Intrinsic mirror symmetry and punctured Gromov-Witten invariants
arXiv:1609.00624
Abstract
This contribution to the 2015 AMS Summer Institute in Algebraic Geometry (Salt Lake City) announces a general mirror construction. This construction applies to log Calabi-Yau pairs (X,D) with maximal boundary D or to maximally unipotent degenerations of Calabi-Yau manifolds. The new ingredient is a notion of "punctured Gromov-Witten invariant", currently in progress with Abramovich and Chen. The mirror to a pair (X,D) is constructed as the spectrum of a ring defined using the punctured invariants of (X,D). An analogous construction leads to mirrors of Calabi-Yau manifolds. This can be viewed as a generalization of constructions developed jointly with Hacking and Keel in the case of log CY surfaces and K3 surfaces.
37 pages, 1 figure. Current version corrects statement in Theorem 2.2: the algebra structure is not expected to be associative in complete generality, and some hypotheses are necessary. This does not affect the application to mirror symmetry constructions