Applications of Pluripotential Theory to Algebraic Geometry

Abstract

Plurisubharmonic functions and positive currents are an essential tool of modern complex analysis. Since their inception by Oka and Lelong in the mid 1940's, major applications to algebraic and analytic geometry have been developed in many directions. One of them is the Bochner-Kodaira technique, providing very strong existence theorems for sections of holomorphic vector bundles with positive curvature via L 2 estimates; one can mention here the foundational work achieved by Bochner, Kodaira, Nakano, Morrey, Kohn, Andreotti-Vesentini, Grauert, Hörmander, Bombieri, Skoda and OhsawaTakegoshi in the course of more than 4 decades. Another development is the theory of holomorphic Morse inequalities (1985), which relate certain curvature integrals with the asymptotic cohomology of large tensor powers of line or vector bundles, and bring a useful complement to the Riemann-Roch formula.We describe here the main techniques involved in the proof of holomorphic Morse inequalities (chapter I) and their link with Monge-Ampère operators and intersection theory (chapter II). The last two chapters III, IV provide applications to the study of asymptotic cohomology functionals and the Green-Griffiths-Lang conjecture. The latter conjecture asserts that every entire curve drawn on a projective variety of general type should satisfy a global algebraic equation; via a probabilistic curvature calculation, holomorphic Morse inequalities imply that entire curves must at least satisfy a global algebraic differential equation.

On semipositivity theorems

“…for every x ∈ X by [D1,Proposition 3.7 (ii)] (see also [D2, (13.12) Theorem (b)] and [D3,Theorem 56 (b)]). Therefore, the Lelong number of ψ ε is zero everywhere.…”

“…. , g N } defined near x 0 and a positive real number c (see [D3,Definition 52]). Since ν(ψ ε , x) = 0 for every x ∈ X, we obtain that ψ ε = −∞ everywhere.…”

Bergman kernel on Riemann surfaces and Kähler metric on symmetric products

“…Estimates of Bergman kernels associated to high tensor-powers of holomorphic line bundles defined on compact complex manifolds has been studied since a long time. Tian [Ti], Zelditch [Ze], [SZ], Demailly [De1] [De2], and more recently, Ma and Marinescu in [MM], have done seminal works in this field. Recently, in [AMM], Auvray, Ma, and Marinescu have extended their estimates of Bergman kernels to noncompact hyperbolic Riemann orbi-surfaces.…”

Stability of solutions to complex Monge–Ampère flows

“…C'est un plaisir de contribuer à ce volume en l'honneur de Jean-Pierre Demailly, dont nous apprécions l'exigence et la générosité. Ses notes de cours [Dem89,Dem13] ont fortement contribué au développement de la théorie du pluripotentiel en France, nous lui en sommes très reconnaissants.…”