How a Jacobi form counts curves on Abelian surfaces and threefolds
We explain how the solution to some natural curve counting problems on Abelian surfaces and threefolds is given by the Fourier coefficients of a certain Jacobi form. In the first half of the talk, we give an elementary description of the curve counting problems and we explain how their solution is elegantly given as the Fourier coefficients of a classical Jacobi form. We explain the key geometric construction that underlies the case of the surface. In the second part of the talk, we delve deeper into curve counting on an Abelian threefold via Donaldson-Thomas theory. We explain how a combination of motivic and toric methods leads to a computation of the Donaldson-Thomas invariants in terms of the topological vertex. The Jacobi form then emerges via a new and surprising trace identity which expresses the vertex in terms of Jacobi forms.
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