The PI will study various topics in Algebraic Geometry, including extension theorems in birational geometry, linear series, regularity conditions in derived categories, and minimal cohomology classes in abelian varieties. In particular, he proposes to further develop extension results for sections of line bundles originating in work of Siu, Takayama and Hacon-McKernan. He also proposes to relate a generic vanishing type filtration on the derived category of a smooth projective variety to the perverse T-structure constructed by Kashiwara, by means of homological and commutative algebra.He will approach the classification of subvarieties of minimal class in abelian varieties via Fourier-Mukai-theoretic methods.
Some of the proposed research problems, like the study of minimal classes on abelian varieties and the extension problem for sections of line bundles, are among the most distinguished in their respective directions. They also concern experts in adjacent fields, and will likely lead to interactions with people outside of the PI's area of expertise. They would be of considerable interest as proved statements, as documented for example by recent applications of extension theorems to the minimal model program. Others, like the results on derived categories, form a newly emerging theory still to be fully understood, but which has already led to numerous applications to both modern and classical questions.