13 research outputs found
Nori 1-motives
Let EHM be Nori's category of effective homological mixed motives. In this
paper, we consider the thick abelian subcategory EHM_1 generated by the i-th
relative homology of pairs of varieties for i = 0,1. We show that EHM_1 is
naturally equivalent to the abelian category M_1 of Deligne 1-motives with
torsion; this is our main theorem. Along the way, we obtain several interesting
results. Firstly, we realize M_1 as the universal abelian category obtained,
using Nori's formalism, from the Betti representation of an explicit diagram of
curves. Secondly, we obtain a conceptual proof of a theorem of Vologodsky on
realizations of 1-motives. Thirdly, we verify a conjecture of Deligne on
extensions of 1-motives in the category of mixed realizations for those
extensions that are effective in Nori's sense
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
Integrality of instanton numbers and p-adic B-model
We study integrality of instanton numbers (genus zero Gopakumar - Vafa invariants) for quintic and other Calabi-Yau manifolds. We start with the analysis of the case when the moduli space of complex structures is one-dimensional; later we show that our methods can be used to prove integrality in general case. We give an expression of instanton numbers in terms of Frobenius map on -adic cohomology ; the proof of integrality is based on this expression