Algebra and geometry seminar



Abstracts
2017/2018

Yohan Brunebarbe
(Zürich)
On the cotangent
bundle of smooth complex algebraic varieties whose fundamental group is
infinite
I will discuss several positivity
properties of the cotangent bundle of the smooth projective complex algebraic
varieties whose fundamental group is infinite, as well as their generalizations
to quasiprojective varieties.
Riccardo Salvati Manni
(Roma “Sapienza”)
Schottky implies Poincaré: an
explicit solution to the weak Schottky problem
For any genus g, we write down a collection of
polynomials in genus g theta constants. They arise by applying a specific SchottkyJung proportionality to an explicit collection of
quartic identities for genus (g1) theta constants. They imply the socalled Poincaré relations, and their common zero locus
contains the locus of Jacobians of genus g curves as an irreducible component.
David Hernandez (IMG – PRG)
Spectra of quantum integrable systems, Langlands
duality and category O
The spectrum of a quantum integrable
system is crucial to understand its properties. Rmatrices give powerful tools
to study such spectra. A better understanding of transfermatrices obtained
from Rmatrices led us to the proof of several results for the corresponding
quantum integrable systems. In particular, their
spectra can be described in terms of "Baxter polynomials". They
appear naturally in the study of a category O of representation of a Borel subalgebra of a quantum
affine algebra. The properties of geometric objects attached to the Langlands dual Lie algebra (the affine opers)
led us to establish new relations in the Grothendieck
ring of this category O, from which one can derive the Bethe Ansatz equations
between the roots of the Baxter polynomials.
This talk is based on joint works with M. Jimbo, E. Frenkel and B. Leclerc.
(Research supported by the European Research Council
under the European Union's Framework Programme H2020 with
ERC Grant Agreement number 647353 Qaffine.)
Claire Voisin (Collège de France)
Segre numbers of
tautological bundles on Hilbert schemes
We establish geometric vanishings in certain ranges
for top Segre classes of tautological bundles of punctual Hilbert schemes of K3
surfaces and K3 surfaces blownup at one point. We show how all these Segre
numbers for any surface and any polarization are formally determined by these
vanishings.
Building on these results, MarianOpreaPandharipande and SzenesVergne
completed the proof of the Lehn conjecture giving a formula for the generating
function determined by these numbers.
Junyan Cao (IMG – PRG)
A decomposition
theorem for projective manifolds with nef anticanonical
bundle
Let X be a
simply connected projective manifold with nef
anticanonical bundle. We prove that X
is a product of a rationally connected manifold and a manifold with trivial
canonical bundle. As an application we describe the MRC fibration
of any projective manifold with nef anticanonical
bundle.
This is a joint work with Andreas Höring.
Sylvain Charpentier
(Columbia)
Rational matrix
differential operators and integrable systems of PDEs
A key feature of integrability
for systems of evolution PDEs ∂_{t}u=F(u),
u=(u_{1},...,u_{k}) is to be part of
an infinite hierarchy of commuting generalized symmetries. In all known
examples, these generalized symmetries are constructed by means of LenardMagri sequences involving a pair of matrix differential
operators (A,B). We show that in the scalar case k=1 a necessary condition for
a pair of differential operators (A,B) to generate a LenardMagri
sequence is that the ratio L=AB^{1} lies in a class of operators which
we call integrable and contains all ratios of
compatible Poisson (or Hamiltonian) structures. We give a sufficient condition
on an integrable pair of matrix differential
operators (A,B) to generate an infinite LenardMagri
sequence when the rational matrix differential operator L= AB^{1} is
weakly nonlocal. If time permits, we will generalize these results to
differentialdifference equations.
Filippo Viviani (Roma 3)
On the cone of
effective cycles on the symmetric products of a curve
I will report on a joint work with F. Bastianelli, A. Kouvidakis and A.
F. Lopez in which we study the cone of (pseudo)effective cycles on symmetric
products of a curve.
We first prove that the diagonal cycles span a face of
the pseudoeffective cone of cycles in any given dimension. Secondly, we look
at the contractibility faces associated to the AbelJacobi morphism towards the
Jacobian and in many cases we are able to compute their dimension. The geometry
of linear series of a curve (e.g. the classical BrillNoether
theory) will play a special role in
our analysis.
Juan Souto (Rennes)
Quasiregular maps: bubbling and extensions
I will recall what are quasiregular
maps and some of the many properties they share with complex functions in one
variable. I will then describe their degenerations, obtaining a picture which
is strongly reminiscent of that in Gromov's
compactness of pseudoholomorphic curves. Understanding these degenerations is
the key new ingredient needed to prove that quasiregular
maps on the sphere admit harmonic extensions to hyperbolic space.
This is join work with Pekka
Pankka.
Christof Geiß (UNAM)
Crystal graphs and semicanonical functions for symmetrizable
Cartan matrices
This is a report on a joint project with B. Leclerc
and J. Schröer.
Associated to a symmetrizable
Cartan matrix, a symmetrizrer
and an orientation we define certain quivers with relations such that the
socalled locally free modules resemble the behavior of a species resp. of its preprojective algebra associated to the same data. The set
of components of maximal dimension for the Efiltered representations of our preprojective algebra have a natural structure of the
crystal B(\infty) associated to the Cartan matrix. In an appropriate
convolution algebra of constructible functions we find
elements which are generically 1 on a given component of this set, and 0 on all
other components. Unfortunately, the generators of our algebra do not fulfill
the Serre relations. However, we conjecture that the
support of the functions which belong to the Serre
ideal does not have maximal dimension.
Rita Pardini
(Pisa)
Sistemi lineari su varietà di
dimensione di Albanese massima
Sia X una varietà
complessa, liscia e proiettiva, sia A una varietà abeliana e sia a:X→A
un morfismo genericamente finito sull'immagine. Dato un fibrato lineare L su X,
si studiano i sistemi lineari L⊗P, dove P è un elemento generale di
Pic^{0}(A) e si mostra che, a meno di cambio base con una opportuna
mappa di moltiplicazione A→A, l'applicazione data da L⊗P è indipendente da
P e induce una fattorizzazione del morfismo a. Se L=K_{X} è il
fibrato canonico di X, questa fattorizzazione è un nuovo oggetto
geometrico intrinsecamente associato a X.
Questo teorema di
fattorizzazione consente di migliorare, sotto opportune ipotesi sul morfismo a,
le disuguaglianze di tipo CliffordSeveri tra gli invarianti numerici di L.
Uno strumento chiave per
ottenere questi risultati è l'introduzione della funzione “rango
continuo”.
(Lavoro in collaborazione
con M.A. Barja e L. Stoppino.)
Nathan Reading (NCSU)
To scatter or to
cluster?
Scattering diagrams arose in the algebraicgeometric
theory of mirror symmetry. Recently,
Gross, Hacking, Keel, and Kontsevich applied
scattering diagrams to prove many longstanding conjectures about cluster
algebras. Scattering diagrams are
certain collections of codimension1 cones, each weighted with a formal power
series. In this talk, I will
introduce cluster scattering diagrams and their connection to cluster algebras,
focusing on rank$2$ (i.e. 2dimensional) examples.
Even 2dimensional cluster scattering diagrams are not
wellunderstood in general. I will
show how the twodimensional "affinetype" cases can be constructed
using cluster algebras and describe a surprising appearance of the Narayana
numbers in the twodimensional affine case. If time allows, I will discuss a
general proof that consistent scattering diagrams cut space into a fan.
Thomas Schick (Göttingen)
Minimal hypersurfaces
and positive scalar curvature
There is a long history to find relations between the
topology of a smooth manifold and its (Riemannian) geometry. The first such is the
GaussBonnet theorem which says that the Euler characterestic
of a compact 2dimensional surface without boundary is (up to a positive
constant) the scalar curvature of that manifold.
Particular conclusion: if the Euler characterestic is not positive (i.e. if the surface is not
a sphere or a real projective plane) then there is not metric such that the
scalar curvature is everywhere positive.
The use of the Dirac operator allows to obtain similar
obstructions to the existence of positive scalar curvature in higher dimension;
but only for spin manifolds (as otherwise this operator doesn't exist).
There is one further approach, invented by Schoen and Yau, which does not rely on the spin condition, but rather
uses minimal hypersurfaces.
We will present this approach and its main
implications.
There are two crucial problems with this approach:
·
in its initial incarnation, it requires regularity
results on minimal hypersurfaces which are available only in dimension less
than 8;
·
it needs a large integral first homology.
We will report on current work which aims to overcome
part of these problems, due to SchoenYau for the
first problem, and developped in joint work with Cecchini for some aspects of the second problem.
Specifically, we will introduce and discuss the case of "enlargeable manfolds" (as introduced by Gromov
and Lawson).
Drazen Adamovic
(Zagreb)
On realizations of
simple affine vertex algebras and their modules
We present some new explicit realizations of simple,
nonrational affine vertex algebras, affine Walgebras and their modules.
We discuss the existence of Whittaker, logarithmic and
indecomposable modules for simple affine vertex algebras.
Giuseppe Pareschi (Roma “Tor Vergata”)
Funzioni di rango coomologico
su varietà abeliane e applicazioni
Su varietà abeliane
è possibile definire la dimensione dei gruppi di coomologia
di fasci (o più generalmente complessi di fasci) algebrici coerenti tensorizzati con potenze razionali di una polarizzazione.
Questo dà luogo a funzioni di Hilbert
razionali che verificano una naturale formula di trasformazione rispetto al
funtore di FourierMukaiPoincaré.
In alcune situazioni queste funzioni sembrano contenere interessanti
informazioni geometriche, cosa che cercherò di mostrare calcolando
alcuni esempi. Si tratta di un lavoro in collaborazione con Zhi
Jiang.
Lorenzo Foscolo (HWU
Edinburgh)
Complete noncompact
G2manifolds from asymptotically conical CalabiYau
3folds
G2manifolds are the Riemannian 7manifolds with G2 holonomy and in many respects can be regarded as
7dimensional analogues of CalabiYau 3folds. In
joint work with Mark Haskins and Johannes Nordström
we construct infinitely many families of new complete noncompact G2 manifolds
(only four such manifolds were previously known). The underlying smooth
7manifolds are all circle bundles over asymptotically conical CalabiYau 3folds. The metrics are circleinvariant and
have an asymptotic geometry that is the 7dimensional analogue of the geometry
of 4dimensional ALF hyperkähler metrics. After
describing the main features of our construction I will concentrate on some
illustrative examples, describing how recent results in CalabiYau
geometry about isolated singularities and their resolutions can be used to
produce examples of complete G2manifolds.
Michel Brion (Grenoble)
Homogeneous vector
bundles on Abelian varieties via representation theory
The objects of the talk are the translationinvariant
vector bundles on an Abelian variety. They form a tensor Abelian category which
is equivalent to the category of coherent sheaves with finite support on the
dual Abelian variety, via the FourierMukai transform.
The talk will present an alternative approach to
homogeneous vector bundles, in terms of representations of a commutative affine
group scheme attached to the Abelian variety. Some natural operations on vector
bundles such as tensor product, dual, pushforward and pullback under
isogenies, can be described in simple terms via this approach.
Sébastien Boucksom
(CMLSCNR)
Valuations, filtrations and Kstability
The notion of Kstability of a polarized projective variety, initially introduced in the context of Kähler geometry as
the algebrogeometric counterpart to the existence of special Kähler metrics of constant curvature, has recently proved to be a key notion in moduli problems.
Checking Kstability of a given variety is however a difficult problem, and I will present in this talk an approach to Kstability
based on valuations and nonArchimedean geometry,
introduced by Jonsson and myself and further developped by Fujita, Li, Odaka.
Henri Guenancia (CNRS+Toulouse)
Polystability of the tangent sheaf of singular varieties and
applications
If X is a compact Kähler
manifold, then the existence of a KählerEinstein
metric on X yields a lot of interesting properties for the tangent bundle T_{X},
among which stability, Chern numbers inequalities or
vanishing/parallelism of holomorphic tensors. I will explain how to generalize
these results to singular varieties in a suitable sense. This relies partially
on joint works with B. Taji and D. Greb & S. Kebekus.
18 July 2018
Speaker (Institution)
Title
Abstract