Past Algebraic and Symplectic Geometry Seminar

22 January 2008
Dominic Joyce
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.
  • Algebraic and Symplectic Geometry Seminar
3 December 2007
Jean-Yves Welschinger
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.
  • Algebraic and Symplectic Geometry Seminar
27 November 2007
Olivier Schiffmann
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.
  • Algebraic and Symplectic Geometry Seminar