Algebraic and Symplectic Geometry Seminar

Let p>0 be a prime number. We shall describe a short Frobenius-theoretic proof of the Adams-Riemann-Roch theorem for the p-th Adams operation, when the involved schemes live in characteristic p and the morphism is smooth. This result implies the Grothendieck-Riemann-Roch theorem for smooth morphisms in positive characteristic and the Hirzebruch-Riemann-Roch theorem in any characteristic. This is joint work with R. Pink.

I will survey some applications of a special kind of stratification of an algebraic stack called a theta-stratification. The goal is to eventually be able to study semistability and wall-crossing 
in a large array of moduli problems beyond the well-known examples. The most general application is to studying the derived category of coherent sheaves on the stack, but one can use this to understand the topology (K-theory, Hodge-structures, etc.) of the semistable locus and how it changes as one varies the stability condition. I will also describe a ``virtual non-abelian localization theorem'' which computes the virtual index of certain classes in the K-theory of a stack with perfect obstruction theory. This generalizes the virtual localization theorem of Pandharipande-Graber and the K-theoretic localization formulas of Teleman and Woodward.

Igusa's p-adic zeta function $Z(s)$ attached to a polynomial $f$ in $N$ variables is a meromorphic function on the complex plane that encodes the numbers of solutions of the equation $f=0$ modulo powers of a prime $p$. It is expressed as a $p$-adic integral, and Igusa proved that it is rational in $p^{-s}$ using resolution of singularities and the change of variables formula. From this computation it is immediately clear that the order of a pole of $Z(s)$ is at most $N$, the number of variables in $f$. In 1999, Wim Veys conjectured that the only possible pole of order $N$ of the so-called topological zeta function of $f$ is minus the log canonical threshold of $f$. I will explain a proof of this conjecture, which also applies to the $p$-adic and motivic zeta functions. The proof is inspired by non-archimedean geometry and Mirror Symmetry, but the main technique that is used is the Minimal Model program in birational geometry. This talk is based on joint work with Chenyang Xu.

Following an idea of Bridgeland, we study the operator on the K-group of algebraic stacks, which takes a stack to its inertia stack.  We prove that the inertia operator is diagonalizable when restricted to nice enough stacks, including those with algebra stabilizers.  We use these results to prove a structure theorem for the motivic Hall algebra of a projective variety, and give a more conceptual definition of virtually indecomposable stack function.  This is joint work with Pooya Ronagh.

Let $G$ be a compact Lie group and $k$ be a field of characteristic $p\ge 0$ such that $H^*(G)$ does not have $p$-torsion. We show that a free Lagrangian orbit of a Hamiltonian $G$-action on a compact, monotone, symplectic manifold $X$ split-generates an idempotent summand of the monotone Fukaya category over $k$ if and only if it represents a non-zero object of that summand. Our result is based on: an explicit understanding of the wrapped Fukaya category through Koszul twisted complexes involving the zero-section and a cotangent fibre; and a functor canonically associated to the Hamiltonian $G$-action on $X$. Several examples can be studied in a uniform manner including toric Fano varieties and certain Grassmannians. 

Given a hypersurface, the Bernstein-Sato polynomial gives deep information about its singularities.  It is defined by a D-module (the algebraic formalism of differential equations) closely related to analytic continuation of the gamma function. On the other hand, given a hypersurface (in a Calabi-Yau variety) one can also consider the Hamiltonian flow by divergence-free vector fields, which also defines a D-module considered by Etingof and myself. I will explain how, in the case of quasihomogeneous hypersurfaces with isolated singularities, the two actually coincide. As a consequence I affirmatively answer a folklore question (to which M. Saito recently found a counterexample in the non-quasihomogeneous case): if c$ is a root of the b-function, is the D-module D f^c / D f^{c+1} nonzero? We also compute this D-module, and for c=-1 its length is one more than the genus (conjecturally in the non-quasihomogenous case), matching an analogous D-module in characteristic p. This is joint work with Bitoun.
 

Over a decade ago Welschinger defined invariants of real symplectic manifolds of complex dimension 2 and 3, which count $J$-holomorphic disks with boundary and interior point constraints. Since then, the problem of extending the definition to higher dimensions has attracted much attention.
  We generalize Welschinger's invariants with boundary and interior constraints to higher odd dimensions using the language of $A_\infty$-algebras and bounding chains. The bounding chains play the role of boundary point constraints. The geometric structure of our invariants is expressed algebraically in a version of the open WDVV equations. These equations give rise to recursive formulae which allow the computation of all invariants for $\mathbb{CP}^n$.
  This is joint work with Jake Solomon.