60 research outputs found
Geometry and arithmetic of Maschke's Calabi-Yau three-fold
Maschke's Calabi-Yau three-fold is the double cover of projective three space branched along Maschke's octic surface. This surface is defined by the lowest degree invariant of a certain finite group acting on a four-dimensional (4D) vector space. Using this group, we show that the middle Betti cohomology group of the three-fold decomposes into the direct sum of 150 2D Hodge substructures. We exhibit 1D families of rational curves on the three-fold and verify that the associated Abel-Jacobi map is non-trivial. By counting the number of points over finite fields, we determine the rank of the Neron-Severi group of Maschke's surface and the Galois representation on the transcendental lattice of some of its quotients. We also formulate precise conjectures on the modularity of the Galois representations associated to Maschke's three-fold (these have now been proven by M. Schutt) and to a genus 33 curve, which parametrizes rational curves in the three-fold
Multi-Centered Invariants, Plethysm and Grassmannians
Motivated by multi-centered black hole solutions of Maxwell-Einstein theories
of (super)gravity in D=4 space-time dimensions, we develop some general
methods, that can be used to determine all homogeneous invariant polynomials on
the irreducible (SL_h(p,R) x G4)-representation (p,R), where p denotes the
number of centers, and SL_h(p,R) is the "horizontal" symmetry of the system,
acting upon the indices labelling the centers. The black hole electric and
magnetic charges sit in the symplectic representation R of the generalized
electric-magnetic (U-)duality group G4. We start with an algebraic approach
based on classical invariant theory, using Schur polynomials and the Cauchy
formula. Then, we perform a geometric analysis, involving Grassmannians,
Pluecker coordinates, and exploiting Bott's Theorem. We focus on non-degenerate
groups G4 "of type E7" relevant for (super)gravities whose (vector multiplets')
scalar manifold is a symmetric space. In the triality-symmetric stu model of
N=2 supergravity, we explicitly construct a basis for the 10 linearly
independent degree-12 invariant polynomials of 3-centered black holes.Comment: 1+29 pages, 6 Table
Modular Forms and Three Loop Superstring Amplitudes
We study a proposal of D'Hoker and Phong for the chiral superstring measure
for genus three. A minor modification of the constraints they impose on certain
Siegel modular forms leads to a unique solution. We reduce the problem of
finding these modular forms, which depend on an even spin structure, to finding
a modular form of weight 8 on a certain subgroup of the modular group. An
explicit formula for this form, as a polynomial in the even theta constants, is
given. We checked that our result is consistent with the vanishing of the
cosmological constant. We also verified a conjecture of D'Hoker and Phong on
modular forms in genus 3 and 4 using results of Igusa.Comment: 25 page
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
Modular symbols and Hecke operators
We survey techniques to compute the action of the Hecke operators on the
cohomology of arithmetic groups. These techniques can be seen as
generalizations in different directions of the classical modular symbol
algorithm, due to Manin and Ash-Rudolph. Most of the work is contained in
papers of the author and the author with Mark McConnell. Some results are
unpublished work of Mark McConnell and Robert MacPherson.Comment: 11 pp, 2 figures, uses psfrag.st
Endothelial Focal Adhesions Are Functional Obstacles for Leukocytes During Basolateral Crawling
Age-dependent changes in geometry, tissue composition and mechanical properties of fetal to adult cryopreserved human heart valves
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