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Hyperbolicity and Holomorphic Jet Differentials in Complex Manifolds

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Overview

Explore complex mathematical concepts in this lecture from Harvard University's Yum-Tong Siu focusing on hyperbolicity and holomorphic jet differentials. Delve into the hyperbolicity problem in function theory, which examines conditions for compact complex manifolds to admit no nonconstant holomorphic map from C, and its parallel in number theory regarding finite rational points. Learn about the construction of jet differentials from position-forgetting maps in abelian varieties and discover how hyperbolicity is achieved for generic hypersurfaces of sufficient degree in complex projective space. Examine the reduction of lower bound degrees for generic n-dimensional hyperbolic hypersurfaces from (n log n)n to polynomial order, with potential reduction to quadratic order, through combined techniques from abelian variety and complex projective space settings. Consider the analogies between hyperbolicity in function theory and number theory, particularly how differentiation in jet differentials translates to difference maps in number theory for abelian varieties.

Syllabus

Yum-Tong Siu, Harvard University: Hyperbolicity and holomorphic jet differentials

Taught by

IMSA

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