Research project Generic vanishing and characterization of semi-abelian varieties

Algebraic geometry is the branch of mathematics which studies geometric objects arising from algebraic ones, such as solutions of polynomial equations like a circle: x^2+y^2-r^2=0.

These are called algebraic varieties. The proposal revolves around a central theme within algebraic geometry: the classification problem.

Given an algebraic variety, it is possible to compute some quantities, called invariants, that altogether give us a sort of piece of the DNA sequence of the variety itself. The aim of the classification problem is to recover meaningful geometric information from the invariants, and to discover the hidden relationships between varieties which share big chunks of their DNA sequence.

For example, a curve of degree d consists of solutions of an equation of degree d (e.g., a circle is a curve of degree 2). If we consider complex solutions, and not just real ones, the curve will look like a series of donuts glued together, as in the picture above.

We can recover g, the number of donuts, by using the formula. Thus, from the degree of the equation defining the curve, we get insights into how it looks like, that is, on its geometry. In this project we aim to establish an innovative approach to the classification problem for smooth quasi-projective varieties. The main goal is to refine techniques from our recent papers and provide new tools that have the potential to revolutionize ongoing research in birational geometry, which is one of the branches of algebraic geometry.