Title: Enumerative Geometry of Determinantal Octic Double Solids
Abstract: The double cover of P^3 branched along a smooth degree 8 surface is a Calabi-Yau threefold originally studied by Herb Clemens in the early 1980s. Its genus zero Gromov-Witten invariants were computed in the early 1990s shortly following the famous calculation of the GW invariants of the quintic threefold by Candelas et al, and these predictions were proven in the mid-1990s by Givental and Lian-Liu-Yau, with the Gromov-Witten invariants themselves having been rigorously defined in the intervening years. The higher genus Gromov-Witten invariants have been computed for the octic double solid for g \le 60 using B-model techniques. These predictions are not yet proven. The integer-valued Gopakumar-Vafa invariants can be inferred from the GW invariants. Intrinsically, GV invariants are invariants of moduli spaces of 1-dimensional sheaves.
A determinantal octic double solid is a double cover of P^3 branched along the degree 8 determinant of a symmetric matrix of homogeneous forms on P^3. B-model techniques can also be used to compute enumerative invariants of determinantal octic double solids up to g \le 32. These threefolds have isolated nodes and do not have a projective small resolution, so the standard algebro-geometric definitions of Gromov-Witten or Gopakumar-Vafa invariants do not apply. Nevertheless, when computational methods for GV invariants are applied to this situation, the results agree with the B-model predictions whenever they can be carried out. There are two developing proposals for rigorously defining these invariants. One is based on moduli spaces of 1-dimensional twisted sheaves on (non-algebraic) small resolutions, with the novel feature that all small resolutions must be considered simultaneously to properly define the invariants. The other proposal is based on moduli spaces of sheaves of modules over a certain locally free sheaf of noncommutative algebras.
This talk is based on joint work with Albrecht Klemm, Thorsten Schimannek, and Eric Sharpe.
