University:
University of Michigan
Course:
MATH 695 | Algebraic Topology I
Academic year:

2019
Views:

71

Pages:

16
Author:

Chanel F

Pluripotential theory in a non-archimedean setting 2011 Pluripotential theory in a non-archimedean setting Plan I Archimedean pluripotential theory. I Non-archimedean pluripotential theory. I Joint work with S. Boucksom and C. Favre. Pluripotential theory in a non-archimedean setting Pluripotential theory on Kähler manifolds I Pluripotential theory = study of plurisubharmonic (psh) fcns. R (X , ω) compact Kähler manifold. Assume X ω n = 1. I Say ϕ : X → [−∞, ∞[ ω-psh if dd c ϕ + ω ≥ 0. I Example: X = Pn , ω = Fubini-Study and I N X |fk |2 1 log , ϕ= 2m k · k2m k=1 where f1 , . . . , fN homo polys on Cn+1 with I T −1 k fk (0) = {0}. Compactness property: the function PSH(X , ω) 3 ϕ 7→ sup ϕ ∈ R X is continuous and proper in the L1 -topology. Pluripotential theory in a non-archimedean setting Geometric Monge-Ampère equation I Monge-Ampère equation: given probability measure µ on X , find ω-psh function ϕ on X such that MA(ϕ) := (ω + dd c ϕ)n = µ. I Thm by Calabi/Yau: uniqueness/existence when µ > 0 smooth. I Thm by Guedj-Zeriahi/Dinew: existence/uniqueness when µ non-pluripolar measure (no mass on {u = −∞}, u psh). I Example: X = P1 , ω = Fubini-Study. Z |z − w | ϕ(z) = log p dµ(w ). 1 + |w |2 I Non-linear problem in dim > 1! Pluripotential theory in a non-archimedean setting Capacity and extremal functions I I To study MA eqn, useful to develop capacity theory. For any subset E ⊆ X , define extremal function uE = sup{ϕ ∈ PSH(X , ω) | ϕ ≤ 0, ϕ|E ≤ −1}. I For E ⊆ X Borel, define the capacity Z Cap(E ) = sup{ MA(ϕ) | ϕ ∈ PSH(X , ω), −1 ≤ ϕ ≤ 0}. E I I Thm: for K ⊆ X compact, Cap(K ) = supp MA(uK∗ ) ⊆ K . Thm: for any E ⊆ X , TFAE: I I I I R K MA(uK∗ ) and E is pluripolar, i.e. E ⊆ {u = −∞}, u psh; ∀ε ∃G ⊇ E open with Cap(G ) < ; uE∗ ≡ 0; E is “negligable”. Pluripotential theory in a non-archimedean setting Key tools I Regularization: can approximate any ϕ ∈ PSH(X , ω) by a decreasing sequence ϕj & ϕ, with ϕj ∈ PSH(X , ω) ∩ C ∞ (X ). I Comparison principle: Z Z MA(ψ) ≤ MA(ϕ) ϕ<ψ ϕ<ψ for ϕ, ψ ∈ PSH(X , ω) sufficiently regular. I “Balayage”: Can solve MA = 0 with Dirichlet boundary condition. Implies supp MA(uK∗ ) ⊂ K . I Countability: X has countable basis for topology. Used to prove e.g. Choquet’s Lemma. Pluripotential theory in a non-archimedean setting Beyond Archimedes I I I I I Try to extend previous analysis to non-archimedean setting. Natural for problems of arithmetic nature. Relevant work by several people: Baker-Rumely, Boucksom-Favre-J, Chambert-Loir, Chinburg-Rumely, Favre-Rivera-Letelier, Gubler, Kontsevich-Tschinkel, Liu, Thuillier, Yuan-Zhang, . . . . Definition. Archimedean=not non-Archimedean. Factoid: Archimedes died 2222 years ago. Pluripotential theory in a non-archimedean setting Non-archimedean analysis I Assume k field equipped with non-archimedean norm: |a + b| ≤ max{|a|, |b|}. Also assume k complete. I Example: C((t)) (Laurent series) 0 < |t| < 1, |c| = 1. I Example: any field k with trivial norm: |a| = 1, a 6= 0. I Can try to develop analytic geometry as over C. I Problem 1: k totally disconnected. I Problem 2: in general, k not locally compact. I Various ways to deal with this. I Approach: replace k (or k n ) by suitable Berkovich space (space of valuations). Pluripotential theory in a non-archimedean setting The Berkovich affine space I I 0 - 0 - I Def: AnBerk is the set of multiplicative seminorms on k[z1 , . . . , zn ] extending the given norm on k. When k = C, AnBerk = Cn by Gelfand-Mazur. Points of A1Berk can be viewed as balls in k. Thus A1Berk admits tree structure. I For n > 1, AnBerk harder to visualize! Pluripotential theory in a non-archimedean setting The valuation space V I I Try to adapt previous results to Berkovich spaces. Look at special case: k = C with trivial valuation. b ⊂ An by Define subsets V ⊂ V Berk b := {seminorms on C[z1 , . . . , zn ] | max |zi | < 1}. V i V := {seminorms on C[z1 , . . . , zn ] | max |zi | = e −1 }. i I Useful for studying singularities at a point in Cn . b ' cone over V. V I V contractible compact Hausdorff space. No countable basis. I V ' limit of simplicial complexes. R-tree when n = 2. I Define class PSH(V) and “do” pluripotential theory on it. b by homogeneity. Functions in PSH(V) extend to V I I Pluripotential theory in a non-archimedean setting Plurisubharmonic functions I View elements of V as semivaluations ν : C[z1 , . . . , zn ] → [0, +∞] I using | · | = e −ν . Define psh fcn as decreasing limit of fcns of the form c max log |fj |, 1≤j≤N I I I T where c > 0, fj ∈ C[z1 , . . . , zn ] and j {fj = 0} = {0}. Here log |f |(ν) := −ν(f ). Can define analogy of L1 -topology (but not metrizable). Thm: the function PSH(V) 3 ϕ 7→ sup ϕ ∈ R− V I is continuous and proper. Proof uses multiplier ideals to construct ϕj & ϕ. Pluripotential theory in a non-archimedean setting Monge-Ampère operator on V I I I First define MA(ϕ) for ϕ = maxj log |fj |. R Suffices to define V ψ MA(ϕ) for ψ = maxi log |gi |. Do this as a local intersection number. Analytically: Z ψ MA(ϕ) = −((dd c ϕ)n−1 ∧ dd c ψ){0}, V where in the RHS we view ϕ and ψ as functions on Cn ! I For general ϕ ∈ PSH(V), define MA(ϕ) = limj MA(ϕj ), where ϕj & ϕ. I Thm. Unique solution to MA(ϕ) = µ for any non-pluripolar measure µ. I Existence proof uses variational approach (Alexandrov; Berman-Boucksom-Guedj-Zeriahi). I Need to develop capacity theory along the way. Pluripotential theory in a non-archimedean setting Capacity and extremal functions I For any subset E ⊆ V, define extremal function uE = sup{ϕ ∈ PSH(V) | ϕ|E ≤ −1}. I For E ⊆ V Borel, define the capacity Z Cap(E ) = sup{ MA(ϕ) | ϕ ∈ PSH(V), ϕ ≥ −1}. E I I Thm: for K ⊆ V compact, Cap(K ) = supp MA(uK∗ ) ⊆ K . Thm: for any E ⊆ V, TFAE: I I I I I R K MA(uK∗ ) and E is pluripolar, i.e. E ⊆ {u = −∞}, u psh; ∀ε ∃G ⊇ E open with Cap(G ) < ; uE∗ ≡ 0; E is “negligable”. Results exactly parallel to the archimedean situation! Pluripotential theory in a non-archimedean setting Key tools I I Many proofs are the same as in the archimedean situation, but a few basic ingredients are different. I Countability: V has no countable basis for topology but C[z1 , . . . , zn ] is noetherian. Used to prove Choquet’s Lemma. I Regularization: can approximate any ϕ ∈ PSH(V) by a decreasing sequence ϕj & ϕ, with ϕj ∈ PSH(V) of nice form. I This was built into the definition of PSH(V). . . but then properness of ϕ → sup ϕ was hard to establish! I In Cn , can regularize using convolution. On V, we use multiplier ideals. Pluripotential theory in a non-archimedean setting Key tools II I Comparison principle: Z Z MA(ψ) ≤ MA(ϕ) ϕ<ψ ϕ<ψ for ϕ, ψ ∈ PSH(V) not too singular. I As in Cn prove this by reduction to the case ϕ, ψ “smooth”. I In Cn , the smooth case uses Stokes’ theorem. On V, uses positivity of certain intersection nos. I “Balayage”: Don’t know how to solve Dirichlet problem locally. Replacement is the orthogonality property of asymptotic Zariski decompositions in the sense of Boucksom-Demailly-Păun-Peternell. Pluripotential theory in a non-archimedean setting Future work I Would like to do pluripotential theory on other Berkovich spaces. I Example: work on all of AnBerk , not just localized at one point. I Example: analytification of projective varieties over C((t)). Pluripotential theory in a non-archimedean setting

Pluripotential Theory

Related Documents

Description

Related Documents

Free up your schedule!

Take 5 seconds to unlock