777 research outputs found
Co-abelian toroidal compactifications of torsion free ball quotients
Let X' be the toroidal compactification of the quotient of the complex 2-ball
by a torsion free lattice G of SU(2,1). We say that X'is co-abelian if there is
an abelian surface, birational to X'. The present work can be viewed as an
illustration for the presence of a plenty of non-compact co-abelian torsion
free toroidal compactifications. More precisely, it shows that all the
admissible values for the volume of a torsion free quotient of the complex
2-ball are attained by co-abelian Picard modular ones over Eisenstein numbers.
The article provides three types of infinite series of finite unramified
coverings of co-abelian, torsion free, Picard modular toroidal
compactifications over Eisenstein numbers, with infinitely increasing volumes.
The first type is supported by mutually birational members with fixed number of
cusps. The second kind is with mutually birational terms and infinitely
increasing number of cusps. The third kind of series relates mutually
non-birational toroidal compactifications with infinitely increasing number of
cusps.Comment: submitted to Duke Journal of Mathematic
Perspectives on the construction and compactification of moduli spaces
In these notes, we introduce various approaches (GIT, Hodge theory, and KSBA)
to constructing and compactifying moduli spaces. We then discuss the pros and
cons for each approach, as well as some connections between them.Comment: 32 pages; notes for the "Compactifying Moduli Spaces" school,
Barcelona (May 2013
Fundamental groups of toroidal compactifications
We compute the fundamental group of a toroidal compactification of a
Hermitian locally symmetric space , without assuming either that
is neat or that it is arithmetic. We also give bounds for the first
Betti number.Comment: Final version. Fixes error pointed out by M. Roessler, leading to
slightly but significantly changed statements: improved notatio
The geometry of Siegel modular varieties
This is a survey article about Siegel modular varieties over the complex
numbers. It is written mostly from the point of view of moduli of abelian
varieties, especially surfaces. We cover compactification of Siegel modular
varieties; classification of the compactified varieties by Kodaira dimension,
etc.; moduli of abelian surfaces and especially applications of the lifting of
Jacobi forms to modular forms; projective models of some special Siegel modular
3-folds; non-principally polarized abelian surfaces; and constructing
degenerating families of abelian varieties.Comment: 66 pages, LaTeX, xyfig. References updated, section VI.2 rewritten
and updated, other minor correction
Chern classes of automorphic vector bundles, II
We prove that the -adic Chern classes of canonical extensions of
automorphic vector bundles, over toroidal compactifications of Shimura
varieties of Hodge type over , descend to classes in the
-adic cohomology of the minimal compactifications. These are invariant
under the Galois group of the -adic field above which the variety and the
bundle are defined.Comment: 28 page
Moduli of K3 Surfaces and Irreducible Symplectic Manifolds
The name "K3 surfaces" was coined by A. Weil in 1957 when he formulated a
research programme for these surfaces and their moduli. Since then, irreducible
holomorphic symplectic manifolds have been introduced as a higher dimensional
analogue of K3 surfaces. In this paper we present a review of this theory
starting from the definition of K3 surfaces and going as far as the global
Torelli theorem for irreducible holomorphic symplectic manifolds as recently
proved by M. Verbitsky. For many years the last open question of Weil's
programme was that of the geometric type of the moduli spaces of polarised K3
surfaces. We explain how this problem has been solved. Our method uses
algebraic geometry, modular forms and Borcherds automorphic products. We
collect and discuss the relevant facts from the theory of modular forms with
respect to the orthogonal group O(2,n). We also give a detailed description of
quasi pull-back of automorphic Borcherds products. This part contains
previously unpublished results. We apply our geometric-automorphic method to
study moduli spaces of both polarised K3 surfaces and irreducible symplectic
varieties.Comment: 71 pages. Fairly minor revisions. To appear in "Handbook of Moduli
Igusa's modular form and the classification of Siegel modular threefolds
In this paper we prove two results concerning the classification of Siegel
modular threefolds. Let A_{1,d}(n) be the moduli space of abelian surfaces with
a (1,d)-polarization and a full level-n structure and let A_{1,d}^{lev}(n) be
the space where one has fixed an additional canonical level structure. We prove
that A_{1,d}(n) is of general type if (d,n)=1 and n ist at least 4. This is the
best possible result which one can prove for all d simultaneously. Let p be an
odd prime and assume that (p,n)=1. Then we prove that the Voronoi
compactification of A_{1,p}^{lev}(n) is smooth and has ample canonical bundle
if and only if n is greater than or equal to 5.Comment: 14 page
Perfect forms and the moduli space of abelian varieties
Perfect quadratic forms give a toroidal compactification of the moduli space
of principally polarized abelian g-folds that is Q-factorial and whose ample
classes are characterized, over any base. In characteristic zero it has
canonical singularities if g is at least 5, and is the canonical model (in the
sense of Mori and Reid) if g is at least 12.Comment: 20 page
Chow-Kunneth decomposition for universal families over Picard modular surfaces
We discuss the existence of an absolute Chow-Kuenneth decomposition for
complete degenerations of families of Abelian threefolds with complex
multiplication over a particular Picard Modular Surface studied by Holzapfel.
In addition to the work of Gordon, Hanamura and Murre we use Relatively
Complete Models in the sense of Mumford-Faltings-Chai of Picard Modular
Surfaces in order to describe complete degenerations of families of abelian
varieties. We furthermore prove vanishing results for cohomology groups of
irreducible representations of certain arithmetic subgroups in SU(2,1) using
the non--compact Simpson type correspondence between the --Higgs
cohomology of the underlying VHS and the --de Rham cohomology resp.
intersection cohomology of local systems.Comment: Final versio
Quasi-modular forms from mixed Noether-Lefschetz theory
The Gromov-Witten theory of threefolds admitting a smooth K3 fibration can be
solved in terms of the Noether-Lefschetz intersection numbers of the fibration
and the reduced invariants of a K3 surface. Toward a generalization of this
result to families with singular fibers, we introduce completed
Noether-Lefschetz numbers using toroidal compactifications of the period space
of elliptic K3 surfaces. As an application, we prove quasi-modularity for some
genus 0 partition functions of Weierstrass fibrations over ruled surfaces, and
show that they satisfy a holomorphic anomaly equation.Comment: 20 pages, final version, minor changes to the proof of Lemma 2
- …