arXiv subjects

Adrian Clingher

Publications and source records attributed to Adrian Clingher.

26 records · Page 2Linked to original sources

arXiv2015

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 $2r-1$ with $r \in \mathbb{N}$. We then show that this generalized Kummer variety is equipped with two fibrations with fibers of genus $2r-1$. 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 $\mathbb{P}^3$.

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arXiv2015

The 14th case VHS via K3 fibrations

We present a study of certain singular one-parameter subfamilies of Calabi-Yau threefolds realized as anticanonical hypersurfaces or complete intersections in toric varieties. Our attention to these families is motivated by the Doran-Morgan classification of variations of Hodge structure which can underlie families of Calabi-Yau threefolds with $h^{2,1} = 1$ over the thrice-punctured sphere. We explore their torically induced fibrations by $M$-polarized K3 surfaces and use these fibrations to construct an explicit geometric transition between an anticanonical hypersurface and a nef complete intersection through a singular subfamily of hypersurfaces. Moreover, we show that another singular subfamily provides a geometric realization of the missing "14th case" variation of Hodge structure from the Doran-Morgan list.

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arXiv2010

Note on a Geometric Isogeny of K3 Surfaces

The paper establishes a correspondence relating two specific classes of complex algebraic K3 surfaces. The first class consists of K3 surfaces polarized by the rank-sixteen lattice H+E_7+E_7. The second class consists of K3 surfaces obtained as minimal resolutions of double covers of the projective plane branched over a configuration of six lines. The correspondence underlies a geometric two-isogeny of K3 surfaces.

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arXiv2010

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.

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arXiv2006

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.

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arXiv2005

On K3 Surfaces with Large Complex Structure

We discuss a notion of large complex structure for elliptic K3 surfaces with section inspired by the eight-dimensional F-theory/heterotic duality in string theory. This concept is naturally associated with the Type II Mumford partial compactification of the moduli space of periods for these structures. The paper provides an explicit Hodge-theoretic condition for the complex structure of an elliptic K3 surface with section to be large. We also discuss certain geometrical implications of this large complex structure condition in terms of the Kodaira types of the singular fibers of the elliptic fibration.

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arXiv2005

Heterotic String Data and Theta Functions

The data required for heterotic string theory with gauge group G, which for anomaly cancellation reasons must either be E8 x E8 or Spin(32)/Z2, consist of the following: a ten-dimensional space-time X and a principal G-bundle P. The fields in the theory consist of: a metric on X, a connection on P and a B-field which is, roughly speaking, a two-form on X. As in a gauge theory, these fields are considered up to gauge equivalence. This means that the connection is considered up to bundle isomorphism, that the metric is considered up to diffeomorphism equivalence, and that the two-form B is considered up to equivalence by adding exact two-forms. The fact that B has a gauge invariance implies that it is not in fact a globally defined two-form but rather must be given as a gerbe-like object. The purpose of this paper is to give a precise definition of the mathematical nature of the B-field following the recent work of Freed and then to work out a description of the moduli space of 8-dimensional Euclidean invariant classical solutions to the string Equations of Motion in the case when space-time is compactified along a two-torus.

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arXiv2003

Mathematics Underlying the F-Theory/Heterotic String Duality in Eight Dimensions

One of the dualities in string theory, the F-theory/heterotic string duality in eight dimensions, predicts an interesting correspondence between two seemingly disparate geometrical objects. On one side of the duality there are elliptically fibered K3 surfaces with section. On the other side, one finds elliptic curves endowed with certain flat connections and complexified Kahler classes. This paper is part of a project aimed at establishing the rigorous mathematical results describing the geometry underlying the classical aspects of this duality. The task involves understanding and comparing the classical moduli spaces on the two sides.

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