Variétés holomorphiquement symplectiques, leurs espaces de modules et formes automorphes – VHSMOD-2009

Theory of irreducible symplectic varieties (ISV) has two aspects, geometric and arithmetic. The geometric aspect includes the description of families of ISVs which possess particular geometric properties or are obtained by specific constructions. From the point of view of moduli theory, different geometric properties or ways of constructing the ISVs correspond to different types of lattice polarization of the ISVs. The period domain of the ISVs being an arithmetic quotient of Siegel's symmetric domain of type IV, the moduli spaces of ISVs with fixed polarization type can be studied in terms of the arithmetic of indefinite lattices and the associated automorphic forms. Automorphic forms with prescribed asymptotics on the boundary are interpreted as pluricanonical forms on the moduli space. The typical results expected are that the moduli space of polarized ISVs is of general type for sufficiently big degree of polarization. Some results of this kind were obtained by Kondo, Gritsenko, Hulek, Sankaran earlier for moduli of K3 and abelian surfaces. These results will be extended not only to moduli of ISVs, but also to other arithmetic quotients of symmetric domains, which are not period domains of known Kähler manifolds, though all of them are moduli of appropriate conformal field theories. The automorphic forms arising in this work are interesting themselves and have other applications in number theory (exact formulas for infinite products), in the theory of Kac-Moody or vertex algebras, and in theoretical physics. Constructions of these forms will be studied, related to the automorphic products of Borcherds type or their pullbacks, theta-blocks and theta-quarks. The questions which relate the geometric and arithmetic aspects are: translate geometric properties of ISVs into numerical properties of their Picard (polarization) lattices; study the orbits of each relevant type of lattices under the monodromy and the orthogonal group; represent each orbit by a geometric construction. One of the objectives of the project is a search of new constructions and generalizations, including singular ISVs, compactifications of open Lagrangian fibrations, fixed loci of finite symplectic automorphism groups in holomorphically symplectic manifolds, and moduli spaces of sheaves over non-symplectic manifolds, on which a symplectic structure is defined via the Atiyah class. More general moduli spaces can be treated on the equal base, for example, noncommutative models of certain Calabi-Yau 4-folds.