A new look at Verdier specialization Abstract: We consider a closed embedding with nonsingular complement, and define
explicitly a constructible function which agrees with the `Verdier specialization' of the
constant 1 in the case of a 1-parameter family with nonsingular general fiber.
We will discuss the relation of this function with the `weighted Chern-Mather class'
defined in previous work, and of some relevance to Behrend's approach to
Donaldson-Thomas invariants. We also give a version of the basic construction
in a quotient of the Grothendieck group of varieties, which can be viewed as a
(very) naive version of the Denef-Loeser motivic Milnor/nearby fiber.
Intersection Space Cohomology and Equivariant Cohomology of Isometric Group Actions Abstract: After reviewing the construction of intersection spaces
associated to stratified singular spaces, we shall present
applications of the resulting cohomology theory to the
calculation of equivariant cohomology. In some detail we
will present a topological argument showing that if the
fundamental group of a closed aspherical manifold acts
isometrically on a closed Riemannian manifold, then the equivariant
real cohomology of this action can be decomposed as a direct
sum of group cohomology contributions with coefficients in the
cohomology modules of the manifold.
Local systems and Bernstein-Sato ideals Abstract: This is preliminary report on work relating five topics:
formality and resonance varieties, structure of characteristic
varieties of local systems, multi-variable nearby cycles,
Bernstein-Sato ideals, and multi-variable monodromy conjecture. The
link is provided by a multi-variable generalization of a result of
Malgrange of Kashiwara relating roots of Bernstein-Sato polynomials
with monodromy eigenvalues.
The Hodge theorem as a derived self-intersection Abstract: The Hodge theorem is one of the most important results in complex
geometry, originally stated and proved by analytic methods. An algebraic
formulation of the Hodge theorem has been known since Grothendieck's work
in the early 1970's. However, the first purely algebraic (and very surprising)
proof was obtained only in 1991 by Deligne and Illusie, using methods involving
reduction to characteristic p. In my talk I shall try to explain their ideas,
and how recent developments in the field of derived algebraic geometry make
their proof more geometric, by allowing us to realize the Deligne-Illusie main
result as a formality result for the derived self-intersection of a subvariety
of a twisted space. This involves a generalization of pioneering results of
Buchweitz-Flenner from the case of the derived self-intersection of the
diagonal subvariety to the case of an arbitrary subvariety.
Lattice point counting via Einstein metrics Abstract: We obtain a growth estimate for the number of lattice points
inside any Q-Gorenstein cone. Our proof uses result of Futaki-Ono-Wang on
Sasaki- Einstein metric for the toric Sasakian manifold associated to the cone,
a Yau's inequality and the Kawasaki-Riemann-Roch formula for orbifolds.
Mordell-Weil groups of isotrivial abelian varieties over function fields and singular curves Abstract: I will discuss recent
results connecting Mordell Weil rank
of an isotrivial abelian variety
with a cyclic holonomy
and the fundamental group
of the complement to the discriminant
provided the discriminant has
singularities in CM class which we
shall introduce in the talk.
Among other things, I give families
of simple Jacobians over field
of rational functions in two variable
for which the Mordell Weil
rank is arbitrary large.
Thom polynomials for quivers Abstract: Consider an oriented graph whose vertices are vector spaces.
Let a "quiver" be a collection of linear maps between these vector spaces
according to the oriented edges of the graph. The group of linear
reparametrizations of the vector spaces at the vertices acts on quivers.
An orbit is called a quiver singularity. To such a quiver singularity we
associate its Thom polynomial, imitating the definition of Thom polynomials
of singularities of map germs. Quiver Thom polynomials generalize various
objects in algebraic combinatorics, e.g. various versions of Schur,
Schubert polynomials . In the lecture we will explore the geometric
meaning of quiver Thom polynomials, as well as their algebraic properties.
I will sketch our proof (with R. Kaliszewski) of Buch's conjecture on the
positivity of certain coefficients of quiver Thom polynomials.
Mathias Schulze :
Logarithmic residues and the Lê-Saito Theorem Abstract: Lê and K. Saito showed that a complex hypersurface is normal crossing in codimension 1 if and only if the local fundamental groups of the complement are abelian. Answering a question of Saito, we show that the latter property can be characterized purely algebraically in terms of logarithmic residues. To this end, we describe a dual version of Saito's logarithmic residue map. The talk is based on joint work with Michel Granger.
Generating series for (equivariant) characteristic classes of
(external and) symmetric products Abstract: First we explain different equivariant characteristic classes of Lefschetz
type for a singular complex quasi-projective variety X acted on by a
finite group G defined for an equivariant constructible or coherent sheaf
(complex), mixed Hodge modules or equivariant relative Grothendieck
groups of algebraic varieties. For a fixed group element g these are
homology classes on the fixed point set Xg, codifying the corresponding
trace of the induced action on the cohomology. Putting them together for
all group elements, one gets equivariant characteristic classes in
delocalized equivariant Borel-Moore homology. These can be used for
calculating invariants of the quotient X/G.
We apply these for the calculation of generating series of equivariant
characteristic classes for external products (of such coefficients)
acted on by the permutation action of the symmetric groups Sn.
Using the Künneth formula, these can be expressed in terms of creation
operators acting on a Fock space given by the delocalized equivariant
Borel-Moore homology of all products Xn (even for X singular).
Pushing down to the symmetric products X(n):=Xn/Sn, the classes live in
the homology Pontrjagin ring of the symmetric products, tensorized with
the ring of symmetric functions generated by all power-sums pn.
Specializing finally all pn to 1, one gets the generating series for
symmetric products in terms of Adams operations.
Manuel Gonzalez Villa :
Arcs and Monodromy of Quasi-ordinary Hypersurfaces Abstract: We explain how motivic zeta function of the quasi-ordinary
hypersurface singularities are determined by their embedded
We discuss applications to the log-canonical threshold, the
Hodge-Steenbrink spectrum and the monodromy conjecture.
We also speculate about possible generalizations. These results are
joint work with P.D. Gonzalez Perez (Universidad Complutense de
Madrid, Spain) and Nero Budur (University of Notre Dame, USA).
Uli Walther :
Jacobians, b-functions, and Milnor fiber cohomology Abstract: We identify some roots of the Bernstein--Sato polynomial using information on the Jacobian ideal of a homogeneous divisor. In the process we also find some cohomology classes on the corresponding Milnor fiber.
Jon Woolf :
Witt groups of perverse sheaves Abstract: Perverse sheaves on a stratified space X with even dimensional
strata form an abelian category Perv(X) with duality. This category has a
Witt group, which we denote W(X), and which is functorial under proper maps
stratified maps. W(X) is isomorphic to the zeroth Balmer-Witt group, and
also to the Youssin cobordism group, of the ambient constructible
If Y is a closed union of strata then there is a canonical decomposition
W(X) = W(Y) ⊕ W(X-Y) which applied inductively decomposes W(X) as a
direct sum of Witt groups of local systems on strata. This can be viewed as
a quadratic version of the decomposition of the Grothendieck group of
There is also an explicit formula for the associated decomposition of
a class in terms of intermediate extensions and restrictions of an
anisotropic representative. In particular, given a proper stratified map
f: Y → X one can obtain a formula for the intersection cohomology signature of Y as a sum of terms localised on the strata of X. For nice X this formula agrees with one due to Cappell and Shaneson, and can be interpreted geometrically. However, by considering examples in which Perv(X) has an explicit quiver description, one sees that in general there are correction terms to their formula.