Geometry of Pseudoholomorphic Curves and Gromov-Witten Invariants

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AbstractAward: DMS-0604874Principal Investigator: Aleksey ZingerThe theory of Gromov-Witten invariants plays a prominent role insymplectic topology, enumerative algebraic geometry, and stringtheory. Via Gromov-Witten theory, string theorists have madecompletely unexpected predictions concerning counts of complex(holomorphic) curves in algebraic manifolds. Some of thesepredictions have been verified mathematically, but most havenot. Many connections between Gromov-Witten theory andenumerative geometry have been discovered independently of stringtheory as well. However, many others remain to be found. The mostfundamental object in the Gromov-Witten theory is the modulispace of (pseudo-) holomorphic maps. The PI has developed anapproach for studying its local structure using analytictechniques of symplectic topology and a separate topologicalapproach for recovering global information from the local data.Combined together, the two approaches have led to a variety ofresults, in enumerative geometry and in symplectic topology, forcounts of curves of low genus, primarily zero and one. Among themis a geometric relation between genus-one invariants of acomplete intersection and those of the ambient projectivespace. One objective of this project is to verify thelong-standing mirror symmetry conjecture for counts of genus-onecurves in the quintic threefold, using this geometricrelation. Another objective is to apply the results to countinggenus-one curves in projective varieties. However, the primaryobjective is to extend the detailed description of the modulispace of genus-one maps already obtained to higher-genus cases,especially genus two.The broader impact of this project is potentially farranging. Its aim is to advance the fundamental understanding ofthe Gromov-Witten theory, which in turn should lead to newapplications in symplectic topology and enumerativegeometry. Furthermore, it should open a way for testing a numberof mathematical predictions of string theory. If some of thesepredictions were shown to fail, string theory would require atleast some modification, perhaps with implications forunderstanding physical phenomena.