Geometry of Pseudoholomorphic Curves and Gromov-Witten Invariants

Abstract Award: DMS-0604874 Principal Investigator: Aleksey Zinger The theory of Gromov-Witten invariants plays a prominent role in symplectic topology, enumerative algebraic geometry, and string theory. Via Gromov-Witten theory, string theorists have made completely unexpected predictions concerning counts of complex (holomorphic) curves in algebraic manifolds. Some of these predictions have been verified mathematically, but most have not. Many connections between Gromov-Witten theory and enumerative geometry have been discovered independently of string theory as well. However, many others remain to be found. The most fundamental object in the Gromov-Witten theory is the moduli space of (pseudo-) holomorphic maps. The PI has developed an approach for studying its local structure using analytic techniques of symplectic topology and a separate topological approach for recovering global information from the local data. Combined together, the two approaches have led to a variety of results, in enumerative geometry and in symplectic topology, for counts of curves of low genus, primarily zero and one. Among them is a geometric relation between genus-one invariants of a complete intersection and those of the ambient projective space. One objective of this project is to verify the long-standing mirror symmetry conjecture for counts of genus-one curves in the quintic threefold, using this geometric relation. Another objective is to apply the results to counting genus-one curves in projective varieties. However, the primary objective is to extend the detailed description of the moduli space of genus-one maps already obtained to higher-genus cases, especially genus two. The broader impact of this project is potentially far ranging. Its aim is to advance the fundamental understanding of the Gromov-Witten theory, which in turn should lead to new applications in symplectic topology and enumerative geometry. Furthermore, it should open a way for testing a number of mathematical predictions of string theory. If some of these predictions were shown to fail, string theory would require at least some modification, perhaps with implications for understanding physical phenomena.