Dates: Inaugural Date: October 30th, 2024, (to be continued on a monthly basis)
Location: University of Miami, Ungar Building Room 528B
Organizers: Damian Brotbek (Université de Lorraine), Bruno De Oliveira (University of Miami), Natalia Garcia-Fritz (Pontificia Universidad Católica de Chile), Steven Lu (Université du Québec à Montréal), Jorge Vitório Pereira (IMPA)
General description: Every month we will have a lecture concerning the geometry of projective manifolds X whose cotangent bundles have positive properties. The complex geometric hyperbolicity properties of X are of special interest to this series, but other geometric features are also to be addressed e.g. topological properties.
Some Guiding Conjectures/Problems:
Conjecture (Orbifold version of Green-Griffiths-Lang Conj.): If (X,D) is an orbifold pair of general type, then there is a proper subvariety Z of X containing all the orbifold entire curves of (X,D).
Problem: What can be said about the existence of nonzero orbifold jet differentials vanishing along an ample divisor on a smooth orbifold pair (X,D) of general type?
Conjecture (Horing-Peternell): The cotangent bundle of a projective manifold X is pseudoeffective if and only if X has symmetric differentials.
Conjecture (Esnault): If a projective manifold has infinite fundamental group, then X has symmetric differentials.
Problem: What are the topological implications of the presence of symmetric differentials on a projective manifold?
Inaugural Event
Wednesday, October 30th, 2024
| 10:00am |
Yum-Tong Siu, Harvard University: Hyperbolicity and holomorphic jet differentials
For function theory the hyperbolicity problem seeks conditions for a compact complex manifold to admit no nonconstant holomorphic map from C. The corresponding problem in number theory is for the set of rational points to be finite or contained in some proper subvariety.
Nonexistence of nonconstant entire curves comes from sufficiently independent holomorphic jet differentials vanishing on some ample divisor. For a complex submanifold of an abelian variety such jet differentials are constructed from the position-forgetting map of the jet space of the submanifold to conclude hyperbolicity unless the submanifold is invariant under some linear translation. For complex hypersurfaces of the complex projective space, hyperbolicity for a generic hypersurface of sufficiently high degree follows from the method of vertical jet differentials on the universal complex hypersurface.
In this talk we will discuss the reduction of the lower bound for the degree of a generic n-dimensional hypersurface to be hyperbolic from the best known bound of the order of (n log n)n to polynomial order in n, possibly even to quadratic in n, by combining techniques from the abelian variety setting and the complex projective space setting. The analogy between hyperbolicity in function theory and in number theory will be considered. For abelian varieties differentiation in jet differentials in function theory is replaced by the difference map in number theory, but similar implementation for the hypersurface setting is not yet clear.
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| 11:15am |
Izzet Coskun, University of Illinois at Chicago: Algebraic hyperbolicity of hypersurfaces and Lang-type conjectures
In this talk, I will survey joint work with Eric Riedl on algebraic hyperbolicity of very general hypersurfaces and the description of Lang-type loci in hypersurfaces. .
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