Enumerative geometry is concerned with counting geometric objects on spaces defined by polynomial equations. The subject, which has roots going back to the ancient Greeks, was revolutionized by string theory in the 90s and has since become a fundamental link between algebraic geometry, representation theory, number theory and physics. The project K3Mod aims to establish and make use of a wide range of new correspondences in enumerative geometry. These link together different enumerative theories and open new perspectives to attack long-standing problems concerning the quantum cohomology of the Hilbert scheme of points on surfaces, modular properties of enumerative invariants of K3 geometries, and string partition functions of Calabi-Yau threefolds.
The geometry of the Hilbert scheme of points on a surface will play a central role. A new correspondence between its Gromov-Witten theory and the Donaldson-Thomas theory of certain threefold families will allow for new insight into its quantum cohomology which will give new connection to representation theory and modular forms. Important examples that will be considered are the K3 surfaces and the elliptic surfaces.
The project will consider the enumerative invariants of K3 geometries also in itself. After elliptic curves, which were treated over 20 years ago with much success and rich results, K3 geometries form the simplest compact Calabi-Yau geometries for which a complete understanding of the Gromov-Witten theory is in reach. This will be considered for K3 surfaces as well as related geometries such as the classical Enriques surfaces, which are quotients of K3 surfaces by fixed-point free involutions, or K3-fibrations.
The string-partition functions of smooth projective threefolds will be studied using correspondences between enumerative theories. For elliptic threefolds, the project will relate Donaldson-Thomas invariants with Jacobi forms, using both degeneration techniques and wallcrossing formulae. This will yield new insight and methods in proving modularity of string partition functions. Further important examples that will be studied are the Chaudhuri–Hockney–Lykken Calabi-Yau threefolds and the Enriques Calabi-Yau threefold.
The proposal will lead to new connections between geometry, modular forms and representation theory. The results provide a new understanding of the interplay between Hilbert schemes, K3 surfaces, and modularity in enumerative geometry.
