Niveau: Supérieur, Doctorat, Bac+8
Pseudoconvex-concave duality and regularization of currents Jean-Pierre Demailly Universite de Grenoble I, Institut Fourier to the memory of Michael Schneider Abstract The goal of this work is to investigate some basic properties of Finsler metrics on holomorphic vector bundles, in the perspective of obtaining geometric versions of the Serre duality theorem. We establish a duality framework under which pseudoconvexity and pseudoconcavity properties get exchanged – up to some technical restrictions. These duality properties are shown to be related to several geometric problems, such as the conjecture of Hartshorne and Schneider, asserting that the complement of a q-codimensional algebraic subvariety with ample normal bundle is q-convex. In full generality, a functorial construction of Finsler metrics on symmetric powers of a Finslerian vector bundle is obtained. The construction preserves positivity of curvature, as expected from the fact that tensor products of ample vector bundles are ample. From this, a new shorter and more geometric proof of a basic regularization theorem for closed (1, 1) currents is derived. The technique is based on the construction of a mollifier operator for plurisubharmonic functions, depending on the choice of a Finsler metric on the cotangent bundle and its symmetric powers. Contents 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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- n?y has
- serre duality
- finsler metric
- strictly plurisubharmonic
- large part through
- closed positive
- normal bundle