For admissible d, we construct a rational map f_d from the moduli space of special Gushel–Mukai fourfolds of discriminant d to the moduli space of degree-d (twisted) polarized K3 surfaces. We use this to count the number of 4-dimensional fibers of Fourier–Mukai partners of very general special Gushel–Mukai fourfolds with associated (twisted) K3 surface. For a very general K3 surface S, the degree of its polarization determines whether S (potentially) lies in the image of some f_d. For twisted K3 surfaces, this is not true. We give a geometric criterion for when a very general twisted K3 surface is (potentially) associated to a Gushel-Mukai fourfold. This is based on an older work with Laura Pertusi and work in progress with Kuan-Wen Lai.

I will present a work in progress with William Sarem, dealing with the problem of giving algebraic characterizations of Baily-Borel-Satake compactifications of quotients of the ball (or other bounded symmetric domains). This discussion is naturally related to constructing holomorphic maps between « stratified holomorphic spaces », that appear as "Satake completions" of either the universal cover of a given quasi-projective variety, or the target bounded symmetric domain.

We will start by discussing the definition of the Lefschetz defect delta(X) for a (smooth, complex) Fano variety X, and by recalling its properties: when delta(X)>3 the variety X is a product, while when delta(X)=3 there is a structure theorem for X. Then we will present some results on the geometry of Fano varieties with Lefschetz defect 2. 

A celebrated conjecture by Debarre, which is now known to be true, states that the cotangent bundle of a general complete intersection in projective space is ample when its codimension is at least half the dimension of the ambient space. Generalizing this concept, Diverio and Trapani conjectured 16 years ago a corresponding threshold for the ampleness of invariant jet differential bundles of higher order and degree. This expectation was naturally supported by both elementary vanishing and non-vanishing theorems, alongside Schneider's classical obstruction. 

In this talk, we present explicit counterexamples showing that this conjecture fails for large weights. Keeping things simple, we will focus on the first fundamental case: we show that on every smooth 3-dimensional hypersurface (in a 4-dimensional projective space), the invariant jet bundle of order 3 is never ample for any sufficiently divisible weight. The mechanism behind this counterexample relies on analyzing the filtration by degree in the highest derivative. The top graded piece yields an extremal quotient bundle whose Schur decomposition cannot support global sections due to Bruckmann-Rackwitz vanishing, contradicting the global generation required by ampleness.

I will report on a joint work with I. Biswas, E. Colombo and A. Ghigi in which, given an etale double cover C'---> C of a smooth complex projective curve C of genus g>6,  we construct a twisted meromorphic bidifferential on CxC and we prove that it gives the second fundamental form of the Prym map. We show that the restriction of this twisted bidifferential to the second infinitesimal neighborhood of the diagonal in CxC induces a projective structure on C, that depends on the double cover. In this way we obtain a section of the space of projective structures on R_g and the (0,1)-component of the differential of this section is proven to be the pullback via the Prym map of the Kaehler form on A_{g-1}.  This generalises a previous result obtained in collaboration with Biswas, Colombo, and Pirola in the case of M_g, showing the existence of a canonical projective structure on every curve of genus g>3, obtained by the second fundamental form of the Torelli map.

Given a quiver Q, consider the stack M of all Q-representations. In the Grothendieck ring of stacks over M, we can define a Hall product by "convolution along extensions". This leads to the definition of the motivic Hall algebra of Q-representations. Choosing a stability condition for Q-representations, we can consider the class of the stack of semistable representations (of some fixed slope) in a completion of the motivic Hall algebra. In this algebra, such a class is invertible, and we present a (conjectural) formula for its inverse. Should we have sufficient time, we present an application to compute the classes of Hilbert schemes of points on Kleinian singularities in the Grothendieck ring of varieties. This is work in progress, joint with Okke van Garderen.

I will talk about joint work with Carolina Tamborini and Matteo Costantini, in which we establish an Arakelov inequality for variations of Hodge structures underlying families of principally polarised complex Abelian varieties. In the equality case, this leads to a numerical characterisation of certain totally geodesic ball quotients inside the moduli space of Abelian varieties. Our result extends work of Möller-Viehweg-Zuo by removing the strong positivity conditions on the log-canonical and the log-cotangent bundle imposed in their statement. For families over compact base spaces, our proof involves showing in a first step that the period map associated with a family of Abelian varieties factors through certain operations of the MMP and then generalising the results of Möller, Viehweg, and Zuo to the resulting singular setting. In the quasiprojective case, we develop a new approach and execute it in dimension two: it first descends the family to a very specific partially ample model and then uses symmetric space theory and semistability considerations to detect totally geodesic ball quotients in the moduli space considered with the locally symmetric orbifold Kähler metric.

We present a basic setup for the study of logarithmic holomorphic symplectic varieties along with some first results in this direction: a local Torelli theorem, deformation theoretic results, some examples and counterexamples. This is joint work with Ben Bakker.

The entropy of an automorphism f of a smooth projective variety X is a nonnegative real number measuring, in a coarse sense, the dynamical complexity of f. For K3 surfaces, automorphisms of zero entropy are precisely those which either have finite order or preserve an elliptic fibration; by contrast, automorphisms of positive entropy exhibit substantially more chaotic dynamics. Following pioneering work of McMullen, Cantat, Nikulin and others, it has become clear that the entropy encodes deep geometric information about the automorphism f, the underlying variety X and the structure of its automorphism group. In this talk, I will explain how dynamical methods allow us to obtain a classification of K3 surfaces with virtually abelian automorphism group. I will then explain how this circle of ideas extends to hyperkähler manifolds, and compute the smallest positive entropy arising from birational automorphisms of hyperkähler manifolds of the known deformation types. These are joint works with S. Brandhorst and G. Martin.

We provide a structure theorem for Kummer type hyperkähler fourfolds equipped with a polarization of Beauville-Bogomolov square 2 and divisibility 2, and discuss several aspects of their moduli space. The key point is a specialization to generalized Kummer fourfolds of genus 2 Jacobians. This is a joint work with Franco Giovenzana and Jieao Song.

Generalized Fermat manifolds (GFM) are defined by a system of Fermat-type equations and extend the classical Fermat manifolds, which are given by a single equation. The aim of this talk is to study certain quotients of these manifolds. In particular, we focus on GFMs that are Calabi–Yau manifolds and investigate specific automorphisms of finite order acting on them.

We show that the resulting quotients are either singular Calabi–Yau varieties or Log Enriques varieties of index 2. Log Enriques varieties have been introduced recently by several authors; they can be viewed as singular analogues of Enriques manifolds, which themselves generalize Enriques surfaces. Moreover, we prove that, with only a few exceptions, these quotients have terminal singularities. 

The results are contained in a joint work in progress with S. Billi (Saarbrücken) and A. Palomino (Talca).

For a Fano variety, it follows from the Mori’s cone theorem that the nef cone is rational polyhedral. As for a variety with trivial canonical class, the Morrison-Kawamata conjecture predicts that the effective nef cone is rational polyhedral up to the action of the automorphism group of the variety. A natural question one can ask is: do we have a unified description of the effective nef cone for a larger class of varieties containing both Fano varieties and varieties with trivial canonical class?

In this talk I will present joint work with Vladimir Lazić and Isabel Stenger providing an answer in dimension two. I will start by introducing the class of klt Calabi-Yau generalised pairs, which includes all smooth varieties with nef anticanonical class. Then I will explain why the automorphism group is no longer sufficient in this broader setting and describe, in dimension two, the group of Cremona isometries that takes its place. Finally, I will discuss how the problem is reduced to considering surfaces obtained by blowing up nine points on the projective plane and illustrate some pathological phenomenon that occurs on this kind of surfaces.