Toric vector bundles: Stability, Cohomology, and Applications.

The topic of this grant is in algebraic geometry, the study of geometric objects defined as the vanishing locus of finitely many polynomial equations, called algebraic varieties. One basic question is the classification of algebraic varieties. Vector bundles are geometric objects associated to algebraic varieties that can be put together to form a new variety, called a moduli space. These moduli spaces are an important tool to construct new varieties from old ones, and to reveal geometric properties of the underlying variety. They have a geometric input data, the Chern class, and it is known only for a few types of varieties for what input data these moduli spaces exist.

I propose to study vector bundles on a class of varieties called toric varieties. While these varieties are very special, they exhibit additional combinatorial structure, that allows their study with a completely new set of tools. They have been a success story serving as examples for conjectures and to develop new theories. Toric varieties carry a special class of vector bundles called toric vector bundles that can be used to study properties of general vector bundles on toric varieties. These toric vector bundles have descriptions in terms of combinatorics and linear algebra, and this proposal will build further bridges between these fields by relating questions originating in algebraic geometry to questions in combinatorics and linear algebra. This will open the door to new cross-fertilization between these fields, by giving access to a much larger toolset and by introducing new research questions to both fields.

The goal of this proposal is to systematically develop the theory of toric vector bundles in order to study questions that are of relevance to algebraic geometry and neighboring fields. One of the main objectives of the proposal is to identify the input data for the existence of moduli spaces on toric varieties. Another main objective is to reveal the fundamental relationship between geometry and algebra intrinsic in the definition of algebraic varieties in the case of toric varieties, by studying, for a given embedding of a toric variety, the numbers of minimal defining equations of a given degree, and the number of minimal higher algebraic relations (syzygies) between these defining equations of a given degree in terms of the geometry of the embedding.

The main impact will be academic. The beneficiaries will be

(1) researchers in algebraic geometry, commutative algebra, combinatorics, and applied algebraic geometry. Their benefits will include

a) a new approach to old problems;
b) a set of new tools to study toric vector bundles;
c) new results on the defining equations of toric varieties and toric vector bundles;
d) an implementation in Macaulay 2 of the results mentioned in c);
e) interactions with applied algebraic geometry, by looking for applications, publishing them when appropriate, and attending conferences;
f) identifying new research questions and conjectures that address the cross-fertilisation between algebraic geometry/commutative algebra and combinatoric/linear algebra;
g) continued interaction with applied algebraic geometry, by looking for applications of my work, publishing them when appropriate, and attending conferences;
h) a workshop bringing together researchers from these areas.

2) the PDRA. They will

a) get exposed to a new area of research;
b) learn valuable skills for tackling problems in combinatorial algebraic geometry;
c) be made aware of opportunities to look for research questions originating in applications'

3) School children, who will benefit from an outreach activity on lattice polygons.

4) Undergraduate students, who will get a chance to experience research by studying questions coming up in my research that have been translated into a language that is accessible to them.

5) Graduate students, who will benefit from a summer school I plan to organise towards the end of the fellowship.