# 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).