Moduli spaces of irreducible symplectic varieties, cubics and Enriques surfaces

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Schedule:
Titles and abstracts Nicolas Addington (Duke University): On derived categories of some moduli spaces of torsion sheaves on K3 A couple of years ago I constructed an autoequivalence of D(Hilb ^{n}(K3)) using the fact the universal sheaf is a ``relative P^{n-1}-object'' over the K3. Related ideas are used in forthcoming work of Markman and Mehrotra to construct a ``non-commutative K3 surface'' associated to any deformation of the Hilbert scheme. In joint work with Ciaran Meachan and Will Donovan, we can construct the analogous autoequivalence for certain moduli spaces M of torsion sheaves. The picture is more geometric than for Hilbert schemes -- the equivalence factors as a product of Namikawa-Kawamata equivalences for Mukai flops and Fourier-Mukai-Arinkin equivalences for Lagrangian torus fibrations. In some cases where M is a Mukai flop of a Hilbert scheme, we can show that the new equivalence and the old one are conjugate by Namikawa-Kawamata, which suggests a possible approach to extending Markman-Mehrotra's construction to general hyperkaehlers. Alexei Bondal (Steklov Mathematical Institute): Lagrangian geometry and harmonic analysis of local systems on full bipartite graph We consider algebraic geometric problems of possible configurations of lines in the complex vector space. A particular case gives Barth-Nieto quintic as the space of solutions. We will show how it is related to representation theory and to harmonic analysis on graphs. We will discuss symplectic and Lagrangian geometry of the problem and reinterpretation in terms of critical points of an appropriate functional similar to Landau-Ginzburg potential in mirror symmetry. This talk is based on joint work with Ilya Zhdanovskiy. Ugo Bruzzo (SISSA): Stacky partial compactifications of moduli spaces of instantons on ALE spaces Moduli spaces of framed sheaves on the complex projective plane are a resolution of singularities of the moduli space of ideal instantons on the 4-sphere. A natural question arises whether similar constructions can be made in the case of other spaces. I will consider instantons on the ALE spaces X of type A _{k}. Here the nontrivial behaviour of the instantons at infinity makes it necessary to consider moduli spaces of framed sheaves on a Deligne-Mumford projective stack whose coarse moduli space is a (singular) toric compactification of X. By instanton counting on this moduli space one can define Nekrasov partition functions, and attempt a proof of the AGT conjecture for ALE spaces.
Olivier Debarre (ENS): Quadratic line complexes In this talk, a quadratic line complex is the intersection, in its Pluecker embedding, of the Grassmannian of lines in an 4-dimensional projective space with a quadric. We study the moduli space of these Fano 5-folds in relation to that of EPW sextics. Gavril Farkas (Humboldt-Universität zu Berlin): Conic bundles and the universal abelian varieties over A _{5}In joint work with Verra we give a very simple unirational parametrization of the universal abelian variety over A _{5} using double
conic bundles in products of projective spaces. As a consequence we obtain
that the boundary divisor of A_{6} is also unirational.Gerard van der Geer (Korteweg-de Vries Instituut): Cycle classes of a stratification on the moduli of K3 surfaces Moduli spaces in positive characteristic can be more accessible that their characteristic 0 counterparts because of the existence of stratifications. We calculate the cycle classes of the strata defined by the height and Artin number on the moduli of K3 surfaces in positive characteristic. The formulas generalize Deuring's famous formula for the number of supertsingular elliptic curves. This is joint work with Torsten Ekedahl. Klaus Hulek (Leibniz Universität Hannover): The Prym map revisited It was already shown by Friedman and Smith that the Prym map does not extend as a regular morphism from the moduli space R _{g+1} of admissible covers to the second Voronoi compactification of the moduli space A_{g} of principally polarized abelian varieties. Due to work of Alexeev, Birkenhake and Hulek as well as Vologodsky it is known that the indeterminacy locus of this map is the closure of the so called Friedman-Smith loci. Motivated by work of Alexeev and Brunyate we investigate the Prym map to other toroidal compactifications of A_{g}, in particular the perfect cone compactification. For this we develop a systematic approach which separates the geometric aspects from combinatorial issues and reduces the problem to the computation of certain monodromy cones.
The motivation for this work comes from the question whether the map which associates to a smooth cubic threefold its intermediate Jacobian can be extended as a regular morphism to suitable compactifications of the space of cubics on the one side and A _{5} on the other side. This is joint work with
Sebastian Casalaina-Martin, Sam Grushevsky and Radu Laza.Ljudmila Kamenova (SUNY Stony Brook): Kobayashi's conjecture for K3 surfaces and hyperkähler manifolds The Kobayashi pseudometric on a complex manifold M is the maximal pseudometric such that any holomorphic map from the Poincare disk to M is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove Kobayashi's conjecture for any hyperkähler manifold that admits a deformation with a Lagrangian fibration, if its Picard rank is not maximal. We shall discuss the proof of Kobayashi's conjecture for K3 surfaces and for certain hyperkähler manifolds. These results are joint with S. Lu and M. Verbitsky. Grzegorz Kapustka (Jagiellonian University): On IHS fourfolds with b _{2}=23In this talk we will discuss the relations between IHS fourfolds with b _{2}=23
and EPW sextics.Ludmil Katzarkov (Universität Wien): Noncommutative Mordell-Lang conjecture In this talk we will develop a parallel between Dynamical Systems and Categories. Alexander Kuznetsov (Steklov Mathematical Institute): Moduli spaces of sheaves on pfaffian cubic fourfolds via derived categories We will discuss the interpretation of the moduli spaces of sheaves on a pfaffian cubic fourfold and their relation to the moduli spaces of sheaves on the associated K3 surface. Manfred Lehn (Johannes Gutenberg Universität Mainz): Cubic curves on cubic fourfolds Generalised twisted cubic curves on smooth cubic fourfold Y that does not contain a plane form a smooth 10-dimensional manifold that admits a contraction M -> Z to an 8-dimensional irreducible hyperkaehler manifold that contains Y as a Langrangian submanifold. Varying Y one obtains a 20-dimensional family of projective 8-dimensional hyperkaehler manifolds. This is a joint project with Christian Lehn, Christoph Sorger and Duco van Straten. Shouhei Ma (Nagoya University): Finiteness of stable orthogonal modular varieties of non-general type We show that there are only finitely many even lattices L of signature (2, n) with n>14, â‰ 16 and containing 2U such that the modular variety associated to the stable orthogonal group of L is not of general type. Eyal Markman (University of Massachusetts): A global Torelli theorem for rigid hyperholomorphic sheaves (Joint work with S. Mehrotra). We construct moduli spaces M of marked triples (X,f,A), where X is an irreducible holomorphic symplectic manifold, f is a marking of X, and A is a stable and infinitesimally rigid reflexive sheaf of Azumaya algebras over the cartesian product X ^{d},
such that the second Chern class of A is invariant under the
diagonal action a finite index subgroup of the monodromy group of X.
We prove a global Torelli theorem: The period map from the
moduli space M to the period domain is a local homeomorphism,
surjective, and generically injective.
The main example is the rank 2n-2 sheaf of Azumaya algebras constructed in arXiv:1105.3223 over the Cartesian square X×X of an irreducible holomorphic symplectic manifold of K3 ^{[n]} deformation type. The characteristic classes of A were used to
prove the standard conjectures when X is algebraic (joint with F. Charles).
The sheaf A is also used to associate to X a generalized (non-commutative)
deformation of the derived category of a K3 surface (joint work in progress
with S. Mehrotra).Daisuke Matsushita (Hokkaido University): On the dual fibration of a Lagrangian fibration We prove that the relative moduli space of locally free sheaves on a Lagrangian fibration over a smooth manifold is smooth and symplectic. As an application, we will discuss a family of intermediate jacobian of cubic threefolds which contain a K3 surface. Riccardo Salvati Manni (Sapienza UniversitÃ di Roma): Coble quartic and the universal Kummer varieties The Kummer surface can be represented as a quartic in P ^{3}
whose coefficients are polynomials of degree 12 in the second order theta
constants.
Such a polynomial of (bi)degree (12,4) is an equation for the
universal Kummer surface.
Using several recent results, we intend to explain as one can extend
this method to higher genera.
In particular we will discuss Coble's hypersurface in genus 3 .
Alessandra Sarti (Université de Poitiers): Lattice theory and moduli spaces of irreducible holomorphic symplectic manifolds I will show recent progress in the description of the moduli spaces of certain irreducible holomorphic symplectic manifolds admitting a lattice polarization. Alexander Tikhomirov (Yaroslavl State Pedagogical University): Degenerations of mathematical instantons The moduli space I(n) of mathematical instantons with c _{2}=n on projective space P^{3} is known to be irreducible
(A.Tikhomirov, 2013) and smooth (M.Jardim and M.Verbitsky, 2014)
for arbitrary n ≥ 1. We study the closure I(n) in the
Gieseker-Maruyama moduli scheme M(n) of semistable torsion-free
rank-2 sheaves with c_{1}=c_{3}=0 and c_{2}=n. We are interested in
those irreducible components of the boundary I(n) - I(n)
which consist of non-locally locally free sheaves. For n=1 this
boundary is well-known and is a quadric divisor in P^{5}. We show
that for any n ≥ 2 this boundary contains at least n-1 irreducible
divisors D_{1},...,D_{n-1}, general points of which, called quasinstantons,
have an explicit description in terms of rational normal cuves and instantons
of charge smaller then n. The picture is completely different from the case
of stable rank-2 bundles with c_{1}=0 on P^{2}, where the boundary of the
moduli space is an irreducible divisor in the Gieseker-Maruyama moduli scheme.
This work is a joint project of M.Jardim, D.Markushevich and A.Tikhomirov.
Misha Verbitsky (Higher School of Economics): Hyperkähler manifolds are non-hyperbolic Using Ratner theorem about ergodic action of arithmetic groups on homogeneous spaces, I prove that holomorphic symplectic Kähler manifolds are never Kobayashi hyperbolic. Claire Voisin (CMLS, École polytechnique): On the fibers of the Abel-Jacobi map We show that for certain 7-nodal quartic double solids X, there does not exist a family of curves for which the Abel-Jacobi map is surjective onto the intermediate Jacobian J(X), with rationally connected fibers. In particular, there is no universal codimension 2 cycle on J(X) × X. This implies that X is not stably rational, while its Artin-Mumford invariant is trivial. Ken-Ichi Yoshikawa (Kyoto University): Equivariant analytic torsion for K3 surfaces with involution I will report recent progresses on the explicit formula for the analytic-torsion invariant for K3 surfaces with involution. Kota Yoshioka (Kobe University): Bridgeland stability and the cone of generalized Kummer manifolds Bridgeland stability is a useful tool to analyse moduli spaces of stable sheaves on abelian surfaces. I would like to explain that the chamber structure of Bridgeland stability is related to the decomposition of positive cone of generalized Kummer manifolds. By using this, I will explain the movable and the nef cones. |