44 research outputs found
The modularity of the Barth-Nieto quintic and its relatives
The moduli space of (1,3)-polarized abelian surfaces with full level-2
structure is birational to a double cover of the Barth-Nieto quintic. Barth and
Nieto have shown that these varieties have Calabi-Yau models Z and Y,
respectively. In this paper we apply the Weil conjectures to show that Y and Z
are rigid and we prove that the L-function of their common third \'etale
cohomology group is modular, as predicted by a conjecture of Fontaine and
Mazur. The corresponding modular form is the unique normalized cusp form of
weight 4 for the group \Gamma_1(6). By Tate's conjecture, this should imply
that Y, the fibred square of the universal elliptic curve S_1(6), and Verrill's
rigid Calabi-Yau Z_{A_3}, which all have the same L-function, are in
correspondence over Q. We show that this is indeed the case by giving explicit
maps.Comment: 30 pages, Latex2
A Picard modular fourfold and the Weyl group W(E6)
We study the geometry of a Picard modular fourfold which parametrizes abelian fourfolds of Weil type for the field of cube roots of unity. We find a projective model of this fourfold as a singular, degree ten, hypersurface X in projective 5-space. The Weyl group W(E6) acts on X and we provide an explicit description of this action. Moreover, we describe various special subvarieties as well as the boundary of X
Determinantal Characterization of Canonical Curves and Combinatorial Theta Identities
We characterize genus g canonical curves by the vanishing of combinatorial
products of g+1 determinants of Brill-Noether matrices. This also implies the
characterization of canonical curves in terms of (g-2)(g-3)/2 theta identities.
A remarkable mechanism, based on a basis of H^0(K_C) expressed in terms of
Szego kernels, reduces such identities to a simple rank condition for matrices
whose entries are logarithmic derivatives of theta functions. Such a basis,
together with the Fay trisecant identity, also leads to the solution of the
question of expressing the determinant of Brill-Noether matrices in terms of
theta functions, without using the problematic Klein-Fay section sigma.Comment: 35 pages. New results, presentation improved, clarifications added.
Accepted for publication in Math. An
Homological Type of Geometric Transitions
The present paper gives an account and quantifies the change in topology
induced by small and type II geometric transitions, by introducing the notion
of the \emph{homological type} of a geometric transition. The obtained results
agree with, and go further than, most results and estimates, given to date by
several authors, both in mathematical and physical literature.Comment: 36 pages. Minor changes: A reference and a related comment in Remark
3.2 were added. This is the final version accepted for publication in the
journal Geometriae Dedicat
Mirror duality and noncommutative tori
In this paper, we study a mirror duality on a generalized complex torus and a
noncommutative complex torus. First, we derive a symplectic version of Riemann
condition using mirror duality on ordinary complex tori. Based on this we will
find a mirror correspondence on generalized complex tori and generalize the
mirror duality on complex tori to the case of noncommutative complex tori.Comment: 22pages, no figure
The F-BAR protein pacsin2 inhibits asymmetric VE-cadherin internalization from tensile adherens junctions
Vascular homoeostasis, development and disease critically depend on the regulation of endothelial cell-cell junctions. Here we uncover a new role for the F-BAR protein pacsin2 in the control of VE-cadherin-based endothelial adhesion. Pacsin2 concentrates at focal adherens junctions (FAJs) that are experiencing unbalanced actomyosin-based pulling. FAJs move in response to differences in local cytoskeletal geometry and pacsin2 is recruited consistently to the trailing end of fast-moving FAJs via a mechanism that requires an intact F-BAR domain. Photoconversion, photobleaching, immunofluorescence and super-resolution microscopy reveal polarized dynamics, and organization of junctional proteins between the front of FAJs and their trailing ends. Interestingly, pacsin2 recruitment inhibits internalization of the VE-cadherin complex from FAJ trailing ends and is important for endothelial monolayer integrity. Together, these findings reveal a novel junction protective mechanism during polarized trafficking of VE-cadherin, which supports barrier maintenance within dynamic endothelial tissue
Quotients of Fermat curves and a Hecke character
We explicitly identify infinitely many curves which are quotients of Fermat curves. We show that some of these have simple Jacobians with complex multiplication by a non-cyclotomic field. For a particular case we determine the local zeta functions with two independent methods. The first uses Jacobi sums and the second applies the general theory of complex multiplication, we verify that both methods give the same result
The modularity of the Barth-Nieto quintic and its relatives
Consiglio Nazionale delle Ricerche - Biblioteca Centrale - P.le Aldo Moro, 7 Rome / CNR - Consiglio Nazionale delle RichercheSIGLEITItal