# Algebraic Geometry Seminar

#### 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).