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 c 1 ( L , � · � ) of a smooth metrized line bundle: ◮ c 1 ( L , � · � ) is a closed (1 , 1)-form on X . ◮ c 1 ( L , � · � ) | U = − dd c log � s � , s ∈ Γ( U , L ), s � = 0. ◮ dd c = 1 2 π ∂∂ is a differential operator. ◮ De Rham class of curvature form is c 1 ( L ) ∈ H 2 ( 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” c 1 ( L , � · � ) n . This is: ◮ smooth volume form on X . ◮ smooth positive measure of mass c 1 ( L ) n if metric is positive. ◮ Calabi-Yau Theorem : if L is ample and µ is a smooth positive measure on X of mass c 1 ( L ) n , then there exists a unique (up to scaling) positive metric � · � on L such that c 1 ( 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: i Spec A i , then X an = � ◮ If X = � i (Spec A i ) an , where (Spec A i ) an = { mult. seminorms on A i 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 X 0 for special fiber. ◮ Picture! ◮ Work mainly with SNC models : ◮ X is regular; ◮ X 0 = � I ∈ I b i E i has simple normal crossing support. ◮ for J ⊂ I , E I := � i ∈ J E i 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 � s � L = 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 ⊆ X 0 . ◮ 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 c 1 ( L , � · � ) n on X = X an of mass c 1 ( L ) n . ◮ When L is an extension of L to an SNC model X with special fiber X 0 = � i ∈ I b i E i , then c 1 ( L , � · � L ) n = � b i ( L| E i ) n δ x i i ∈ I is an atomic measure. ◮ In general, c 1 ( 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 c 1 ( L ) n supported on some dual complex ∆ X . Then there exists a semipositive continuous metric � · � on L such that c 1 ( 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 c 1 ( L , � · � ) n = µ becomes Monge-Amp` ere equation ( ω + dd c ϕ ) n = µ, where ω = c 1 ( 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 O X . ◮ Gubler: Stone-Weierstrass = ⇒ the set D ( X ) of model functions is dense in C 0 ( X ). ◮ A closed (1 , 1) -form ω ∈ Z 1 , 1 ( X ) on X is a numerical class of an R -line bundle L on some model X . ◮ A semipositive form ω ∈ Z 1 , 1 + ( X ) is the class of a nef line bundle on some model X . + ( X ) is a semipositive form, define ω n as positive ◮ If ω ∈ Z 1 , 1 (atomic) Radon measure on X using intersection theory. ◮ Have operator dd c : D ( X ) → Z 1 , 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 ∈ Z 1 , 1 + ( X ). ◮ A model function ϕ is ω -psh if ω + dd c ϕ is semipositive. ◮ A general function ϕ : X → R ∪ {−∞} is ω -psh if it is a limit of 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 n 1 � ϕ ( ω + dd c ϕ ) j ∧ ω n − j . � E ( ϕ ) := n + 1 X j =0 ◮ Easy to see that E is concave and Gateaux differentiable with E ′ ( ϕ ) = ( ω + dd c ϕ ) n . ◮ If ϕ ∈ PSH( X , ω ) ∩ D ( X ) maximizes the functional � F µ ( ψ ) := E ( ψ ) − ψ µ X over D ( X ) then ( ω + dd c ϕ ) n = µ as desired. ◮ Problem: no reason for there to be a maximizer! ◮ Idea: extend E and F µ 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 F µ = E − µ has maximizer ϕ ∈ PSH( X , ω ). ◮ Would like to say ( ω + dd c ϕ ) n = µ . ◮ Question: how to define ( ω + dd c ϕ ) 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 + dd c ϕ ) n . ◮ If ϕ minimizes E − µ , this yields ( ω + dd c ϕ ) n = µ , as desired. ◮ Differentiability follows from: ◮ Orthogonality : if f ∈ D ( X ), then (picture) � ( f − P ( f ))( ω + dd c P ( 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 + dd c ϕ ) 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