Uniform Hörmander estimates for flat nontrivial line bundles

Abstract: Hörmander’s $L^2$-estimates for the $\bar{\partial}$ operators on holomorphic line bundles are of fundamental importance in complex analytic geometry, whose conventional proof crucially relies on the positivity of the line bundle. In this talk, we prove the $L^2$-estimates for the solutions to the $\bar{\partial}$ equation that hold uniformly for all flat nontrivial line bundles on compact Kähler manifolds, whose main feature is the quantitative description of the blow-up behaviour as the line bundle approaches the trivial one. A key ingredient in the proof is the observation that line bundles with vanishing first Chern classes are topologically trivial and can be identified with the trivial bundle with the "perturbed" $\bar{\partial}$ operator which we define in terms of coordinates on the Picard variety. This is a joint work with Takayuki Koike.

algebraic geometryanalysis of PDEsalgebraic topologycomplex variablesdifferential geometrygeneral topologygeometric topologyK-theory and homologymetric geometrysymplectic geometry

Audience: researchers in the topic

CRM - Séminaire du CIRGET / Géométrie et Topologie

Series comments: Hybrid seminar of geometry and topology. Laboratory : CIRGET - www.cirget.uqam.ca The homepage of the seminar is www.cirget.uqam.ca/fr/seminaires.html

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