
Flatness is a subtle algebraic notion that expresses continuity of the fibres of a mapping, but it has remained geometrically elusive. The aim of this talk is to understand the notion of flatness in explicit geometric terms. We show that nonflatness of a morphism of schemes of finite type with a regular target of dimension n manifests in the existence of the socalled vertical components in the nfold fibred power of the morphism (i.e., components with a nowhere dense image). This leads to an effective algorithm for flatness over regular affine algebras, by means of Grobner bases.
This work was done with E. Bierstone and P. D. Milman.
Can we find the smallest class of singularities S with the following properties:
Coherent systems are the analogous for higher classical linear systems. That is, a coherent system of type (n,d,k) is a pair (E,V) where E is a holomorphic bundle of rank n and degree d and V is a linear subspace of its space of holomorphic sections of dimension k. There is a stability notion for a pair (E,V), distinct from the stability of the bundle E. The natural definition of such stability depends on a real parameter a and leads to a finite family of moduli spaces of astable coherent systems. In this talk we will describe such moduli spaces for certain values of (n,d,k).
I will present joint work with Angelo Vistoli and Zinovy Reichstein on essential dimension of the moduli stacks of curves and of abelian varieties. The main technical result used to compute this essential dimension is a "genericity theorem" which reduces the computation of the essential dimension of a smooth DeligneMumford stack to the essential dimension of its generic gerbe. I will sketch the proof of this result.
There are two primary theories of "curve counting" on a CalabiYau threefold, DonaldsonThomas theory and GromovWitten theory. Both theories extend to CalabiYau orbifolds. A three dimensional CalabiYau orbifold X has a canonical CalabiYau resolution Y. The four curve counting theories DT(X), DT(Y), GW(X), and GW(Y) are expected to be equivalent. We give an overview of these theories and conjectural equivalences.
In this talk we present a generalized algebraic index formula for certain class of realanalytic vector fields with nonalgebraically isolated singularities. In the algebraically isolated case, we compute the index with the library PHindex writing for the software Singular, using the EisenbudLevineKhimshiashvili's algebraic index formula.
Let C be an smooth complex algebraic curve of genus g ³ 8. Denote by K_{C} the canonical line bundle on C. Let L = g^{1}_{g2} a pencil on C free of base points such that the residual g^{2}_{g} of the g^{1}_{g2} determines a birational map onto a plane curve of degree g and geometric genus g with d = [((g1)(g2))/2] nodes as singularities. Consider the Petri map m_{L} : H^{0}(C,L) ÄH^{0}(C,K_{C} ÄL^{1}) ® H^{0}(C,K_{C}). We show that m_{L} is not injective if and only there exists a curve F of degree g5 containing d1 nodes of G.
Now consider the Severi Variety V^{g,g,d} of reduced and irreducible plane curves of degree g and genus g having d nodes as singularities. Let V_{g} : = {G Î V^{g,g,d} : d1 nodes lie on a curve of degree g5}. Let f: V^{d}_{g,g} ® M_{g} the natural morphism to the moduli space of curves M_{g}. We show that the image f(V_{g}) is a divisor in M_{g}. We discuss the irreducibility of f(V_{g}) in some cases.
Given a holomorphic mapgerm f : (C^{n},0) ® (C,0) with a critical value at 0 Î C, there are two equivalent ways of defining its Milnor fibration. The first is:
 (1) 
 (2) 
We show that there is a canonical decomposition of the whole ball B_{e} into real analytic hypersurfaces X_{q} that spin around their "axis" V_{e} = f^{1}(0) ÇB_{e} forming a kind of "openbook" with singular binding. Using this decomposition we improve, or refine, Milnor's fibration theorem in several directions.
This is joint work with J. Seade and J. Snoussi.
We consider a particular family of singular CalabiYau 3folds parametrized by U = P/1  {q_{1},...,q_{6}}. The singular locus of the fiber over u consists of exactly 100 nodes for every u Î U.
We study the monodromy action and in particular the variation of Hodge structure associated to the smooth family of the desingularizations.
Holomorphic Poisson manifolds are highly constrained compared to their relatively flexible smooth counterparts. I shall describe some new results concerning the structure and classification of holomorphic Poisson manifolds.
The moduli spaces of instantons on the foursphere and of calorons (instantons on the circle times R^{3}) turn out to be describable in terms of holomorphic maps from the Riemann sphere into a flag manifold of a loop group. These in turn, like for their finite dimensional cousins, admit a poles and principal parts description that allows one to describe the moduli and prove, for example, a topological stability theorem for the moduli.
Joint work with Michael Murray.
Let X be a algebraic variety equipped with an action of a reductive algebraic group G. Also let L be a finite dimensional subspace of rational functions invariant under G. In this talk we discuss various convex bodies that one can associate to (X,L) which encode information about number of solutions of generic systems of equations from L plus information on multiplicities of irreducible representations appearing in powers L^{k}. Similarly one associates convex bodies to a projective Gvariety X and a Glinearized line bundle L on X. These far generalize the notion of Newton convex polytope in toric geometry as well as GelfandCetlin polytopes associated to irreducible representations of GL(n) (i.e., flag variety of GL(n)).
This is a joint work with A. G. Khovanskii.
In this talk, I will examine the geometry of moduli spaces of stable bundles on Kodaira surfaces, which are nonKaehler compact surfaces that can be realised as torus fibrations over elliptic curves. These moduli spaces are interesting examples of holomorphic symplectic manifolds whose geometry is similar to the geometry of Mukai's moduli spaces on K3 and abelian surfaces.
The Chern classes of complex manifolds are important invariants that appear in many fields of geometry and topology. From the topological viewpoint, these are closely related to the local index of PoincaréHopf for vector fields, and generalizations of it to the case of sections of certain fibre bundles associated to the tangent bundle. It is natural to ask whether one has similar notions for complex analytic varieties. This is a question that goes back to the work of M. H. Schwartz in the 1960s, MacPherson in the 1970s, and many others after that. In this talk I will speak about work done mostly with JeanPaul Brasselet and Tatsuo Suwa. We relate this problem with that of studying indices of vector fields on singular varieties, and the various generalizations one has for singular varieties of the concept of tangent bundle.