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.
This project investigated connections between two seemingly different geometric correspondences. The first, known as the Kuga-Satake correspondence, relates a very special family of surfaces called K3 surfaces, to another well-studied family of geometric objects called abelian varieties. Although the objects being matched by this correspondence are geometric, the correspondence is almost entirely an algebraic one. One of the grand problems of the last eighty years in mathematics, the Hodge Conjecture, is to show that these seemingly purely algebraic correspondences have a geometric interpretation. To date, this conjecture has only been verified in a few special cases, and many mathematicians feel that a solution will require genuinely new approaches. The second correspondence, Mirror Symmetry, is a relationship that was first discovered by string theorists and is still in the process of being made mathematically precise. Mirror symmetry relates complicated algebraic geometry of one space to a much simpler construction involving the Kaehler geometry of its "mirror" space. Mirror symmetry and the Kuga-Satake correspondence have very different flavors, but work by the PI and collaborators suggested that in the case of K3 surfaces, there is a relationship between the two. This link is particularly intriguing because it offers a first hint that mirror symmetry could offer a fresh perspective on the Hodge Conjecture. This project developed that link by focusing on specific examples where the delicate and often abstract geometric structures involved could be made explicit. In one example, the PI with collaborators D. Karp, D. Moore, D. Skjorshammer, and U. Whitcher investigated a highly symmetric family of K3 surfaces and related the Kuga-Satake construction for this family to a mirror-symmetric phenomenon known as "mirror moonshine." In another example, the PI collaborated with N. Ilten and V. Przyjalkowski to explain an initially surprising property of a class of three dimensional objects (Fano threefolds of Picard number one). This work showed that that property was a mirror reflection of the Kuga-Satake correspondence for the families of K3 surfaces mirror to the Fano threefolds. This relationship will be explored further in forthcoming work by the PI with C. Doran, L. Katsarkov, and V. Przyjalkowski. A more general look at the nested families of K3 surfaces for these relationships may hold is a work in progress of the PI with C. Doran and A. Clingher.
