Modular Forms and Geometry of Modular Varieties

IMPA, Rio de Janeiro, May 4-8, 2015.

Many moduli spaces in algebraic geometry can be constructed as quotients of homogeneous domains by arithmetic groups. Among the best known examples are the moduli spaces of principally polarized Abelian varieties or of polarized K3 surfaces. The existence of automorphic forms with special properties often encodes much information about the geometry of the moduli spaces. For example special automorphic forms can often be used to determine whether certain moduli spaces are of general type or have negative Kodaira dimension. For the construction of forms with special properties, Borcherds modular forms play an essential role. At the same time automorphic forms can be often used to describe the Picard group of moduli spaces or, more generally, modular varieties. In this activity we want to explore some of the interactions between modular forms and the geometry of modular varieties.

Confirmed Specialists taking part of this concentration period are:

 

Schedule

Monday, May 4
Auditorium 01

9.30 - 10.30
Fundamental groups of toroidal compactifications
G.K. Sankaran

11.00 - 12.15
Moduli of polarized Enriques surfaces
K. Hulek

 

Tuesday, May 5
Auditorium 01

9.30 - 11.00
Analytic torsion invariant for K3 surfaces with involution
K.-I. Yoshikawa

 

Wednesday, May 6
Auditorium 01

9.30 - 10.45
The monodromy group of generalized Kummer varieties
G. Mongardi

11.15 - 12.15
Polarization types of Lagrangian fibrations
B. Wieneck

 

Thursday, May 7
Auditorium Ricardo Mañé

9.30 - 10.30
Finiteness of 2-reflective lattices of signature (2,n)
S. Ma

11:00 - 12.15
A simple vector valued modular form with applications in algebraic geometry and modular forms
R. Salvati-Manni

 

Titles and Abstracts

Finiteness of 2-reflective lattices of signature (2,n)
Shouhei Ma
Tokyo Inst. Technology

A modular form for an even lattice L of signature (2,n) is said to be 2-reflective if its zero divisor is set-theoretically contained in the Heegner divisor defined by the (-2)-vectors in L. We prove that there are only finitely many even lattices with n>6 which admit 2-reflective modular forms. In particular, there is no such lattice in n>29. This answers a conjecture of Gritsenko and Nikulin in n>6.

 

Fundamental groups of toroidal compactifications
Gregory Sankaran
U. Bath

I shall describe joint work with Azniv Kasparian (Sofia) in which we compute the fundamental group of a toroidal compactification of a hermitian locally symmetric space D/Γ, without assuming either that Γ is neat or that it is arithmetic.

 

Monodromy of irreducible symplectic manifolds
Giovanni Mongardi
U. Milano

Exploiting recent results on the ample cone of irreducible symplectic manifolds, we provide a different point of view for the computation of their monodromy groups. In particular, we give the final step in the computation of the monodromy group for generalised Kummer manifolds and we prove that the monodromy of O'Grady's ten dimensional manifold is smaller than what was expected.

 

Polarization types of Lagrangian fibrations
Benjamin Wieneck
Leibniz U. Hannover

The generic fiber of a Lagrangian fibration on an irreducible holomorphic symplectic manifold is an abelian variety. This fact is used to construct a deformation invariant, called the polarization type, for every Lagrangian fibration which is essentially a polarization type of a polarized generic fiber. Conjecturally this invariant should only depend on the deformation class of the total space. Indeed for K3^[n]-type fibrations it is always a principal polarization which can be shown using methods developed by E. Markman.

 

Analytic torsion invariant for K3 surfaces with involution
Ken-ichi Yoshikawa
Kyoto U

I will report a recent progress on the structure of the analytic invariant of K3 surfaces with involution, which was introduced about 10 years ago. I will give an explicit formula for the invariant for all deformation types. This is a joint work with Shouhei Ma. If time permits, I will explain the role of this invariant in mirror symmetry and some conjectural consequences.

 

Moduli of polarized Enriques surfaces
Klaus Hulek
Leibniz U. Hannover

In this talk I will discuss moduli spaces of polarized and numerically polarized Enriques surfaces. Over the complex numbers these spaces can be related to homogeneous forms and modular forms can be used to gain insight into the geometry of these moduli spaces. This is joint work with V. Gristsenko.

 

A simple vector valued modular form with applications in algebraic geometry and modular forms
Riccardo Salvati Manin
U. Roma La Sapienza

I will report on joint work with Dalla Piazza, Fiorentino , Grushevsky and Perna. We use the image, under the Gauss map, of smooth two-torsion points of the theta divisor of a ppav to construct holomorphic differential forms on Siegel modular varieties and to characterize the locus of decomposable abelian varieties.