20 research outputs found
Extending Torelli map to toroidal compactifications of Siegel space
It has been known since the 1970s that the Torelli map ,
associating to a smooth curve its jacobian, extends to a regular map from the
Deligne-Mumford compactification to the 2nd Voronoi
compactification .
We prove that the extended Torelli map to the perfect cone (1st Voronoi)
compactification is also regular, and moreover
and share a common Zariski open
neighborhood of the image of . We also show that the map to the
Igusa monoidal transform (central cone compactification) is NOT regular for
; this disproves a 1973 conjecture of Namikawa.Comment: To appear in Inventiones Mathematica
The Schottky problem in genus five
In this paper, we present a solution to the Schottky problem in the spirit of
Schottky and Jung for genus five curves. To do so, we exploit natural incidence
structures on the fibers of several maps to reduce all questions to statements
about the Prym map for genus six curves. This allows us to find all components
of the big Schottky locus and thus, to show that the small Schottky locus
introduced by Donagi is irreducible.Comment: 20 page
Comparing Perfect and 2nd Voronoi decompositions: the matroidal locus
We compare two rational polyhedral admissible decompositions of the cone of
positive definite quadratic forms: the perfect cone decomposition and the 2nd
Voronoi decomposition. We determine which cones belong to both the
decompositions, thus providing a positive answer to a conjecture of V. Alexeev
and A. Brunyate. As an application, we compare the two associated toroidal
compactifications of the moduli space of principal polarized abelian varieties:
the perfect cone compactification and the 2nd Voronoi compactification.Comment: 27 pages, 2 figures, final version, to appear in Mathematische
Annale
Complete moduli of cubic threefolds and their intermediate Jacobians
The intermediate Jacobian map, which associates to a smooth cubic threefold
its intermediate Jacobian, does not extend to the GIT compactification of the
space of cubic threefolds, not even as a map to the Satake compactification of
the moduli space of principally polarized abelian fivefolds. A much better
"wonderful" compactification of the space of cubic threefolds was constructed
by the first and fourth authors --- it has a modular interpretation, and
divisorial normal crossing boundary. We prove that the intermediate Jacobian
map extends to a morphism from the wonderful compactification to the second
Voronoi toroidal compactification of the moduli of principally polarized
abelian fivefolds --- the first and fourth author previously showed that it
extends to the Satake compactification. Since the second Voronoi
compactification has a modular interpretation, our extended intermediate
Jacobian map encodes all of the geometric information about the degenerations
of intermediate Jacobians, and allows for the study of the geometry of cubic
threefolds via degeneration techniques. As one application we give a complete
classification of all degenerations of intermediate Jacobians of cubic
threefolds of torus rank 1 and 2.Comment: 56 pages; v2: multiple updates and clarification in response to
detailed referee's comment
Local Calabi-Yau manifolds of type \tilde{A} via SYZ mirror symmetry
We carry out the SYZ program for the local Calabi--Yau manifolds of type
by developing an equivariant SYZ theory for the toric
Calabi--Yau manifolds of infinite-type. Mirror geometry is shown to be
expressed in terms of the Riemann theta functions and generating functions of
open Gromov--Witten invariants, whose modular properties are found and studied
in this article. Our work also provides a mathematical justification for a
mirror symmetry assertion of the physicists Hollowood--Iqbal--Vafa.Comment: v3: 43 pages, 12 figures, improved expositio