15-16 October, 2012

Samuel Boissière(Poitiers), Alessandro Chiodo (Paris 6), Laurent Manivel (Grenoble) and Etienne Mann(Montpellier)

within the ANR programme TheoryGW and the GDR Algebraic Geometry and Complex geometry

Let X be an algebraic variety with Gorenstein singularities. We define the notion of a wonderful resolution of singularities of X by analogy with the theory of wonderful compactifications of semi-simple linear algebraic groups. We prove that if X has rational singularities and has a wonderful resolution of singularities, then X admits a categorical crepant resolution of singularities. As an immediate corollary, we get that all determinantal varieties defined by the minors of a generic square/symmetric/skew-symmetric matrix admit categorical crepant resolution of singularities. We also discuss notions of minimality for a categorical resolution of singularities and we explore some links between minimality and crepancy for such resolutions.

(II) Correspondences for the quintic

I will discuss recent work with A. Pixton concerning the relationship between stable maps (Gromov-Witten theory) and sheaves (Donaldson-Thomas theory) for 3-folds. The first talk will be about descendents. For stable maps, descendents involve the cotangent line. For DT theory, descendents are obtained from the Chern characters of the universal sheaf. The goal of the talk is to explain a descendent correspondence for toric geometries. The second talk will use descendents, relative invariants, and degeneration to prove correspondences in compact non-toric settings --- including the quintic 3-fold in projective 4 space.

We will write down a Landau-Ginzburg model for an arbitrary Grassmannian X in terms of Pluecker coordinates on a dual Grassmannian. This LG-model gives rise to a vector bundle with Gauss-Manin connection, and we relate this vector bundle explicitly with a bundle on the A-model side of X: the trivial bundle over C with fibre H*(X) and its small Dubrovin-Givental connection. To show that our map between the vector bundles takes one flat connection to the other we make heavy use of the cluster structure of the homogeneous coordinate ring of the dual Grassmannian.

In the past years several Gromov-Witten-like invariants have been constructed: stable quotient invariants, quasi-maps invariants, stable (un-)ramified maps, etc. The moduli spaces which lead to these invariants are obtained by allowing different types of degenerate maps (e.g by allowing a stable map to acquire base-points, or by allowing the target space to degenerate). One is then tempted to expect all these invariants to be closely related. There are by now several theorems and conjectures in this direction. In this seminar I will discuss a few examples in which we can prove the expected relationships between various GW-type invariants by geometric methods. In particular, one can understand geometrically that GWI are equal to stable quotient invariants.