Conférence finale de l'ANR FanoHK

It is known since Wierzba–Wisniewski's work in 2003 that a birational map between holomorphic symplectic fourfolds is a composition of Mukai flops. Hu and Yau conjectured that in any dimension, a birational map is a composition of Mukai elementary transformations in codimension two, that is, maps which locally look like a product of a 4-dimensional flop with the identity on a polydisc, ``up to codimension three or higher''. Call a birational map f from X to X' a ``Hu-Yau transformation'' , if there are proper closed subsets Z, Z' of codimension three or higher such that f induces a Mukai transformation in codimension two between X\Z and X'\Z'. We show that any birational map between irreducible holomorphic symplectic manifolds is a product of Hu-Yau transformations, and give an example showing that a stronger version of the conjecture cannot be true: it is not possible to decompose a given map into a product of Mukai elementary transformation after discarding some codimension three closed subsets from the source and the target. Joint work with A. Soldatenkov and M. Verbitsky.

Vector bundles on curves and quiver representations have moduli theories that are very similar. Inspired by this connection, I will recall the canonical 4-term sequence on quiver moduli and its relationship to the Kodaira-Spencer morphism. This sequence can be used to prove a conjecture of Schofield, establish the rigidity of quiver moduli, describe their infinitesimal symmetries, and show that the universal representation furnishes a fully faithful Fourier-Mukai functor. These are all statements that have already been established for moduli of vector bundles on curves. The stacky perspective on these moduli spaces and Teleman quantization will play an important role in the proof. This is joint work with Ana-Maria Brecan, Hans Franzen, Gianni Petrella, and Markus Reineke.

In this talk I will report on a joint project with Michele Bolognesi, Daniele Faenzi and Laurent Manivel. In this project we investigated the geometric relationship between some theta representations constructed by Vinberg, and families of curves of low genus. Given a theta representation, one can construct a family of curves, and by using degeracy loci constructions, also the family of corresponding Jacobians and/or various moduli spaces of vector bundles on the curve itself. This process also allows to recover the so-called Coble hypersurfaces, which are hypersurfaces whose singular locus is exactly the Jacobian of the curve.

In fact, we will be mostly interested in the family parametrizing Coble hypersurfaces, which is the "Gopel" variety associated to the chosen theta representation. I will present some results concerning the birational geometry of these Gopel varieties, focusing on specific examples (hyperelliptic curves, genus two and genus three).

A database of Fano fourfolds described as zero loci of homogeneous vector bundles, coming with numerical invariants such as Hodge numbers, can be obtained under Macaulay2.

I will give a short presentation of a web interface that allows a flexible and accessible search in the database, that provides moreover a page for each entry, where it is possible to detail a geometric description, allowing a future collaborative encyclopedia of such fourfolds. This is a collaboration with E. Fatighenti, L. Manivel, F. Tanturri, and the web-designer R. Duran.

We will present some classification results for (smooth, complex) Fano 4-folds X with Picard number rho(X)>6. First of all, if rho(X)>9, then X is a product of del Pezzo surfaces; this is sharp, since we know one family of Fano 4-folds with rho(X)=9 that is not a product of surfaces. In the range rho(X)=7,8,9, we will explain some partial classification results, based on a detailed and explicit study of the geometry of X using birational geometry in the framework of the MMP. In particular, if rho(X)>6 and X has no small contractions, then either X is a product of surfaces, or rho(X)=7,8,9 and X is a blow-up of a cubic 4-fold along rho(X)-1 planes that intersect pairwise at a point.

We exhibit examples of slope-stable and modular vector bundles on a hyperkähler manifold of K3^[2]-type. These are obtained by performing standard linear algebra constructions on the examples studied by O’Grady of (rigid) modular bundles on the Fano varieties of lines of a general cubic 4-fold and the Debarre-Voisin hyperkähler. Interestingly enough, these constructions are almost never infinitesimally rigid, and more precisely we show how to get (infinitely many) 20 and 40 dimensional families. This is a joint work with Claudio Onorati. Time permitting, I will also present a joint work with Alessandro D'Andrea and Claudio Onorati on a connection between discriminants of vector bundles on smooth and projective varieties and representation theory of GL(n).

A Nikulin surface is a minimal resolution of a quiotient of a K3 surface S through a symplectic involution i or equivalently a component of the fixed locus of the induced symplectic action of i on S^[2].

We describe K3 surfaces that are fixed loci of a symplectic involution on hyper-Kahler manifolds that are deformations of S^[2]; we call them generalised Nikulin surfaces. This is a joint work in progress with C.Camere, A.Garbagnati and M. Kapustka

We will present a lattice theoretic criterion for a pair of hyper-Kähler fourfolds of K3^[2]-type and Picard rank 1 to be derived equivalent. The criterion is analogous to the Mukai-Orlov criterion for K3 surfaces. It is proven by means of deforming Fourier-Mukai kernels of so-called BKR equivalences between Hilbert schemes of points on K3 surfaces along twistor paths. We will then investigate explicit examples. This is joint work with Grzegorz Kapustka.

Together with Debarre, we proved that the polarized Hodge structures of the primitive cohomology of double EPW sextics are isomorphic (up to Tate twist) to the vanishing cohomology of Gushel–Mukai fourfolds. On the other hand, Kapustka, Kapustka, and Mongardi showed that the primitive cohomology of double EPW cubes are isomorphic to the primitive cohomology of double EPW sextics. In the talk I will explain a direct argument proving that the primitive cohomology of double EPW cubes are isomorphic (up to Tate twist) to the vanishing cohomology of GM fourfolds, using a 20-nodal octic surface to relate them.

This is a joint work with Kapustka, Kapustka, and Mongardi.

The Tate-Shafarevich group of an elliptic K3 surface S with a section parametrizes twists of S, that is, other elliptic K3 surfaces that are étale locally isomorphic to S. By a result of Artin and Tate, this group is in fact isomorphic to Br(S). The notion of twist has been recently generalized (in several ways) to higher-dimensional hyperkähler manifolds admitting a Lagrangian fibration. I will present joint work with D. Huybrechts in which we study the Tate-Shafarevich group of fibrations in Jacobians of curves of arbitrary genus on K3 surfaces, and its relation to the (special) Brauer group.

It is known that the Hilbert square Hilb(S) of a general polarized genus 16 K3 surface S is isomorphic to a Debarre-Voisin 4-fold associated to some 3-form, but an explicit construction of the 3-form is still missing. In this project, we construct a potential 3-form following the work of Frédéric Han, and provide some evidence of the validity of such a 3-form. This is done by incorporating the role of the Fourier-Mukai partner S’, which is closely related to Mukai’s projective models for genus 16 K3 surfaces.

We construct an explicit complex Gushel-Mukai threefold and prove that it is not rational by showing that its intermediate Jacobian has a faithful action of PSL(2,11). Along the way, we construct Gushel-Mukai varieties of various dimensions with rather large (finite) automorphism groups. This is joint work with Olivier Debarre.

The non (stable) rationality of very general smooth Fano complete intersections \(X^n\subset\mathbb P^{n+c}\) of dimension \(n\geq 3\) and fixed type \((d_1,\ldots, d_c)\) has been recently proved in many cases by several authors. Then one may ask under which conditions there might exist rational examples of that type/dimension like for cubic fourfolds or for complete intersections of three quadrics in \(\mathbb P^7\).

After briefly recalling the state of the art, we shall present some geometric considerations of birational nature suggesting a neat statement, which collects the actual knowledge of the subject and which might provide a different perspective/interpretation on the very general (stable) irrationality results.

A famous theorem of Shokurov states that a general anticanonical divisor of a smooth Fano threefold is a smooth K3 surface. In a joint work with Andreas Höring we proved that for four-dimensional Fano manifolds the behaviour is completely opposite: if the anticanonical base locus is a normal surface, all the anticanonical divisors are singular.

In this talk I will present our follow-up result, namely the classification of smooth Fano fourfolds with scheme-theoretic base locus a smooth surface: they form 22 families. I will also mention a result on elliptic Calabi-Yau threefolds that we obtained as a technical step in our study.

We study semiorthogonal decompositions of derived categories of smooth projective varieties and their behavior under birational transformations. Motivated by Kuznetsov’s conjecture on cubic fourfolds and Kontsevich’s vision of canonical decompositions, we introduce the notion of G-atomic theory: canonical, mutation-equivalence classes of G-invariant semiorthogonal decompositions compatible with derived contractions with respect to a group G-action. We prove the existence of such a theory in dimension ≤ 2 for any group G, thereby establishing Kontsevich’s conjecture in this case. This framework refines and extends previous work by Auel–Bernardara and yields a complete birational classification of geometrically rational surfaces over perfect fields in terms of their atoms (canonical building blocks of the derived category). Connections are drawn to the atomic decompositions of quantum cohomology by Katzarkov–Kontsevich–Pantev–Yu. This is joint work in progress with Alexey Elagin and Julia Schneider.

I will discuss an ascending filtration on the Chow group of zero-cycles on a smooth projective variety obtained roughly by considering the successive kernels of the iterates of some modified diagonal embedding of the variety. This filtration is particularly relevant in the case of abelian varieties and of hyper-Kähler varieties, where it is expected to be opposite to the conjectural Bloch-Beilinson filtration. In the case of abelian varieties, it can in fact be described explicitly in terms of the Beauville decomposition, while in the case of hyper-Kähler varieties, I conjecture (and prove in some cases) that it coincides with a filtration introduced earlier by Claire Voisin. As a by-product we obtain in joint work with Olivier Martin a criterion involving second Chern classes for two effective zero-cycles on a moduli space of stable objects on a K3 surface to be rationally equivalent, generalising a result of Marian-Zhao.

A Fano Manifold X over an algebraically closed field K of characteristic 0 is rationally connected, hence its Chow group of zero-cycles is trivial. When the field K is not algebraically closed, the group of zero-cycles of degree 0 on X is of torsion but is nontrivial in general. The CH₀-group of X is generated by L-points of X for some finite extensions L of K. I will discuss an improvement of a theorem by Coray on the degrees of points of cubic surfaces, boundedness results for zero-cycles on del Pezzo surfaces, inpired by Colliot-Thélène's work, and if time permits, unboundedness results for zero-cycles on higher dimensional Fano manifolds.