Past Algebraic and Symplectic Geometry Seminar

There are two basic theories of curve counting on Calabi-Yau threefolds. Donaldson-Thomas theory arises by considering curves as subschemes; Gromov-Witten theory arises by considering curves as the image of maps. Both theories can also be formulated for orbifolds. Let X be a dimension three Calabi-Yau orbifold and let

Y --> X be a Calabi-Yau resolution. The Gromov-Witten theories of X and Y are related by the Crepant Resolution Conjecture. The Gromov-Witten and Donaldson-Thomas theories of Y are related by the famous MNOP conjecture. In this talk I will (with some provisos) formulate the remaining equivalences: the crepant resolution conjecture in Donaldson-Thomas theory and the MNOP conjecture for orbifolds. I will discuss examples to illustrate and provide evidence for the conjectures.

This is the second of two seminars this afternoon describing a generalization of Donaldson-Thomas invariants, joint work of Yinan Song and Dominic Joyce. (Still work in progress.)

Behrend showed that conventional Donaldson-Thomas invariants can be written as the Euler characteristic of the moduli space of semistable sheaves weighted by a "microlocal obstruction function" \mu.

In previous work, the speaker defined Donaldson-Thomas type invariants "counting" coherent sheaves on a Calabi-Yau 3-fold using

Euler characteristics of sheaf moduli spaces, and more generally, of moduli spaces of "configurations" of sheaves. However, these invariants are not deformation-invariant.

We now combine these ideas, and insert Behrend's microlocal obstruction \mu into the speaker's previous definition to get new generalized Donaldson-Thomas invariants. Microlocal functions \mu have a multiplicative property implying that the new invariants transform according to the same multiplicative transformation law as the previous invariants under change of stability condition.

Then we show that the invariants counting pairs in the previous seminar are sums of products of the new generalized Donaldson-Thomas invariants. Since the pair invariants are deformation invariant, we can deduce by induction on rank that the new generalized Donaldson-Thomas invariants are unchanged under deformations of the underlying Calabi-Yau 3-fold.

This is the first of two seminars this afternoon describing a generalization of Donaldson-Thomas invariants, joint work of Yinan Song and Dominic Joyce. We shall define invariants "counting" semistable coherent sheaves on a Calabi-Yau 3-fold. Our invariants are invariant under deformations of the complex structure of the underlying Calabi-Yau 3-fold, and have known transformation law under change of stability condition.

This first seminar constructs an auxiliary invariant "counting" stable pairs (s,E), where E is a Gieseker semistable coherent sheaf with fixed Hilbert polynomial and s : O(-n) --> E for n >> 0 is a morphism of sheaves, and (s,E) satisfies a stability condition. Using Behrend-Fantechi's approach to obstruction theories and virtual classes we prove this auxiliary invariant is unchanged under deformation of the underlying Calabi-Yau 3-fold.

The geometric Langlands program aims at a "spectral decomposition" of certain derived categories, in analogy with the spectral decomposition of function spaces provided by the Fourier transform. I'll explain such a geometrically-defined spectral decomposition of categories for a particular geometry that arises naturally in connection with integrable systems (more precisely, the quantum Calogero-Moser system) and representation theory (of Cherednik algebras). The category in this case comes from the moduli space of vector bundles on a curve equipped with a choice of ``mirabolic'' structure at a point. The spectral decomposition in this setting may be understood as a case of ``tamely ramified geometric Langlands''. In the talk, I won't assume any prior familiarity with the geometric Langlands program, integrable systems or Cherednik algebras.

I plan to discuss some aspects the mysterious relationship between the symmetries of toroidal compactifications of M-theory and helices on del Pezzo surfaces.

Let G be a connected reductive algebraic group over an

algebraically closed field of characteristic 0. A normal

irreducible G-variety X is called spherical if a Borel

subgroup of G has an open orbit on X. It was conjectured by F.

Knop that two smooth affine spherical G-varieties are

equivariantly isomorphic provided their algebras of regular

functions are isomorphic as G-modules. Knop proved that this

conjecture implies a uniqueness property for multiplicity free

Hamiltonian actions of compact groups on compact real manifolds

(the Delzant conjecture). In the talk I am going to outline my

recent proof of Knop's conjecture (arXiv:math/AG.0612561).

This is the second of two talks, and probably will not be comprehensible unless you came to last week's talk.

A Kuranishi space is a topological space equipped with a Kuranishi structure, defined by Fukaya and Ono. Kuranishi structures occur naturally on many moduli spaces in differential geometry, and in particular, in moduli spaces of stable $J$-holomorphic curves in symplectic geometry.

Let $Y$ be an orbifold, and $R$ a commutative ring. We define four topological invariants of $Y$: two kinds of Kuranishi bordism ring $KB_*(Y;R)$, and two kinds of Kuranishi homology ring $KH_*(Y;R)$. Roughly speaking, they are spanned over $R$ by isomorphism classes $[X,f]$ with various choices of relations, where $X$ is a compact oriented Kuranishi space, which is without boundary for bordism and with boundary and corners for homology, and $f:X\rightarrow Y$ is a strong submersion. These theories are powerful tools in symplectic geometry.

Today we discuss the definition of Kuranishi homology, and the proof that weak Kuranishi homology is isomorphic to the singular homology.

A Kuranishi space is a topological space equipped with a Kuranishi structure, defined by Fukaya and Ono. Kuranishi structures occur naturally on many moduli spaces in differential geometry, and in particular, in moduli spaces of stable $J$-holomorphic curves in symplectic geometry.

Let $Y$ be an orbifold, and $R$ a commutative ring. We shall define four topological invariants of $Y$: two kinds of Kuranishi bordism ring $KB_*(Y;R)$, and two kinds of Kuranishi homology ring $KH_*(Y;R)$. Roughly speaking, they are spanned over $R$ by isomorphism classes $[X,f]$ with various choices of relations, where $X$ is a compact oriented Kuranishi space, which is without boundary for bordism and with boundary and corners for homology, and $f:X\rightarrow Y$ is a strong submersion. Our main result is that weak Kuranishi homology is isomorphic to the singular homology of $Y$.

These theories are powerful tools in symplectic geometry for several reasons. Firstly, using them eliminates the issues of virtual cycles and perturbation of moduli spaces, yielding technical simplifications. Secondly, as $KB_*,KH_*(Y;R)$ are very large, invariants defined in these groups contain more information than invariants in conventional homology. Thirdly, we can define Gromov-Witten type invariants in Kuranishi bordism or homology groups over $\mathbb Z$, not just $\mathbb Q$, so they can be used to study the integrality properties of Gromov-Witten invariants.

This is the first of two talks. Today we deal with motivation from symplectic geometry, and Kuranishi bordism. Next week's talk discusses Kuranishi homology.

I will associate, to every pair of smooth transversal

Lagrangian spheres in a symplectic manifold having vanishing first Chern

class, its Floer cohomology groups. Hamiltonian isotopic spheres give

rise to isomorphic groups. In order to define these Floer cohomology

groups, I will make a key use of symplectic field theory.

We provide a realization of Cherednik's double affine Hecke

algebras (for GL_n) as a convolution algebra of functions on moduli spaces

of coherent sheaves on an elliptic curve. As an application we give a

geometric construction of Macdonald polynomials as (traces of) certain

natural perverse sheaves on these moduli spaces. We will discuss the

possible extensions to higher (or lower !) genus curves and the relation

to the Hitchin nilpotent variety. This is (partly) based on joint work

with I. Burban and E. Vasserot.