Tag - Algebraic geometry
This will be an attempt to summarize what one might expect about Homological Mirror Symmetry in the presence of an anticanonical divisor... and the (much smaller, but more reliable) subset of things I can prove about that situation.
Consider a positively monotone (Fano) closed symplectic manifold M and a symplectic simple crossings divisor D in it. Assume that the Poincare dual of the anti-canonical class is a positive
rational linear combination of the classes [Di], where Di are the components of D with their symplectic orientation. A choice of such coefficients, called the weights, (roughly speaking) equips M - D with a Liouville structure. I will start by discussing results relating the components of D with their symplectic orientation. A choice of such coefficients, called the weights, (roughly speaking) equips M - D with a Liouville structure. I will start by discussing results relating the symplectic cohomology of M - D with quantum cohomology of M. These results are particularly sharp when the weights are all at most 1 (hypothesis A). Then, I will discuss certain rigidity results (inside M) for
skeleton type subsets of M - D, which will also demonstrate the geometric meaning of hypothesis A in examples.
One formulation of mirror symmetry predicts (omitting a few adjectives) a 1-1 correspondence between equivalence classes of certain lattice polygons and deformation families of certain del Pezzo surfaces.
Lattice polygons corresponding to smooth Del Pezzo surfaces are called T-polygons, and these have been classified by Kasprzyk-Nill-Prince using combinatorial methods. I will sketch a new geometric proof of their classification result.
Quasimaps provide an alternate curve counting system to Gromov-Witten theory, which are related by wall-crossing formulae. Relative (or logarithmic) Gromov-Witten theory has proved useful for constructions in mirror symmetry, as well as for determining ordinary Gromov-Witten invariants via the degeneration formula. Different versions of this theory rely on various technologies, including expansions (or accordions) as well as logarithmic structures. I will discuss how to use a hybrid of these approaches to produce a proper moduli space parametrizing quasimaps relative a smooth divisor in any genus.
Many of the algebraic structures we construct from geometric data are represented 'motivically', that is, their structure constants are obtained by pushing and pulling 'coefficients' (e.g. functions, sheaves) along diagrams like X ← W → Y. In this quasi-survey talk, I will explain how many of the convenient properties of the algebraic categories we like to work in (e.g. vector spaces, dg-categories) are already present in categories of correspondences themselves. This explains their frequent appearance in the study of universal homology theories.
While Gorenstein codimension 3 varieties are well understood from Buchsbaum-Eisenbud's structure theorem, the picture is less clear for codimension 4. In this talk we describe some constructions of stable curves corresponding to the possible Betti tables for Artin Gorenstein algebras of regularity and codimension four, as outlined in a paper of Schenck, Stillman and Yuan. These constructions use techniques from liaison theory and the Tom and Jerry formats of Brown and Reid.
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