On the geometry of genus 1 Gromov-Witten invariants

On the geometry of genus 1 Gromov-Witten invariants

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Aleksey Zinger, Stony Brook University
Fine Hall 214

The mirror symmetry principle of string theory has led to astounding predictions for counts of holomorpic curves, especially for a quintic threefold (a degree 5 hypersurface in $P^4$). There has been much success in verifying these predictions in genus $0$, in part due to a good undertanding of the geometry of genus $0$ GW-invariants. In this talk, I will give an overview of geometric properties of genus $1$ GW-invariants, including a relation between GW-invariants of a hypersurface and of the ambient projective space. These properties mimic well-known genus $0$ properties. Taken together, they provide a method for computing genus $1$ GW-invariants of all complete intersections and have led to the verification of the 1993 BCOV mirror symmetry prediction for genus $1$ GW-invariants of a quintic threefold.