Overview

Fano manifolds are complex projective manifolds having positive first Chern class, and play a key role in higher dimensional algebraic geometry. The positivity condition on the first Chern class has far reaching geometric and arithmetic implications. For instance, Fano manifolds are covered by rational curves, and families of Fano manifolds over one dimensional bases always admit sections.

In recent years, there has been great effort into defining suitable higher analogs of the Fano condition. Higher Fano manifolds are expected to enjoy higher versions of several of the nice properties of Fano manifolds. For instance, they should be covered by higher dimensional rational varieties, and families of higher Fano manifolds over higher dimensional bases should admit sections (modulo Brower obstruction).

An interesting notion of 2-Fano manifolds was introduced by De Jong and Starr 15 years ago. These are complex projective manifolds with positive first and second Chern characters. Similarly, one can define higher Fano manifolds in terms of positivity of higher Chern characters. The geometry of higher Fano manifolds has been fairly investigated, and in several special cases they are shown to enjoy the expected properties. In particular, there is a classification of 2-Fano manifolds of high index due to Araujo and Castravet.

Questions

During this 3-day workshop, we expect to make progress on the following problems concerning higher Fano manifolds.

• Classification of 3-Fano manifolds of high index.

• Classification of toric 2-Fano manifolds.

• Describing the geometry of rational curves on higher Fano manifolds.

Objectives

In addition to making progress on the theory of higher Fano manifolds, we expect that this workshop will strengthen the network of women mathematicians from around the globe working in algebraic geometry.