Séminaire Méditerranéen de Géométrie Algébrique, 10–11/09/2024 home
Speakers:
Vladimiro Benedetti (Université Côte-d'Azur)
Juan Carlos Naranjo Del Val (Universidad de Barcelona)
Joao-Pedro Dos Santos (Institut Montpellierain Alexander Grothendieck)
Sorin Dumitrescu (Université Côte-d'Azur)
Fulvio Gesmundo (Paul Sabatier University)
Lucas Li Bassi (Università di Genova)
Yingqi Liu (University of Chinese Academy of Sciences)
Victor Lozovanu (Università di Genova)
Jinxing Xu (University of Science and Technology of China)
Schedule:
10/9
9:30 welcome coffee and registration
10:15 Naranjo
11:25 Li Bassi
14:30 Gesmundo
15:30 coffee break
16:00 Liu
17:10 Dumitrescu
20:00 social dinner at PEsciolino
11/9
9:45 Dos Santos
10:45 coffee break
11:25 Xu
14:30 Lozovanu
15:30 coffee break
16:00 Benedetti
Titles and abstracts:
Vladimiro Benedetti - Quantum K-théory of symplectic Grassmannians
From a complex projective variety X, one can construct different algebraic structures encoding various information. In this talk we will focus on the Grothendieck ring, which encodes the behaviour of coherent sheaves on X. This ring can be seen as a generalization of the intersection ring in cohomology; one can also define a "quantum Grothendieck ring", which is both a deformation of the classical one and a finer avatar of the quantum cohomology ring. In order to understand the structure of this ring, one needs to understand the rationality properties of the moduli space of rational curves on X. In this talk, I will present some techniques allowing to compute the quantum Grothendieck ring of a particular class of homogeneous varieties, the symplectic Grassmannians. It is a joint work with Nicolas Perrin and Weihong Xu.
Juan Carlos Naranjo Del Val - Simplicity of Jacobians attached to cyclic hyperelliptic coverings
We study an explicit $(2g-1)$-dimensional family of Jacobian varieties of dimension $\frac{d-1}2 (g-1)$. This appears naturally when considering unramified cyclic coverings of prime degree $d$ of hyperelliptic curves of genus $g\ge 2$. By using a deformation argument, we prove that the generic element of the family is simple. Furthermore, we completely describe their endomorphism algebra, and we show that they admit a rank $\frac{d-1}2-1$ group of non-polarized automorphisms. We apply this results to give a generic Torelli theorem for the Prym varieties of cyclic coverings of hyperelliptic curves of odd prime degree under some slight numerical restrictions. This result generalizes in several directions previous results on genus 2.
This is a joint work with A. Ortega, G.P. Pirola and I. Spelta.
Joao-Pedro Dos Santos - Distributions with locally free tangent sheaf
In 2008 F. Cukierman and J. V. Pereira published a study of the locus $\mathrm{Dec}$, where the tangent sheaf of a family of foliations in $\mathbf{P}^n$ is decomposable, i.e. a sum of line bundles. They concluded that, once the singular locus has sufficiently large codimension, $\mathrm{Dec}$ turns out to be open. I shall report on work done with J. V. Pereira on the theme.
In it, we study the locus $\mathrm{LF}$ of points of a family of distributions where the tangent sheaf is locally free. Through general Commutative Algebra, we show that $\mathrm{LF}$ is open provided that singularities have codimension at least three.
This, in turn, is applied to families in $\mathbf{P}^n$ and in $\mathcal B$, the variety of Borel subgroups of a simple group. With the help of a theorem putting in bijection irreducible components of the space of subalgebras of a given semi-simple Lie algebra and its nilpotent orbits, we conclude that the space of foliations of rank two, on $\mathbf{P}^n$ and $\mathcal B$, may have quite many irreducible components.
Sorin Dumitrescu - Holomorphic foliations with no transversely projective structure
This talk deals with holomorphic foliations on compact complex manifolds.
According to a result of Ghys nonsingular codimension one holomorphic foliations on
compact complex torii are either translation invariant or turbulent. We prove that on
the product of two elliptic curves a generic nonsingular turbulent holomorphic foliation
does not admit any transversely holomorphic projective structure. The talk is based on
a joint paper with Indranil Biswas (University Shiv Nadar, New Delhi).
Fulvio Gesmundo - The subrank of a tensor in geometry and invariant theory
Tensor subrank is a notion dual to the classical notion of tensor rank. It measures how much a tensor can be diagonalized by applying linear maps on its tensor factors. In this talk, I will introduce this notion, and explain its connections to the classical geometry and invariant theory of small orbits in tensor spaces. If time permits, I will discuss applications in computational complexity and in the resource theory of entanglement.
Lucas Li Bassi - Lines on a singular cubic fourfold
Since their discovery, the construction of irreducible holomorphic
symplectic (IHS) manifolds has generated great interest in the mathematical community. One of the most well-known methods to construct
an example in dimension 4 is by considering the Fano variety of lines
on a smooth cubic fourfold. Beauville and Donagi proved that this is
an IHS fourfold of type K3[2]. This construction becomes even more
intriguing when considering mildly singular cubic fourfolds, such as cubic fourfolds Y that are triple covers of P4 branched over a singular
cubic threefold. In this case, F(Y), the Fano variety of lines on Y, is
birational to an IHS manifold of type K3[2]. This fact has been used by
Boissière–Camere–Sarti and myself to study certain compactifications
of the moduli spaces of irreducible holomorphic symplectic manifolds
with an order three non-symplectic automorphism. However, to achieve
these results, the authors did not fully consider the rich geometry of
the singular variety F(Y).
I will present some results obtained in collaboration with Samuel Boissière
and Paola Comparin that explain how the geometry of F(Y) provides a
deeper understanding of the relationship between cyclic cubic fourfolds
and IHS manifolds of type K3[2] with a non-symplectic automorphism
of order three.
Yingqi Liu - Geometry of Hermitian cubes: after Freudenthal, Vinberg
and Bahargava
The orbit space of the action of GL_2 × SL_2^{\times 3} on (C2)^{\otimes 4} naturally classifies codimension
two linear sections of P^1 × P^1 × P^1 inside its Segre embedding. There are two perspectives to realize
its GIT quotient space, which is known as wP(1, 2, 2, 4): one is Vinberg’s theory of θ-groups, the
other one is Bahargava’s story on hypercubes. Sitting in the third row of geometric Freudenthal’s
magic square, one gets four more subadjoint varieties apart from P^1 × P^1 × P^1. They are related to
representations on Hermitian cubes. In this talk, I will explain how the story extends in both ways to
Hermitian cubes. As a main result, we get the same GIT quotient space of nonsingular codimension
two linear sections of the above five subadjoint varieties.
Victor Lozovanu - Infinitesimal positivity aspects
Going back to the early nineties Demailly had the idea to quantify positivity aspects of global data infinitesimally at a given point of an algebraic variety. The picture turned out to be incredibly rich and well structured. In this talk we will try to explain how one can futher enrich this line of research by studying positivity aspects of line bundles at a point through the convex geometry of Newton-Okounkov bodies. This talk is based on a current joint work with M. Fulger.
Jinxing Xu - Commuting scheme, Chevalley restriction theorem and Hitchin morphism
The classical Chevalley restriction theorem asserts that for a semisimple complex Lie group G, the ring of G-invariant polynomials on the Lie algebra g is isomorphic through restriction to the ring of Weyl group invariant polynomials on the Cartan subalgebra. In studying the Hitchin morphism of principal G-Higgs bundles over higher dimensional varieties, Tsao-Hsien Chen and Ngo Bao Chau proposed a conjectured multi-variable version of the Chevalley restriction theorem, and they related it to the image of the Hitchin morphism. I will explain their program and show a proof of this generalised Chevalley restriction theorem for classical groups. Based on joint work with Lei Song and Xiaopeng Xia.
Séminaire Méditerranéen de Géométrie Algébrique, 10–11/09/2024 home