K3 Surfaces, Derived Categories, and Cubic Fourfolds

University Of Oregon Eugene, Eugene OR

Investigators

Abstract

Algebraic geometry occupies a central place in modern mathematics, interacting with number theory, representation theory, homotopy theory, complex analysis, mathematical physics, and other fields. The PI studies concrete problems in algebraic geometry using the derived category, which first appeared in the 1960s as a bookkeeping device for homological algebra but which in recent decades has emerged as a fundamental geometric invariant in its own right, and as a conduit of ideas from string theory to algebraic geometry. This project contains a strong component aimed to the training of students at different levels in algebraic geometry. The research program consists of three related projects. The first aims to unify a number of examples of "K3 categories," sometimes billed as "non-commutated K3 surfaces," that arise in connection with birational and hyperkaehler geometry, in a common construction. The second aims to show that a derived Torelli theorem does not hold for Calabi-Yau threefolds, using an invariant called BCOV torsion to distinguish between two spaces with the same Hodge structure. The third deals with several questions of a more classical flavor about four-dimensional cubic hypersurfaces. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.