International Research Fellowship Program: K3 Surfaces, Normal Forms and the Kuga-Satake Hodge Conjecture

Lewis Jacob M, Shoreline WA

Investigators

Abstract

0965183 Lewis The International Research Fellowship Program enables U.S. scientists and engineers to conduct nine to twenty-four months of research abroad. The program's awards provide opportunities for joint research, and the use of unique or complementary facilities, expertise and experimental conditions abroad. This award will support a twenty-four-month research fellowship by Dr. Jacob M. Lewis to work with Dr. Ludmil Katzarkov at the University of Vienna in Austria. The Kuga-Satake Hodge conjecture posits an algebraic correspondence between two very different geometric objects?a K3 surface and an abelian variety. This conjecture?a special case of the Hodge conjecture, one of the Clay Mathematical Institute?s famous Millenium Problems?is widely believed to be true, but remarkably few examples are known. This project uses Mirror Symmetry for K3 surfaces to shed new light on the classification of Fano varieties and on the Kuga-Satake Hodge conjecture. Mirror symmetry, a surprising mathematical duality first predicted by string theorists, often relates complicated algebraic geometry of one space to a much simpler construction involving the Kähler geometry of the ?mirror? space. Recent work in progress by the PI with collaborators suggests that most of the known cases of the Kuga-Satake Hodge conjecture admit a re-interpretation in terms of mirror symmetry. As part of this project, in addition to reinterpreting the known cases of the conjecture, new examples are being developed, with the ultimate goal of providing a general framework for the problem. This work requires deep knowledge of homological mirror symmetry, in which Dr. Katzarkov and members of his research group are experts. It also requires facility with toric varieties, mirror maps, and hypergeometric functions, which are objects of study in the PI's research. The project also deals with other questions related to mirror symmetry for K3 surfaces, including the use of certain families of K3 surfaces in the classification of Fano threefolds with Picard number one. The Hodge conjecture is one of the great open problems in algebraic geometry, and indeed in all of mathematics. While the Kuga-Satake Hodge conjecture is a strict sub-problem of the full Hodge conjecture, it is nevertheless of great interest. Similarly, mirror symmetry is developing importance in algebraic geometry; it needs further study but is already leading to fascinating discoveries. Finding a role for mirror symmetry within the Hodge conjecture and classification problems provides a fresh perspective to these imposing areas of research.