# Homological Mirror Geometry (16w5062)

Arriving in Banff, Alberta Sunday, March 6 and departing Friday March 11, 2016

## Organizers

(University of South Carolina)

(University of Miami)

(University of Alberta)

## Objectives

#### Overview

Following Kontsevich's homological mirror symmetry proposal [33], substantial effort has gone into understanding mirror symmetry in terms of equivalences of derived categories of coherent sheaves with appropriate Fukaya-type categories of Lagrangians. Originally stated for Calabi-Yau manifolds, the scope of these conjectures have since been extended to include Fano varieties [34] and manifolds of general type [29], where the mirror theory is controlled by the singularity theory of a holomorphic function (Landau-Ginzburg model). Various special cases of homological mirror symmetry have been proved [37, 38, 42, 6, 7, 39, 21, 1], however, a complete picture remains elusive.

On the other hand, after the pioneering works of Bondal and Orlov in the mid 90's [12], derived categories of coherent sheaves on varieties can be studied by examining their decomposition properties under the presence of a birational morphism (e.g. a blow up). This way of thinking has led to a steady stream of developments in the understanding of derived categories guided by birational geometry. Notable examples include Kuznetsov's reformulation of the conjectural non-rationality of cubic fourfolds in terms of their derived categories [35] and Bridgeland's introduction of Fourier-Mukai techniques to the study and construct threefold flops [13].

This workshop aims to gather researchers working at the intersection of these themes: the interplay between the birational geometry of varieties with the symplectic geometry of their mirrors, often mediated by categorical considerations. Cutting-edge developments at this intersection include the study of the minimal model program for moduli spaces of sheaves via Bridgeland stability conditions, comparisons between derived categories through Geometric Invariant Theory, the classification of Fano varieties via their Landau-Ginzburg mirrors, and the construction of generalized Theta functions on log Calabi Yau manifolds through mirror symmetry.

The flurry of recent activity in these topics, all connecting birational geometry with mirror symmetry, suggests now is a truly opportune moment to bring together experts and early career researchers. In addition to allowing senior scholars to explore interactions with other large scale research programs, this is a prime opportunity to expose early career researchers to a wider range of interconnected ideas and themes and afford them a deeper perspective on their own emerging research.

Additionally, the organizers have already established a track record of successful and diversely populated conferences including a 3-month thematic period on the Geometry of D-branes at the Erwin Schrödinger Institute, String-Math 2014, Commutative Algebra-Algebraic Geometry in the Southeast, and an upcoming AMS Special Session on Mirror Symmetry.

#### Objectives

This workshop will bring together experts in birational geometry, derived categories, and symplectic geometry to foster an environment of cross-polination, with a focus on the following rapidly developing areas of investigation.

#### VGIT and mirror symmetry

Since its birth, Mumford's Geometric Invariant Theory (GIT) has been a central tool in the construction of moduli spaces in algebraic geometry. First thoroughly investigated by Dolgachev and Hu [17} and Thaddeus [43], the variation of the auxiliary choices made in GIT (VGIT) provides a robust framework for studying interesting birational maps. On the other hand, since the seminal work of Bondal and Orlov, the close ties between birational geometry and derived categories have been manifest.

Recent work of Ballard, Favero, Katzarkov [8] and Halpern-Leistner [22] has completed this circle of ideas. Developing a novel idea originating with work of Kawamata [31], and further pursued by Herbst, Hori, Page [24] and Segal [41], these authors produce semi-orthogonal decompositions relating the derived categories of different GIT quotients by analyzing stratifications of their unstable loci. This provides a new perspective on many classical results in the study of derived categories and unifies many previously disparate results.

Under mirror symmetry, we have semi-orthogonal decompositions of Fukaya(-Seidel) categories. What is the underlying symplectic geometry that creates them? In the toric case, VGIT can be viewed as a degeneration of a Landau-Ginzburg model, as studied by Diemer, Katzarkov, and Kerr [16]. There it was shown that certain degenerations of Laudau-Ginzburg models admit semiorthogonal decompositions of their corresponding Fukaya-Seidel categories of vanishing Lagrangians which conjecturally match the semi-orthogonal decompositions of a mirror variety discussed above. This exemplifies the rich structure present in moduli spaces of Landau-Ginzburg models as recently exposed in [28, 30]. The duality between VGIT and symplectic Morse theory suggests additional structure beyond Kontsevich's original homological mirror symmetry (HMS) proposal which directly incorporates the underlying birational geometry. However, at present only a few classes of varieties exist where this phenomenon has been fully explored.

Objective 1 Develop and contextualize VGIT as an additional level of structure in HMS.

#### Autoequivalences from VGIT

One of the first major insights born from homological mirror symmetry was that the abundance of symplectomorphisms in symplectic geometry must be reflected as well in algebraic geometry. Work of Seidel and Thomas on spherical twists [40] blazed a trail for researchers studying autoequivalences of derived categories. Spherical twists are understood to be mirror to autoequivalences induced by symplectic Dehn twists on the symplectic side of mirror symmetry, but correspond to highly novel operations on an algebraic mirror. The examples constructed by Seidel-Thomas have recently been greatly expanded in scope, yielding many types of exotic autoequivalences of derived categories. We note, for example, the results on Grassmannian twists and mixed braid group actions of Donovan and Segal [18, 19] and the $\mathbf{Q}$-massless autoequivalences of Addington and Aspinwall [2].

A unifying homological approach for studying these examples is the notion of a spherical functor as introduced by Anno [3, 4]. Perhaps the deepest understanding of such autoequivalences, though, comes from the work of Herbst, Hori, and Page on parallel transport of D-branes over a mirror moduli space. Recent work of Halpern-Leistner and Shipman [23] interpreted this mathematically and proved general relationships between spherical functors, VGIT wall-crossings, and categorical mutations. The use of VGIT to study the autoequivalence groups of derived categories merits further investigation, as new examples of geometric interest continue to arise, and the scope of the theory does not seem to be fully realized.

Objective 2 Assemble experts in symplectic and algebraic geometry to enhance the study of autoequivalences through the VGIT/HMS narrative.

#### Birational geometry of Bridgeland moduli spaces

Inspired by Douglas' $\Pi$-stability for D-branes [20], Bridgeland introduced a notion of stability for triangulated categories [14]; a deep categorification of classical GIT stability. A long held expectation, inspired by mirror symmetry, is that the Bridgeland stability manifold is the correct interpretation of the so-called "stringy Kahler moduli space". Recently, work of Arcara, Bertram, Coskun, and Huizenga [5], Bayer and Macr i [10,11], and of Toda [44, 45] has shown that varying the Bridgeland stablity condition can reproduce a run of the minimal model program for surfaces, threefolds, and moduli spaces of sheaves associated to surfaces. In the specific case of the Hilbert scheme of points on the projective plane from [5], the Bridgeland moduli spaces can be constructed explicitly as GIT quotients and variation of GIT stability is equivalent to variation of Bridgeland stability. While this is far too clean to happen in general (indeed, precious few examples of Bridgeland stability manifolds have been explicitly constructed), the techniques for studying derived categories via GIT discussed above should have an analog for Bridgeland stability. Bringing together experts from both sides would hopefully allow for creation of a general framework, currently quite undeveloped, to understand how derived categories of moduli spaces, or even general varieties, behave under variation of stability.

Objective 3 Open the dialogue between experts on varying Bridgeland stability conditions and experts on VGIT with a focus on the effect on derived categories.

#### Classifying Fano varieties via LG mirrors

Another area of focus for the workshop will be the mirror symmetry underlying the classification of Fano manifolds. The fundamental insight which drives this topic is Golyshev's observation that the famous 17 families of Picard rank 1 Fano threefolds correspond bijectively to certain modular families of K3 surfaces [25], expected to be the mirror Landau-Ginzburg models. The subject has since developed into an ambitious program aiming to classify Fano varieties in higher dimensions in terms of their conjectural Landau-Ginzburg mirrors, via explicit calculations at the level of quantum cohomology and the mirror quantum differential equations. This year, Coates, Corti, Galkin, and Kaspryzk [15] demonstrated explicit Landau-Ginzburg mirrors for all three dimensional Fano manifolds, and their work suggests that mirror symmetry may aid in understanding classification problems in higher dimensions, where essentially nothing is known beyond non-explicit finiteness results. We also note that at the level of homological mirror symmetry, very little is proven about Fano manifolds beyond the surface (del Pezzo) case \cite{AKO06], and we hope that the interaction of experts from these respective topics will advance mutual understanding.

Objective 4 Integrate new developments from the broader mirror symmetry field into the classification of Fano varieties via Landau-Ginzburg mirrors program.

#### Mirror symmetry via the Gross-Siebert Program

The SYZ proposal [SYZ96] for mirror symmetry as dualization of torus fibrations has provided incredible inspiration in mathematics. Unfortunately, a completely general and rigorous mathematical framework for SYZ has been diffcult to achieve. Currently, most promising approach is the Gross-Siebert Program [GS03, GS06, GS10, GS11] proceeding via toric degenerations. While work remains to capture all of mirror symmetry in this framework, it already has found significant application. As an example, Gross, Hacking, and Keel recently proposed an explicit mirror construction for a large class of rational surfaces, associated to cluster varieties [26, 27]. Their results make contact with many interesting topics in birational geometry: their mirror families naturally fiber over the toric variety of the effective cone of the surface, and a detailed study of the behavior of the fibers nearby the point corresponding to the nef cone was shown to reproduce many fundamental results (as well as solve long standing conjectures) in the deformation theory of surface singularities. Their work has resulted in a number of far-reaching conjectures, the scope of which is perhaps still not fully appreciated. In particular, they conjecture that their results, though not explicitly categorical in nature, may aid in proofs of certain open problems in homological mirror symmetry. Some evidence for this was given in recent work of Pascale
[36] who verified some conjectures of Gross, Hacking, Keel at the degree 0 level of symplectic homology. We hope that a discussion of the techniques introduced in these works with experts in related fields will yield ideas for implementation of such proofs of new cases of homological mirror symmetry.

Objective 5 Centralize the role of the Gross, Hacking, Keel framework in the modern HMS discussion.

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