Algebra and geometry seminar
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 quasi-projective 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 Schottky-Jung proportionality to an explicit collection of quartic identities for genus (g-1) theta constants. They imply the so-called 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. R-matrices give powerful tools to study such spectra. A better understanding of transfer-matrices obtained from R-matrices 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 blown-up 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, Marian-Oprea-Pandharipande and Szenes-Vergne 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 ∂tu=F(u), u=(u1,...,uk) is to be part of an infinite hierarchy of commuting generalized symmetries. In all known examples, these generalized symmetries are constructed by means of Lenard-Magri 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 Lenard-Magri 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 Lenard-Magri sequence when the rational matrix differential operator L= AB-1 is weakly non-local. If time permits, we will generalize these results to differential-difference 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 pseudo-effective cone of cycles in any given dimension. Secondly, we look at the contractibility faces associated to the Abel-Jacobi 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 Brill-Noether 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 pseudo-holomorphic 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 so-called 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 E-filtered 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 Pic0(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=KX 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 Clifford-Severi 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 algebraic-geometric 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 codimension-1 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. 2-dimensional) examples.
Even 2-dimensional cluster scattering diagrams are not well-understood in general. I will show how the two-dimensional "affine-type" cases can be constructed using cluster algebras and describe a surprising appearance of the Narayana numbers in the two-dimensional 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 Gauss-Bonnet theorem which says that the Euler characterestic of a compact 2-dimensional 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 Schoen-Yau 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, non-rational affine vertex algebras, affine W--algebras 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 Fourier-Mukai-Poincaré. 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 non-compact G2-manifolds from asymptotically conical Calabi-Yau 3-folds
G2-manifolds are the Riemannian 7-manifolds with G2 holonomy and in many respects can be regarded as 7-dimensional analogues of Calabi-Yau 3-folds. In joint work with Mark Haskins and Johannes Nordström we construct infinitely many families of new complete non-compact G2 manifolds (only four such manifolds were previously known). The underlying smooth 7-manifolds are all circle bundles over asymptotically conical Calabi-Yau 3-folds. The metrics are circle-invariant and have an asymptotic geometry that is the 7-dimensional analogue of the geometry of 4-dimensional ALF hyperkähler metrics. After describing the main features of our construction I will concentrate on some illustrative examples, describing how recent results in Calabi-Yau geometry about isolated singularities and their resolutions can be used to produce examples of complete G2-manifolds.
Michel Brion (Grenoble)
Homogeneous vector bundles on Abelian varieties via representation theory
The objects of the talk are the translation-invariant 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 Fourier-Mukai 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, push-forward and pull-back under isogenies, can be described in simple terms via this approach.
Sébastien Boucksom (CMLS-CNR)
Valuations, filtrations and K-stability
The notion of K-stability of a polarized projective variety, initially introduced in the context of Kähler geometry as the algebro-geometric counterpart to the existence of special Kähler metrics of constant curvature, has recently proved to be a key notion in moduli problems. Checking K-stability of a given variety is however a difficult problem, and I will present in this talk an approach to K-stability based on valuations and non-Archimedean 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 Kähler-Einstein metric on X yields a lot of interesting properties for the tangent bundle TX, 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