Abstracts
| Agostini |
Syzygies of abelian surfaces
Syzygies or minimal free resolutions give a way to study the
geometry of a projective variety via the relations between its equations:
very often, we can use unexpected relations to single out interesting loci
in the moduli space. In recent years, much work has been done for
curves and K3 surfaces. In this talk we want to present this circle of
ideas in the case of abelian surfaces.
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| Balestrieri |
Iterating the algebraic étale-Brauer set
In arithmetic geometry, a classical task is to study the set of rational points on varieties over number fields; a fruitful approach consists in trying to locate the rational points inside the adelic points by considering certain "obstruction" sets. In this talk, we explain how to iterate the algebraic étale-Brauer obstruction set of a smooth, projective, geometrically connected variety over any number field and show that, when the geometric étale fundamental group of the variety is finite, the iteration doesn't yield any new information. This gives evidence towards some conjectures by Colliot-Thélène and Skorobogatov on the arithmetic behaviour of rational points on rationally connected varieties and K3 surfaces, and answers certain cases of a question by Poonen about iterating the étale-Brauer set.
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| Bei |
Degenerating Hermitian metrics and spectral geometry of the canonical bundle
Let (X, h) be a compact and irreducible Hermitian complex space of complex dimension m. In this talk we will report about some recent results concerning the Hodge-Kodaira Laplacian acting on the space of L^2 sections of the canonical bundle of reg(X), the regular part of X. More precisely, given any L^2 closed extension D of the Dolbeault operator acting on the space of smooth (m, 0)-forms with compact support in reg(X), we will consider the operator D*D, which is a self-adjoint extension of the Hodge-Kodaira Laplacian. We will show that D*D has discrete spectrum and moreover we will provide an estimate for the growth of its eigenvalues. Finally we will give some applications in the setting of complex projective surfaces.
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| Onorati |
Moduli spaces of generalised Kummer varieties are not connected
Using the recent computation of the monodromy group of irreducible holomorphic symplectic (IHS) manifolds deformation equivalent to generalised Kummer varieties, we count the number of connected components of the moduli space of both marked and polarised such manifolds. After recalling basic facts about IHS manifolds, their moduli spaces and parallel transport operators, we show how to construct a monodromy invariant which translates this problem in a combinatorial one and eventually solve this last problem.
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| Stella |
Kac-Moody groups, generalized minors, and quiver representations
Both the representation theories of Kac-Moody groups and quivers, in
the non-finite types, present a tripartite structure.
Representations of a Kac-Moody group G comes naturally in three classes
(positive, zero, and negative level representations) according to the scalar
by which the center of G acts.
Indecomposable representation of a quiver Q are either preprojective,
postinjective, or regular depending on where they sit in the associated
Auslander-Reiten quiver.
We connect these two trialities using cluster algebras. By identifying the
ring of coordinates of an appropriate double Bruhat cell of G as a cluster
algebra we show how cluster variables coming from preprojective (resp.
postinjective and regular) representations of Q can be interpreted as
generalized minors of G arising from positive level (resp. negative level
and 0 level) representations.
No prior knowledge of cluster algebras will be assumed and only simple
notions of representation theory will be used.
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| Viviani |
The cohomology of the Hilbert scheme and of the compactified Jacobians of a singular curve
We generalize the classical MacDonald formula for smooth curves to reduced curves with planar singularities. More precisely, we show that the cohomologies of the Hilbert schemes of points on a such a curve are encoded in the cohomologies of the fine compactified Jacobians of its connected subcurves, via the perverse Leray filtration. The proof uses the equigeneric stratification of the semiuniversal deformation space of the curve to reduce to the case of nodal curves, where everything can be computed explicitly. I will also try to explain the relationship of this result with the Hitchin fibration, PT/BPS enumerative invariants and knot theory. This is a joint work with Luca Migliorini and Vivek Schende.
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