Abstract:

 Landau-Ginzburg (LG) models consist of the data of a quotient stack X and a regular complex-valued function W on X. Here, geometry is encapsulated in the singularity theory of W. One can find that LG models are deformations of many Calabi-Yau varieties in some sense. For example, if the Calabi-Yau is a hypersurface in a smooth projective variety Z cut out by a polynomial f, then one can take X to be the canonical bundle of Z with function W=uf, where u is the bundle coordinate—when the hypersurface is smooth, the critical locus of uf will indeed just be the hypersurface. Exoflops were introduced by Aspinwall as a way to effectively find new birational models of the quotient stack to get new geometries. They effectively create new GIT problems of partial compactifications of X, expanding the tractable birational geometries related to Z. We will explain this technique, provide some foundational results about this, and then provide some new applications proven recently for Calabi-Yau varieties with nontrivial scaling symmetry groups. This talk contains results from a series of joint works (some in progress) with D. Favero (UMinn), C. Doran (Bard/Alberta), A. Malter (Birmingham).