Algebraic Geometry Seminar

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Stability of the tangent bundle, Gromov-Witten invariants, and weak approximation in positive characteristic.

Speaker: Jason Starr, Stony Brook

Location: Warren Weaver Hall 805

Date: Tuesday, October 15, 2019, 3:30 p.m.

Synopsis:

Over the function field of a curve over an algebraically 
closed field, for a complete intersection of type ( d_1 , ... , d_r ) in 
projective n-space, Tsen proved existence of at least one rational point 
assuming positivity of the integer i := n + 1 - ( d_1 + ... + d_r ), 
which equals the "Fano index". There are singular examples with only one 
rational point.  In the smooth case, are there at least two points?

A smooth projective variety is a "Fano manifold with Picard rank 1 and 
Fano index 1" if the first Chern classes of ample line bundles are 
precisely the positive integer multiples of the first Chern class of the 
tangent bundle.  Following earlier work of Bogomolov, Miles Reid proved 
that such varieties have stable tangent bundle in characteristic 0.  
Using results around "decomposition of the diagonal", we prove the same 
provided the characteristic p is prime to certain Gromov-Witten 
invariants. Combined with work of Zhiyu Tian and work of Tian and 
Runhong Zong, such a Fano manifold defined over the function field of a 
curve satisfies "weak approximation" (in the sense of the 
Hassett-Tschinkel Conjecture) at all places of potentially good 
reduction.  In particular, for complete intersections, these 
Gromov-Witten invariants are explicit: weak approximation holds, and 
thus there are infinitely many rational points, provided i > 1 and the 
characteristic is larger than max ( d_1 , ... , d_r ).

This is joint work with Zhiyu Tian (Beijing International Centre for 
Mathematical Research, Peking U.) and Runhong Zong (Nanjing University 
of China).