Speakers

This is a joint work with Misha Verbitsky. Let X be an irreducible holomorphic symplectic manifold and z a Beauville-Bogomolov negative integral (1,1)-class. Using the ergodicity of the monodromy action and recent results by Bakker and Lehn, we prove some deformation-invariance statements for the locus covered by rational curves of class z in X.

Two K3 surfaces with maximal Picard number, X_3 and X_4, were considered by Vinberg and called by him 'the most algebraic K3 surfaces'. We investigate IHS manifolds related to them. In a joint work with B. van Geemen, G. Kapustka, M. Kapustka and J. Wisniewski we show that Hilb^2(X_4) is a double cover of a very special EPW sextic, which leads to an example of a complete family of 20 incident planes in the 5-dimensional projective space. In a joint project in progress with G. Kapustka look for similar phenomena for Hilb^2(X_3). In particular, we investigate Vinberg's algorithm for finding the fundamental polyhedron of the action of the group generated by roots in a hyperbolic lattice.

Subvarieties of Grassmannians (and especially Fano varieties) obtained from section of homogeneous vector bundles are far from being classified. A case of particular interest is given by the Fano varieties of K3 type, for their deep links with hyperk\"ahler geometry. This talk will be mainly devoted to the construction of some new examples of such varieties. This is a work in progress with Giovanni Mongardi.

Intersection cohomology of the moduli space of Higgs bundles on a smooth projective curve Let X be a smooth projective curve of genus g over $\mathbb{C}$. The character variety $\mathcal{M}_B$ parametrizing conjugacy classes of representations from the fundamental group of X into $SL(2,\mathbb{C})$ is an affine irreducible singular projective variety. The Non Abelian Hodge theorem states that there is a real analytic isomorphism between $\mathcal{M}_B$ and the quasi projective singular variety $\mathcal{M}_{Dol}$ which parametrizes semistable Higgs bundles of rank 2 and degree 0 on X. During the seminar I will present desingularizations of these moduli spaces which are analogous to the one which construct the O’ Grady six dimensional example. This will allow me to study the intersection cohomology of $\mathcal{M}_{Dol}$, by mean of the Decomposition Theorem.

Theory of Newton-Okounkov convex bodies extends methods of toric geometry to non-toric varieties. The Newton-Okounkov convex body of a line bundle on a variety depends on a valuation on the field of rational functions. For classical groups SL(n), SO(n) and Sp(2n), we define uniformly geometric valuations on the corresponding complete flag varieties. The valuation in every type comes from a natural coordinate system on the open Schubert cell, and is combinatorially related to the Gelfand-Zetlin pattern in the same type. For SL(n) and Sp(2n), we identify the corresponding Newton-Okounkov polytopes with the Feigin-Fourier-Littelmann-Vinberg polytopes. For SO(n), we compute low-dimensional examples and formulate open problems. All necessary definitions will be given in the talk.

I'll discuss some work in progress in collaboration with M.A. de Cataldo and J. Heinloth. Let C be a nonsingular projective curve of genus >1, and let n and d be two coprime integers. Given the moduli space M of stable Higgs bundles (E,\Phi), where E is a vector bundle on C of rank n and degree d, and \Phi: E \to E\otimes K_C is the Higgs field, there is an associated Hitchin fibration \chi: M \to A, where A is the affine space parameterizing the spectral curves C_a \subset T^\vee C associated to Higgs bundles. The map \chi is projective, hence, by the decomposition theorem of Beilinson Bernstein Deligne and Gabber, R\chi_* Q_M splits in a direct sum of (shifted) intersection cohomology complexes. We determine completely the supports of those intersection cohomology complexes which intersect the locus A_red parameterizing reduced spectral curves. We show in particular that, for every partition \lambda=(\lambda_1, ... , \lambda_r) of n, there is a string of summands appearing in the decomposition of R\chi_* Q_M which is supported on the locus A_\lambda where the spectral curve splits into r reducible components mapping to C with degrees \lambda_1, ... , \lambda_r. The proof relies on the technique of "higher discriminants", developed in collaboration with V. Shende, and on some results on the universal family of compactified jacobians, due to V. Shende, F. Viviani and myself.

Given a smooth cubic fourfold, one can associate a fibration whose fibres are twisted intermediate Jacobians of smooth linear sections. Such a variety can be compactified and the resulting projective variety is an irreducible holomorphic symplectic manifold of OG10 type. In this talk we recall the construction of these varieties and we explain how to construct a distinguished polarisation. This is then used to show that the Picard group always contains a hyperbolic plane. Finally, we point out some possible consequences to the geometry of OG10 manifolds.

I will discuss finite groups acting by automorphisms and birational automorphisms. We will see that they are bounded provided that the base field is a function field, and also make some observations on their structure in the general case.

Algebraic version of LeBrun-Salamon conjecture states that every projective Fano smooth variety with contact structure on tangent bundle is homogeneous space. One possible way to look for counterexamples could be to first ease the smoothness assumption. Unfortunately it is not obvious how we should define contact structure on singular variety, especially if we want to preserve relations between contact and symplectic structures. I will present some results related to this problem, along with examples based on toric geometry.

Amplitude Modulation (AM) and Frequency Modulation (FM) are two different technologies of broadcasting radio signals. AM works by modulating the amplitude of the signal with constant frequency. In FM technology the information is encoded by varying the frequency of the wave with amplitude being constant. Given a complex projective variety X with an ample line bundle L and an action of an algebraic torus we can study it in two complementary ways: (1) by examining amplitude of L on curves on X (AM technology) and (2) by understanding weights of a linearization of L on fixed point components of the action of the torus (FM technology).