The dual Lagrangian fibration of compact hyper-Kähler manifolds

A compact hyper-Kähler manifold is a higher dimensional generalization of a K3 surface. An elliptic fibration of a K3 surface correspondingly generalizes to the so-called Lagrangian fibration of a compact hyper-Kähler manifold. It is known that an elliptic fibration of a K3 surface is always "self-dual" in a certain sense. This turns out to be not the case for higher-dimensional Lagrangian fibrations. In this talk, I will propose a construction for the dual Lagrangian fibration of all currently known examples of compact hyper-Kähler manifolds, and try to justify this.

Studying the birational geometry of Fano varieties using holomorphic forms

One of the best invariants for studying the birational geometry of a variety is its holomorphic forms. Unfortunately, in characteristic 0, low degree hypersurfaces (or more generally Fano varieties) do not have any holomorphic forms. For this reason, many problems about birational geometry of these varieties are quite difficult and interesting. E.g. (1) determining if the birational automorphism group is infinite or finite, (2) studying the possible rational endomorphisms of a Fano variety, and (3) understanding the rationality/nonrationality of a Fano variety. Surprisingly, Kollár showed that in characteristic p>0, certain Fano varieties admit many global (n-1)-forms, and introduced a specialization method for using these forms in characteristic p to control the birational geometry of characteristic 0 Fano varieties. In this talk, we show how this method gives answers to problems (1)-(3). This is joint work with Nathan Chen.

Geometric local systems on very general curves and isomonodromy

I'll show that a very general n-pointed curve of genus g does not carry any non-isotrivial local systems of geometric origin of rank less than 2\sqrt{g+1}, and explain how this resolves conjectures of Esnault-Kerz and Budur-Wang. The main input is an answer to questions of Biswas, Heu, and Hurtubise on stability properties of isomonodromic deformations of flat vector bundles. This is joint work with Aaron Landesman.

Pathologies of the volume function

If L is a line bundle on a variety X, then it is a basic result that h⁰(mL) grows roughly polynomially in m. In birational geometry, it is frequently useful to instead fix an ample divisor A and consider the growth of h⁰(mL+A) as m increases. I will show that the behavior of this growth can be quite strange.

Intrinsic constructions of moduli spaces via affine grassmannians

For a projective variety X, the moduli problem of coherent sheaves on X is naturally parametrized by an algebraic stack M, which is a geometric object that naturally encodes the notion of families of sheaves. In this talk I will explain a GIT-free construction of the moduli space of Gieseker semistable pure sheaves which is intrinsic to the moduli stack M. This approach also yields a Harder-Narasimhan stratification of the unstable locus of the stack. Our main technical tools are the theory of Theta-stability introduced by Halpern-Leistner, and some recent techniques developed by Alper, Halpern-Leistner and Heinloth. In order to apply these results, one needs to prove some monotonicity conditions for a polynomial numerical invariant on the stack. We show monotonicity by defining a higher dimensional analogue of the affine grassmannian for pure sheaves. If time allows, I will also explain some applications of these ideas to other moduli problems. This talk is based on joint work with Daniel Halpern-Leistner and Trevor Jones, as well as work with Tomas Gomez and Alfonso Zamora.

A quadratically enriched residual intersection formula

Excess intersections occur when the zero locus of s polynomial functions has smaller codimension than s. Removing a large codimension subscheme of the zero locus results in a residual intersection. A famous example is Chasles’ Theorem that there are 3264 smooth conics in the complex projective plane tangent to five generally chosen conics over the complex numbers. Without taking into account an excess intersection caused by double lines, Bézout’s theorem suggests that there would be 6^5 = 7776. Once the double lines are removed, the residual intersection consists of 3264 smooth conics.

We study enumerative results for residual intersections over non algebraically closed fields using duality for coherent sheaves. Duality for coherent sheaves on a smooth scheme is related to the cotangent space. For regular embeddings, coherent duality is related to the conormal bundle. Eisenbud and Ulrich have recently given coherent duality results in the setting of certain residual intersections. We twist their description of the dualizing object, globalizing their result and identifying an exceptional pushforward in Hermitian K-theory. We use this to give a residual intersection formula enriched in quadratic forms. We give several examples in enumerative geometry over an arbitrary field of characteristic not 2, including Chasles’ Theorem. This is joint work with Tom Bachmann.

Open Mirror Symmetry for Landau-Ginzburg models

A Landau-Ginzburg (LG) model is a triplet of data (X, W, G) consisting of a regular function $W:X \to \mathbb{C}$ from a quasi-projective variety X with a group G acting on X leaving W invariant. An enumerative theory developed by Fan, Jarvis, and Ruan gives the analogue of Gromov-Witten invariants for an LG model. These invariants are now called FJRW invariants. We define a new open enumerative theory for certain Landau-Ginzburg LG models. Roughly speaking, this involves computing specific integrals on certain moduli of disks with boundary and interior marked points. One can then construct a mirror Landau-Ginzburg model to a Landau-Ginzburg model using these invariants.

This allows us to prove a mirror symmetry result analogous to that established by Cho-Oh, Fukaya-Oh-Ohta-Ono, and Gross for mirror symmetry for toric Fano manifolds. Essentially, given the original Landau-Ginzburg model, one constructs its mirror LG model using these open invariants. Then, one can compute oscillatory integrals on the mirror and deduce the (closed) FJRW invariants of the original LG model explicitly. This provides a Landau-Ginzburg version of a phenomenon found in mirror symmetry for toric Fano manifolds established by Cho-Oh, Fukaya-Oh-Ohta-Ono, and Gross. If time permits, I will explain some key features that this enumerative geometry enjoys (e.g., topological recursion relations and wall-crossing phenomena). This is joint work with Mark Gross and Ran Tessler.