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Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds

herausgegeben von: Radu Laza, Matthias Schütt, Noriko Yui

Verlag: Springer New York

Buchreihe : Fields Institute Communications

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

In recent years, research in K3 surfaces and Calabi–Yau varieties has seen spectacular progress from both arithmetic and geometric points of view, which in turn continues to have a huge influence and impact in theoretical physics—in particular, in string theory. The workshop on Arithmetic and Geometry of K3 surfaces and Calabi–Yau threefolds, held at the Fields Institute (August 16-25, 2011), aimed to give a state-of-the-art survey of these new developments. This proceedings volume includes a representative sampling of the broad range of topics covered by the workshop. While the subjects range from arithmetic geometry through algebraic geometry and differential geometry to mathematical physics, the papers are naturally related by the common theme of Calabi–Yau varieties. With the big variety of branches of mathematics and mathematical physics touched upon, this area reveals many deep connections between subjects previously considered unrelated.

Unlike most other conferences, the 2011 Calabi–Yau workshop started with 3 days of introductory lectures. A selection of 4 of these lectures is included in this volume. These lectures can be used as a starting point for the graduate students and other junior researchers, or as a guide to the subject.

Inhaltsverzeichnis

Frontmatter

Introductory Lectures

Frontmatter

K3 and Enriques Surfaces

Abstract

This is a note on my introductory lectures on K3 and Enriques surfaces in the workshop “Arithmetic and Geometry of K3 surfaces and Calabi–Yau threefolds” held at the Fields Institute. No new results are included.

Shigeyuki Kondō

Transcendental Methods in the Study of Algebraic Cycles with a Special Emphasis on Calabi–Yau Varieties

Abstract

We review the transcendental aspects of algebraic cycles, and explain how this relates to Calabi–Yau varieties. More precisely, after presenting a general overview, we begin with some rudimentary aspects of Hodge theory and algebraic cycles. We then introduce Deligne cohomology, as well as the generalized higher cycles due to Bloch that are connected to higher K-theory, and associated regulators. Finally, we specialize to the Calabi–Yau situation, and explain some recent developments in the field.

James D. Lewis

Two Lectures on the Arithmetic of K3 Surfaces

Abstract

In these lecture notes we review different aspects of the arithmetic of K3 surfaces. Topics include rational points, Picard number and Tate conjecture, zeta functions and modularity.

Matthias Schütt

Modularity of Calabi–Yau Varieties: 2011 and Beyond

Abstract

This paper presents the current status on modularity of Calabi–Yau varieties since the last update in 2003. We will focus on Calabi–Yau varieties of dimension at most three. Here modularity refers to at least two different types: arithmetic modularity and geometric modularity. These will include: (1) the modularity (automorphy) of Galois representations of Calabi–Yau varieties (or motives) defined over $\mathbb{Q}$ or number fields, (2) the modularity of solutions of Picard–Fuchs differential equations of families of Calabi–Yau varieties, and mirror maps (mirror moonshine), (3) the modularity of generating functions of invariants counting certain quantities on Calabi–Yau varieties, and (4) the modularity of moduli for families of Calabi–Yau varieties. The topic (4) is commonly known as geometric modularity.

Discussions in this paper are centered around arithmetic modularity, namely on (1), and (2), with a brief excursion to (3).

Noriko Yui

Research Articles: Arithmetic and Geometry of K3, Enriques and Other Surfaces

Frontmatter

Explicit Algebraic Coverings of a Pointed Torus

Abstract

This note contains an application of the algebraic study by Schütt and Shioda of the elliptic modular surface attached to the commutator subgroup of the modular group. This is used here to provide algebraic descriptions of certain coverings of a j-invariant 0 elliptic curve, unramified except over precisely one point.

Ane S. I. Anema, Jaap Top

Elliptic Fibrations on the Modular Surface Associated to Γ 1(8)

Abstract

We give all the elliptic fibrations of the K3 surface associated to the modular group Γ₁(8).

M. J. Bertin, O. Lecacheux

Universal Kummer Families Over Shimura Curves

Abstract

We give a number of examples of an isomorphism between two types of moduli problems. The first classifies elliptic surfaces over the projective line with five specified singular fibers, of which four are fixed and one gives the parameter; the second classifies K3 surfaces with a specified isogeny to an abelian surface with quaternionic multiplication.

Amnon Besser, Ron Livné

Numerical Trivial Automorphisms of Enriques Surfaces in Arbitrary Characteristic

Abstract

We extend to arbitrary characteristic some known results on automorphisms of complex Enriques surfaces that act identically on the cohomology or the cohomology modulo torsion.

Igor V. Dolgachev

Picard–Fuchs Equations of Special One-Parameter Families of Invertible Polynomials

Abstract

In this article we calculate the Picard–Fuchs equation of hypersurfaces defined by certain one-parameter families associated to invertible polynomials. For this we deduce the Picard–Fuchs equation from the GKZ system. As consequences of our work and facts from the literature, we show a relation between the Picard–Fuchs equation, the Poincaré series and the monodromy in the space of period integrals.

Swantje Gährs

A Structure Theorem for Fibrations on Delsarte Surfaces

Abstract

In this paper we study a special class of fibrations on Delsarte surfaces. We call these fibrations Delsarte fibrations. We show that after a specific cyclic base change, the fibration is the pullback of a fibration with three singular fibers and that this second-base change is completely ramified at two points where the fiber is singular. As a corollary we show that every Delsarte fibration of genus 1 with nonconstant j-invariant occurs as the base change of an elliptic surface from Fastenberg’s list of rational elliptic surfaces with γ < 1.

Bas Heijne, Remke Kloosterman

Fourier–Mukai Partners and Polarised $$\mathop{\mathrm{K3}}\nolimits$$ Surfaces

Abstract

The purpose of this note is twofold. We first review the theory of Fourier–Mukai partners together with the relevant part of Nikulin’s theory of lattice embeddings via discriminants. Then we consider Fourier–Mukai partners of $\mathop{\mathrm{K3}}\nolimits$ surfaces in the presence of polarisations, in which case we prove a counting formula for the number of partners.

K. Hulek, D. Ploog

On a Family of K3 Surfaces with $$\mathcal{S}_{4}$$ Symmetry

Abstract

The largest group which occurs as the rotational symmetries of a three-dimensional reflexive polytope is S ₄. There are three pairs of three- dimensional reflexive polytopes with this symmetry group, up to isomorphism. We identify a natural one-parameter family of K3 surfaces corresponding to each of these pairs, show that S ₄ acts symplectically on members of these families, and show that a general K3 surface in each family has Picard rank 19. The properties of two of these families have been analyzed in the literature using other methods. We compute the Picard–Fuchs equation for the third Picard rank 19 family by extending the Griffiths–Dwork technique for computing Picard–Fuchs equations to the case of semi-ample hypersurfaces in toric varieties. The holomorphic solutions to our Picard–Fuchs equation exhibit modularity properties known as “Mirror Moonshine”; we relate these properties to the geometric structure of our family.

Dagan Karp, Jacob Lewis, Daniel Moore, Dmitri Skjorshammer, Ursula Whitcher

K 1 ind of Elliptically Fibered K3 Surfaces: A Tale of Two Cycles

Abstract

We discuss two approaches to the computation of transcendental invariants of indecomposable algebraic K ₁ classes. Both the construction of the classes and the evaluation of the regulator map are based on the elliptic fibration structure on the family of K3 surfaces. The first computation involves a Tauberian lemma, while the second produces a “Maass form with two poles”.

Matt Kerr

A Note About Special Cycles on Moduli Spaces of K3 Surfaces

Abstract

We describe the application of the results of Kudla–Millson on the modularity of generating series for cohomology classes of special cycles to the case of lattice polarized K3 surfaces. In this case, the special cycles can be interpreted as higher Noether–Lefschetz loci. These generating series can be paired with the cohomology classes of complete subvarieties of the moduli space to give classical Siegel modular forms with higher Noether–Lefschetz numbers as Fourier coefficients. Examples of such complete families associated to quadratic spaces over totally real number fields are constructed.

Stephen Kudla

Enriques Surfaces of Hutchinson–Göpel Type and Mathieu Automorphisms

Abstract

We study a class of Enriques surfaces called of Hutchinson–Göpel type. Starting with the projective geometry of Jacobian Kummer surfaces, we present the Enriques’ sextic expression of these surfaces and their intrinsic symmetry by $G = C_{2}^{3}$. We show that this G is of Mathieu type and conversely, that these surfaces are characterized among Enriques surfaces by the group action by $C_{2}^{3}$ with prescribed topological type of fixed point loci. As an application, we construct Mathieu type actions by the groups $C_{2} \times \mathfrak{A}_{4}$ and $C_{2} \times C_{4}$. Two introductory sections are also included.

Shigeru Mukai, Hisanori Ohashi

Quartic K3 Surfaces and Cremona Transformations

Abstract

We prove that there is a smooth quartic K3 surface automorphism that is not derived from the Cremona transformation of the ambient three-dimensional projective space. This gives a negative answer to a question of Professor Marat Gizatullin.

Keiji Oguiso

Invariants of Regular Models of the Product of Two Elliptic Curves at a Place of Multiplicative Reduction

Abstract

The divisor class group, (co)homology, and Picard group of the closed fibers of various regular proper models of the product of two elliptic curves at a place of multiplicative reduction are computed. The variation of the isomorphism class of the closed fiber with the variation of the elliptic curves is discussed. The higher direct images of the sheaf, $\mathbb{Z}/n$, are computed when n is prime to the residue characteristic.

Chad Schoen

Research Articles: Arithmetic and Geometry of Calabi-Yau Threefolds and Higher Dimentional Varieties

Frontmatter

Dynamics of Special Points on Intermediate Jacobians

Abstract

We prove some general density statements about the subgroup of invertible points on intermediate jacobians; namely those points in the Abel–Jacobi image of nullhomologous algebraic cycles on projective algebraic manifolds.

Xi Chen, James D. Lewis

Calabi–Yau Conifold Expansions

Abstract

We describe examples of computations of Picard–Fuchs operators for families of Calabi–Yau manifolds based on the expansion of a period near a conifold point. We find examples of operators without a point of maximal unipotent monodromy, thus answering a question posed by J. Rohde.

Slawomir Cynk, Duco van Straten

Quadratic Twists of Rigid Calabi–Yau Threefolds Over ℚ

Abstract

We consider rigid Calabi–Yau threefolds defined over $\mathbb{Q}$ and the question of whether they admit quadratic twists. We give a precise geometric definition of the notion of a quadratic twists in this setting. Every rigid Calabi–Yau threefold over $\mathbb{Q}$ is modular so there is attached to it a certain newform of weight 4 on some Γ ₀(N). We show that quadratic twisting of a threefold corresponds to twisting the attached newform by quadratic characters and illustrate with a number of obvious and not so obvious examples. The question is motivated by the deeper question of which newforms of weight 4 on some Γ ₀(N) and integral Fourier coefficients arise from rigid Calabi–Yau threefolds defined over $\mathbb{Q}$ (a geometric realization problem).

Fernando Q. Gouvêa, Ian Kiming, Noriko Yui

Counting Sheaves on Calabi–Yau and Abelian Threefolds

Abstract

We survey the foundations for Donaldson–Thomas invariants for stable sheaves on algebraic threefolds with trivial canonical bundle, with emphasis on the case of abelian threefolds.

Martin G. Gulbrandsen

The Segre Cubic and Borcherds Products

Abstract

We shall construct a five-dimensional linear system of holomorphic automorphic forms on a three-dimensional complex ball by applying Borcherds theory of automorphic forms. We shall show that this linear system gives the dual map from the Segre cubic threefold to the Igusa quartic threefold.

Shigeyuki Kondō

Quasi-modular Forms Attached to Hodge Structures

Abstract

The space D of Hodge structures on a fixed polarized lattice is known as Griffiths period domain, and its quotient by the isometry group of the lattice is the moduli of polarized Hodge structures of a fixed type. When D is a Hermitian symmetric domain, then we have automorphic forms on D, which according to Baily–Borel theorem, they give an algebraic structure to the mentioned moduli space. In this article we slightly modify this picture by considering the space U of polarized lattices in a fixed complex vector space with a fixed Hodge filtration and polarization. It turns out that the isometry group of the filtration and polarization, which is an algebraic group, acts on U and the quotient is again the moduli of polarized Hodge structures. This formulation leads us to a notion of quasi-automorphic forms which generalizes quasi-modular forms attached to elliptic curves.

Hossein Movasati

The Zero Locus of the Infinitesimal Invariant

Abstract

Let $\nu$ be a normal function on a complex manifold X. The infinitesimal invariant of $\nu$ has a well-defined zero locus inside the tangent bundle TX. When X is quasi-projective, and $\nu$ is admissible, we show that this zero locus is constructible in the Zariski topology.

G. Pearlstein, Ch. Schnell

Titel: Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds
herausgegeben von: Radu Laza
Matthias Schütt
Noriko Yui
Verlag: Springer New York
Electronic ISBN: 978-1-4614-6403-7
Print ISBN: 978-1-4614-6402-0
DOI: https://doi.org/10.1007/978-1-4614-6403-7