Schedule for: 16w5062 - Homological Mirror Geometry

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

One consequence of the homological mirror symmetry conjecture predicts that many varieties will have ``hidden symmetries" in the form of autoequivalences of their derived categories of coherent sheaves which do not correspond to any automorphism of the underlying variety. In fact the fundamental groupoid of a certain "complexified Kaehler moduli space" conjecturally acts on the derived category. When the space in question is the cotangent bundle of a flag variety, actions of this kind have been studied intensely in the context of geometric representation theory and Kahzdan-Lusztig theory. We establish the conjectured group action on the derived category of any variety which arises as a symplectic or hyperkaehler reduction of a linear representation of a compact Lie group. Our methods are quite explicit and essentially combinatorial -- leading to explicit generators for the derived category and an explicit description of the complexified Kaehler moduli space. The method generalizes the work of Donovan, Segal, Hori, Herbst, and Page which studies grade restriction rules in specific examples associated to ``magic windows." Based on joint work with Steven Sam.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

For a Fano hypersurface in P^n, the derived category decomposes into an exceptional collection and a category of matrix factorizations. For a complete intersection of k hypersurfaces of degree d, it decomposes into an exceptional collection and a sort of bundle of categories of matrix factorizations over P^{k-1}. What about a complete intersection of hypersurfaces of unequal degrees d_1...d_k? Do we get a similar bundle over weighted P^{k-1}, with weights d_1...d_k? Not really: it is better to view it as a categorical resolution of the category of matrix factorizations of some higher-dimensional, singular hypersurface. The prototypical example is Kuznetsov's degree-6 K3 surface resolving the category of matrix factorizations of a nodal cubic 4-fold. We will discuss several other examples and state some general results. This is joint work with Paul Aspinwall.

When a 3-fold contains a floppable curve, there is an associated equivalence between the derived categories of the 3-fold and its flop. If the curve is reducible, there may exist multiple such flop functors, one for each irreducible component. I will explain joint work with Michael Wemyss, showing how this leads to new actions of braid-type groups on the derived category, and give an update on related results.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We present a new construction of “mirror pairs” of Calabi-Yau manifolds. On one side of the mirror correspondence are Calabi-Yau manifolds fibered in codimension one by Calabi-Yau submanifolds, for example elliptic fibered K3 surfaces or K3 surface fibered Calabi-Yau threefolds. On the other side are so-called “Tyurin degenerations”, i.e., smoothings of pairs of quasi-Fano varieties whose common intersection Calabi-Yaus are mirror to the fibers; these correspond to Type II Kulikov degenerations in the K3 surface case and Kawamata-Namikawa smoothings in the case of Calabi-Yau threefolds. Evidence that the construction produces mirror pairs comes from several directions: The fibered Calabi-Yaus are constructed by “gluing” the pair of Landau-Ginzburg models mirror to the pair of quasi-Fano varieties, and we establish mirror symmetry of Euler and Hodge numbers. Our construction is compatible with the Batyrev-Borisov mirror construction, wherein a bipartite nef partition produces the structures on both sides and the singular fibers of the fibration encode properties of the Landau-Ginzburg models mirror to the two quasi-Fano varieties. In the case of elliptic fibered K3 surfaces, the KSBA compactification of moduli of pairs suggests a broad correspondence between Type II degenerations of a lattice-polarized K3 surface and elliptic fibrations on its Dolgachev-Nikulin mirror. A complete classification of Calabi-Yau threefolds fibered by mirror quartic K3 surfaces leads to explicit constructions of candidate mirror threefolds and their Tyurin degenerations, showing that our construction is not limited to threefolds constructed as toric complete intersections. Finally, we show that in the context of homological mirror symmetry, “non-commutative” K3 fibrations should be mirror to Tyurin degenerations along loci in moduli disjoint from points of maximal unipotent monodromy.

"Descendent Gromov-Witten invariants play a central role in canonical deformations of Landau-Ginzburg models as well as the multiplication rules of generalized theta functions, both relevant for (homological) mirror symmetry. In a joint work with Travis Mandel, I prove that tropical Gromov-Witten invariants with psi class conditions coincide with descendent log Gromov Witten invariants for smooth toric varieties whenever non-superabundance is given. We use toric degenerations a la facon de Siebert-Nishinou and we expect that our approach will be generalizable to Mumford or Gross-Siebert type degenerations."

In this talk we will introduce the notion of Donaldson Uhlenbeck Yau correspondence. A connection with sheaves of categories will be discussed.

In a series of collaborations with Hiromichi Takagi, I have been studying certain complete intersection Calabi-Yau spaces, which are nicely related to determinantal varieties in projective spaces. After summarizing relations to the linear duality (due to Kuznetov), I will focus on the mirror symmetry of these Calabi-Yau spaces. In particular, I will describe conifold transitions explicitly for the case of mirror family obtained in our CMP paper (2014, vol.329, 1171--1218).

We analyze singularities in the parameter space of the gauged linear sigma model and show how they coincide with the GKZ A-determinant in the noncompact case. We show that this requires logarithmic coordinates to work correctly. The same analysis gives a natural picture for generic monodromy in the derived category around components of the discriminant in terms of specific spherical functors.

In this talk we will describe progress towards a generalization of mirror symmetry pertinent for heterotic strings. Whereas ordinary mirror symmetry relates, in its simplest incarnations, pairs of Calabi-Yau manifolds, the heterotic generalization relates pairs of holomorphic vector bundles over (typically distinct) Calabi-Yau's, satisfying certain consistency conditions. We will also outline the corresponding analogue of quantum cohomology, known as quantum sheaf cohomology, describing results for deformations of tangent bundles of toric varieties and Grassmannians, and we will discuss (0,2) Landau-Ginzburg Toda-like mirrors to deformations of tangent bundles of products of projective spaces.

I will give an update on applications of stability conditions for surfaces within algebraic geometry.

It has been known since Bondal and Orlov's work on semi-orthogonal decompositions that for blow-ups, projective bundles and certain flips f : X ---> Y, one may decompose the derived category of D(X) = < D(Y), C >. In this talk I will describe the mirror LG model to C when f is a birational cobordism with trivial center. Diemer-Katzarkov-K. conjectured that this was a Fukaya-Seidel category FS (W) of a potential W from a higher dimensional pair of pants to the punctured plane. I will explain a recent proof of this conjecture. The classical version of HMS for weighted projective spaces of arbitrary dimension then will be observed as a corollary.

This talk is based on a joint work with T. Logvinenko, and gives some background for his talk on "P-functors". One of the major problems of working with triangulated categories is that the cone of a map between functors is not well defined, and in constructions such as that of P-twists, we need not just a cone, but a convolution of a three-term complex. In this talk, we will discuss Bondal and Kapranov's pretriangulated categories, where such convolutions exist naturally. We are also going to introduce the construction of a twisted line bundle over a DG category, a version of which is going to be instrumental in the definition of P-functors.

\(P^n\) objects are a class of objects in derived categories of algebraic varieties first studied by Huybrechts and Thomas. They were shown to give rise to derived autoequivalences in a similar fashion to Seidel-Thomas spherical objects. It was also shown that they could sometimes be produced out of spherical objects by taking a hyperplane section of the ambient variety. In this talk, based on work in progress with Rina Anno, we’ll first recall the basics on spherical and \(P^n\) objects, and then explain how to generalise the latter to the notion of P-functors between (enhanced) triangulated categories. We'll also discuss a closely related notion of a non-commutative line bundle over such category, inspired by a construction of Ed Segal

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.