58 research outputs found
Lattice Polarized K3 Surfaces and Siegel Modular Forms
The goal of the present paper is two-fold. First, we present a classification
of algebraic K3 surfaces polarized by the lattice H+E_8+E_7. Key ingredients
for this classification are: a normal form for these lattice polarized K3
surfaces, a coarse moduli space and an explicit description of the inverse
period map in terms of Siegel modular forms. Second, we give explicit formulas
for a Hodge correspondence that relates these K3 surfaces to principally
polarized abelian surfaces. The Hodge correspondence in question underlies a
geometric two-isogeny of K3 surfaces
Normal forms for Kummer surfaces
We determine normal forms for the Kummer surfaces associated with abelian
surfaces of polarization of type , , , , and
. Explicit formulas for coordinates and moduli parameters in terms of
Theta functions of genus two are also given. The normal forms in question are
closely connected to the generalized Riemann identities for Theta functions of
Mumford's.Comment: 49 page
Modular Invariants for Lattice Polarized K3 Surfaces
We study the class of complex algebraic K3 surfaces admitting an embedding of
H+E8+E8 inside the Neron-Severi lattice. These special K3 surfaces are
classified by a pair of modular invariants, in the same manner that elliptic
curves over the field of complex numbers are classified by the J-invariant. Via
the canonical Shioda-Inose structure we construct a geometric correspondence
relating K3 surfaces of the above type with abelian surfaces realized as
cartesian products of two elliptic curves. We then use this correspondence to
determine explicit formulas for the modular invariants.Comment: 29 pages, LaTe
The Sen Limit
F-theory compactifications on elliptic Calabi-Yau manifolds may be related to
IIb compactifications by taking a certain limit in complex structure moduli
space, introduced by A. Sen. The limit has been characterized on the basis of
SL(2,Z) monodromies of the elliptic fibration. Instead, we introduce a stable
version of the Sen limit. In this picture the elliptic Calabi-Yau splits into
two pieces, a P^1-bundle and a conic bundle, and the intersection yields the
IIb space-time. We get a precise match between F-theory and perturbative type
IIb. The correspondence is holographic, in the sense that physical quantities
seemingly spread in the bulk of the F-theory Calabi-Yau may be rewritten as
expressions on the log boundary. Smoothing the F-theory Calabi-Yau corresponds
to summing up the D(-1)-instanton corrections to the IIb theory.Comment: 41 pp, 1 figure, LaTe
Six line configurations and string dualities
We study the family of K3 surfaces of Picard rank sixteen associated with the
double cover of the projective plane branched along the union of six lines, and
the family of its Van Geemen-Sarti partners, i.e., K3 surfaces with special
Nikulin involutions, such that quotienting by the involution and blowing up
recovers the former. We prove that the family of Van Geemen-Sarti partners is a
four-parameter family of K3 surfaces with
lattice polarization. We describe explicit Weierstrass models on both families
using even modular forms on the bounded symmetric domain of type . We also
show that our construction provides a geometric interpretation, called
geometric two-isogeny, for the F-theory/heterotic string duality in eight
dimensions. As a result, we obtain novel F-theory models, dual to non-geometric
heterotic string compactifications in eight dimensions with two non-vanishing
Wilson line parameters.Comment: 42 pages; minor typos corrected in version
Special function identities from superelliptic Kummer varieties
We prove that the factorization of Appell's generalized hypergeometric series
satisfying the so-called quadric property into a product of two Gauss'
hypergeometric functions has a geometric origin: we first construct a
generalized Kummer variety as minimal nonsingular model for a product-quotient
surface with only rational double points from a pair of superelliptic curves of
genus with . We then show that this generalized Kummer
variety is equipped with two fibrations with fibers of genus . When
periods of a holomorphic two-form over carefully crafted transcendental
two-cycles on the generalized Kummer variety are evaluated using either of the
two fibrations, the answer must be independent of the fibration and the
aforementioned family of special function identities is obtained. This family
of identities can be seen as a multivariate generalization of Clausen's
Formula. Interestingly, this paper's finding bridges Ernst Kummer's two
independent lines of research, algebraic transformations for the Gauss'
hypergeometric function and nodal surfaces of degree four in .Comment: 46 pages, 2 figure
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