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Niveau: Supérieur, Licence, Bac+2

Holomorphic Morse inequalities and entire curves on projective varieties Jean-Pierre Demailly Universite de Grenoble I, Institut Fourier March 2012 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 L2 estimates; one can mention here the foundational work achieved by Bochner, Kodaira, Nakano, Morrey, Kohn, Andreotti-Vesentini, Grauert, Hormander, Bombieri, Skoda and Ohsawa- Takegoshi 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-Ampere 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.

Holomorphic Morse inequalities and entire curves on projective varieties Jean-Pierre Demailly Universite de Grenoble I, Institut Fourier March 2012 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 L2 estimates; one can mention here the foundational work achieved by Bochner, Kodaira, Nakano, Morrey, Kohn, Andreotti-Vesentini, Grauert, Hormander, Bombieri, Skoda and Ohsawa- Takegoshi 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-Ampere 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.

- plurisubharmonic functions via bergman kernels
- rham cohomology
- vector bundles
- compact complex
- morse inequalities
- atiyah-bott-patodi proof
- laplace-beltrami operators

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Publié par | profil-vieg-2012 |

Nombre de lectures | 19 |

Langue | English |

Poids de l'ouvrage | 1 Mo |

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