” using forms and weights rather

than line bundles and metrics.

◮ Given reference model metric · 0 on L, write

· = · 0 · e−ϕ for some function ϕ ∈ C 0(X).

◮ Equation c1(L, · )n = µ becomes Monge-Amp`

ere equation (ω + ddcϕ)n = µ, where ω = c1(L, · 0) is the curvature form (to be defined).

◮ Solve the Monge-Amp`

ere equation using variational method going back to Alexandrov (1937). We follow approach by Berman-Boucksom-Guedj-Zeriahi (2009) over C.

◮ In this approach, also use singular metrics: ϕ ∈ C 0(X).

Mattias Jonsson Singular metrics and Calabi-Yau

Forms, positivity and model functions

◮ Introduce notions parallel to the complex case. ◮ A model function is a model metric on OX. ◮ Gubler: Stone-Weierstrass =

⇒ the set D(X) of model functions is dense in C 0(X).

◮ A closed (1, 1)-form ω ∈ Z1,1(X) on X is a numerical class of

an R-line bundle L on some model X.

◮ A semipositive form ω ∈ Z1,1 + (X) is the class of a nef line

bundle on some model X.

◮ If ω ∈ Z1,1 + (X) is a semipositive form, define ωn as positive

(atomic) Radon measure on X using intersection theory.

◮ Have operator ddc : D(X) → Z1,1(X). Image consists of line

bundles that are numerically trivial on X.

Mattias Jonsson Singular metrics and Calabi-Yau

The class of ω-psh functions

◮ Fix a semipositive form ω ∈ Z1,1 + (X). ◮ A model function ϕ is ω-psh if ω + ddcϕ is semipositive. ◮ A general function ϕ : X → R ∪ {−∞} is ω-psh if it is a limit

• f ω-psh model functions.

◮ Topology on PSH(X, ω): uniform conv. on dual complexes. ◮ Thm. The convex set PSH(X, ω)/R is compact. ◮ Thm. Any ω-psh function is a decreasing pointwise limit of

ω-psh model functions.

◮ Proofs use techniques from alg. geometry (multiplier ideals).

Mattias Jonsson Singular metrics and Calabi-Yau

Energy of model functions

◮ Define the energy of a model function ϕ as

E(ϕ) := 1 n + 1

n

• j=0
• X

ϕ(ω + ddcϕ)j ∧ ωn−j.

◮ Easy to see that E is concave and Gateaux differentiable with

E ′(ϕ) = (ω + ddcϕ)n.

◮ If ϕ ∈ PSH(X, ω) ∩ D(X) maximizes the functional

Fµ(ψ) := E(ψ) −

• X

ψ µ

• ver D(X) then (ω + ddcϕ)n = µ as desired.

◮ Problem: no reason for there to be a maximizer! ◮ Idea: extend E and Fµ to PSH(X, ω).

Mattias Jonsson Singular metrics and Calabi-Yau

Energy of ω-psh functions

◮ Define the energy of ϕ ∈ PSH(X, ω) as

E(ϕ) := inf{E(ψ) | ψ ∈ PSH(X, ω) ∩ D(X), ψ ≥ ϕ}.

◮ Get an usc functional E : PSH(X, ω) → R ∪ {−∞}. ◮ Hypotheses =

⇒ µ continuous functional on PSH(X, ω).

◮ Thus Fµ = E − µ has maximizer ϕ ∈ PSH(X, ω). ◮ Would like to say (ω + ddcϕ)n = µ. ◮ Question: how to define (ω + ddcϕ)n? ◮ Answer: follow complex approach by Bedford-Taylor, extended

by Guedj-Zeriahi.

◮ Problem: PSH(X, ω) is a convex set, not a vector space! ◮ Solution: use envelopes.

Mattias Jonsson Singular metrics and Calabi-Yau

Envelopes, differentiability and orthogonality

◮ Given any function f : X → R ∪ {−∞} define

P(f ) := sup{ϕ ∈ PSH(X, ω) | ϕ ≤ f }∗, the ω-psh envelope of f . Picture!

◮ Differentiability: if E(ϕ) > −∞, then the function

D(X) ∋ f → E(P(ϕ + f )) ∈ R is Gateaux differentiable, with derivative (ω + ddcϕ)n.

◮ If ϕ minimizes E − µ, this yields (ω + ddcϕ)n = µ, as desired. ◮ Differentiability follows from: ◮ Orthogonality: if f ∈ D(X), then (picture)

• (f − P(f ))(ω + ddcP(f ))n = 0.

◮ This in turn reduces to the “BDPP” thm in alg. geometry.

Mattias Jonsson Singular metrics and Calabi-Yau

Regularity

◮ At this point, have found ϕ ∈ PSH(X, ω) such that

E(ϕ) > −∞ and (ω + ddcϕ)n = µ.

◮ Following Ko

lodziej, can use capacity estimates to show that ϕ is in fact continuous.

◮ Unclear if there is an analogue of Yau’s Theorem (µ and ϕ

smooth).

Mattias Jonsson Singular metrics and Calabi-Yau