# SCIENTIFIC PROGRAMS AND ACTIVITIES

April 30, 2016

## August 16-25, 2011 Workshop on Arithmetic and Geometry of K3 surfaces and Calabi-Yau threefolds

to be held at the University of Toronto and the Fields Institute

Organizers:
Charles F. Doran, University of Alberta, and PIMS
Shigeyuki Kondõ, Nagoya University

James D. Lewis, University of Alberta
Matthias Schütt, Leibniz University Hannover
Noriko Yui, Queen’s University (corresponding organizer)

This workshop is supported by:
The Fields Institute
NSF (1100007 plus individual grants)
JSPS Grant-in-Aid (S), No. 22224001
DFG GRK 1463

### Pedagogical Lectures:

C. Doran (University of Alberta, Canada)
Periods, Picard-Fuchs Equations, and Calabi-Yau Moduli

We introduce and explore the transcendental theory of Calabi-Yau manifolds and its interplay with explicit algebraic moduli. The focus in each lecture will be on Calabi-Yau manifolds of sequentially higher dimension (elliptic curves, K3 surfaces, and Calabi-Yau threefolds). Special attention will be given to hypersurfaces and complete intersections in toric varieties.

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S. Kondo (Nagoya University, Japan)
K3 and Enriques surfaces

In this introductory lecture, I shall give a survey on moduli and automorphisms of K3, Enriques surfaces. A related lattice theory and the theory of automorphic forms will be included.

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R. Laza (Stony Brook University, USA)
Degenerations of K3 surfaces and Calabi-Yau threefolds

In these lectures we will review the degenerations of K3 surfaces and Calabi-Yau threefolds from a geometric and Hodge theoretic point of view. In the first lecture we will focus on K3 surfaces, and we will review the period and its compactifications. In the second lecture, we will discuss the behavior of the period map near the boundary and the connection to mirror symmetry.

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J. Lewis (University of Alberta, Canada)
Lectures in Transcendental Algebraic Geometry:  Algebraic Cycles with a Special Emphasis on Calabi-Yau Varieties

These lectures serve as an introduction to algebraic cycle groups and their regulators for projective algebraic manifolds. 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 generalized cycles  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.

Lecture Notes

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M. Schuett (University of Hannover, Germany)
Arithmetic of K3 surfaces

We will review various aspects of the arithmetic of K3 surfaces. Topics will include rational points, Picard number and Tate conjecture, zeta functions and modularity.

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Modularities of Calabi--Yau varieties: 2011 and beyond

This paper presents the current status on modularities of Calabi--Yau varieties since the last update in 2001. We will focus on Calabi-Yau varieties of dimension at most three. Here modularities refer to at least two different types: arithmetic modularity, and geometric modularity. These will include:
(1) the modularity of Galois representations of Calabi--Yau varieties (or motives) defined over $\QQ$ or number fields,
(2) the modularity of solutions of Picard-Fuchs differential equations of families of Calabi-Yau varieties, and the modularity of mirror maps (mirror moonshine),
(3) the modularity of generating functions of various invariants counting some quantities on Calabi-Yau varieties, and
(4) the modularity of moduli for families of Calabi-Yau varieties.
The topic (4) is commonly known as the geometric modularity.

In this series of talks, I will concentrate on arithmetic modularity, namely, on the topic (1), and possibly on the topics (2) and (3) if time permits.

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### Invited Speaker Abstracts:

M. Artebani
Examples of Mori dream Calabi-Yau threefolds

Let $Z$ be a Mori dream space, i.e. a normal projective variety having finitely generated Cox ring $R(Z)$, and let $X$ be a hypersurface of $Z$. In a joint work with A. Laface we provided a necessary
and sufficient condition for the Cox ring $R(X)$ to be isomorphic to $R(Z)/(f)$, where $f$ is a defining section for $X$. In this talk, after presenting this result, two applications to Calabi-Yau 3-folds will be given. Firstly, we will show that there are five families of Calabi-Yau hypersurfaces insmooth toric Fano fourfolds whose Cox ring is a
polynomial ring with one relation. As a second application, we will compute the Cox ring of the generic quintic 3-fold containing a plane.

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X. Chen (University of Alberta, Canada)
Rational self-maps of K3 surfaces and Calabi-Yau manifolds

It is conjectured that a very general K3 surface does not have any nontrivial dominant rational self-maps. I'll give a proof for this conjecture and also show the same holds for a very general Calabi-Yau complete intersection in projective spaces of higher dimensions by induction.

Slides

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A. Clingher (Washington University in St. Louis, USA)
On K3 Surfaces of High Picard Rank

I will report on a classification of a certain class of K3 surfaces of Picard rank 16 or higher. In terms of periods, the moduli space of these objects is a quotient of a four-dimensional bounded symmetric domain of type IV. Explicit normal forms will be presented, as well as a discussion of modular forms associated with this family.

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S. Cynk (Jagiellonian University, Poland)
Arithmetically significant Calabi-Yau threefolds

From the point of view of their arithmetic the most interesting Calabi-Yau threefolds are those with small Hodge number $h^{1,2}$, especially the rigid ones. I will discuss the most important constructions of such Calabi-Yau threefolds, f.i. the Kummer construction, fiber product of rational elliptic surfaces with section and their refinements.

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I. Dolgachev (University of Michigan, USA)
Quartic surfaces and Cremona transformations

I will discuss the following question: when a birational automorphism of a quartic surface is a restriction of a Cremona transformation of the ambient space.

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N. Elkies (Harvard University, USA)
Even lattices and elliptic fibrations of K3 surfaces I, II

Abstract: Given a K3 surface $X$, any elliptic fibration with zero-section has an essential lattice $L$ (orthogonal complement of a hyperbolic plane) whose genus depends only on the Neron-Severi lattice $NS(X)$.
The Kneser-Nishiyama gluing method and related techniques often makes it feasible to list all possible $L$, or all $L$ satisfying some additional condition such as nontrivial torsion or large Mordell-Weil rank, and to give explicit equations when one equation for $X$ is known. We illustrate with several examples:

(a) Of the 13 elliptic fibrations of Euler's surface $E_a: xyz(x+y+z)=a$, nine can be defined over $Q(a)$, all with Mordell-Weil rank zero. This may both explain why Euler found it unusually hard to find families of solutions in $Q(a)$ and suggest how he did eventually find one such family. Over an algebraically closed field, the $E_a$ all become isomorphic with the "singular" K3 surface (Picard number $20$, maximal in characteristic zero) with $disc(NS(X)) = -4$.

(b) If $NS(X)$ has rank $20$ and consists entirely of classes defined over $Q$, then $|disc(NS(X))|$ is at most $163$. We use this to show that no elliptic fibration can have attain the maximum of $18$ for the
Mordell-Weil rank of an elliptic K3 surface over $C(t)$; this together with an explicit rank $17$ surface over $Q(t)$ (with $\rho=19$) answers a question of Shioda (1994).

(c) Certain families of K3 surfaces with Picard number $19$ are parametrized by Shimura modular curves; this makes it possible to give explicit equations and CM coordinates on these curves that were previously
inaccessible, and to find the genus $2$ curves with quaternionic multiplication that the Shimura curves parametrize.

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R. Girivaru (University of Missouri--St. Louis, USA)
Extension theorems for subvarieties and bundles

Given a subvariety (respectively a vector bundle) on a smooth hyperplane section of a smooth projective variety, it is of interest to know when it is the restriction of a subvariety (resp a bundle) on the ambient variety. I will present some results on this theme.

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J. W. Hoffman (Louisiana State University, USA)
Picard groups of Siegel modular threefolds and theta lifting

This is a joint work with Hongyu He.
A Siegel modular threefold is a quotient of the Siegel upper half space of genus 2 by a subgroup of finite index in Sp(4, Z). These spaces are moduli spaces for abelian varieties with additional structure, and are examples of Shimura varieties. We discuss the structure of the Picard groups of these; they are groups generated by algebraic cycles of codimension one. We show that these Picard groups are generated by special cycles in the sense of Kudla-Millson. These special cycles are identified with the classically defined Humbert surfaces. The key points are: (1) the theory of special cycles relating geometric cycles to automorphic forms coming from theta-lifting; (2) Weissauer's theorems describing the Picard groups via automorphic forms; (3) results of Howe about the oscillator representation.

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K. Hulek (University of Hannover, Germany)
Abelian varieties with a singular odd $2$-torsion point on the theta divisor

We study the (closure of the) locus of intermediate Jacobians of cubic threefolds in the perfect cone compactification of the moduli space of principally polarized abelian fivefolds for which we obtain an expression in the tautological Chow ring. As a generalization we consider the locus of principally polarized abelian varieties with a singular odd $2$-torsion point on the theta divisor and their degenerations. This is joint work with S. Grushevsky.

Lecture Notes

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M. Kerr (Washington University in St. Louis, USA)
Higher Chow cycles on families of K3 surfaces

This talk is a tale of two cycles, both supported on singular fibers of families of elliptically fibered K3's. The first lives on a cover of the $H+E8+E8$-polarized family of Clingher and Doran, and we discuss a direct evaluation of the real regulator (part of joint work with Chen, Doran, and Lewis). The resulting function is related to a kind of "Maass cusp formwith pole". For the second cycle, we explain how to use a bit of Tauberian theory to compute the transcendental regulator.

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J. Keum (KIAS, Korea)
Finite groups acting on K3 surfaces in positive characteristic

A remarkable work of S. Mukai [1988] gives a classification of finite groups which can act on a complex K3 surface leaving invariant its holomorphic 2-form (symplectic automorphism groups). Any such group turns out to be isomorphic to a subgroup of the Mathieu group $M_{23}$ which has at least 5 orbits in its natural action on the set of 24 elements. A list of maximal subgroups with this property consists of 11 groups, each of these can be realized on an explicitly given K3 surface. Different proofs of Mukai's result were given by S. Kond\={o} [1998] and G. Xiao [1996].None of the 3 proofs extends to the case of K3 surfaces over algebraically closed fields of positive characteristic $p$.In this talk I will outline a recent joint work with I. Dolgachev on extending Mukai's result to the positive characteristic case.In positive characteristic case we first have to handle wild automorphisms, the ones whose orders are divisible by the characteristic $p$.It turns out that no wild automorphism of a K3 surface exists in characteristic $p > 11$. Then a classification of finite groups will be given which may act symplectically on a K3 surface in positive characteristic.

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R. Kloosterman (Humboldt Universitaet zu Berlin, Germany)
Mordell-Weil ranks, highest degree syzygies and Alexander polynomials

We discuss an approach to calculate the Mordell-Weil rank for elliptic threefold. We apply this method to a class of elliptic threefolds with constant $j$-invariant 0.

It turns out that in this particular case there is a strong connection between
1. the number of highest degree syzygies of the ideal of a certain
subscheme of the singular locus of the discriminant curve,
2. the Mordell-Weil rank of the fibration,
3. the exponent of $(t^2-t+1)$ in the Alexander polynomial of
the discriminant curve.
We used the connection between 1 and 2 to find a nontrivial upper bound for the Mordell-Weil rank.

As an application we use the connection between 1 and 2 to describe all degree 18 plane curves, with only nodes and cusps as singularities, such that its deformation space has larger dimension than expected. (In this case the associated elliptic threefold is a degeneration of a Calabi-Yau elliptic threefold.)

We then show that one can recover the Alexander polynomial of any even degree $d$ plane curve $C=Z(f(z_0,z_1,z_2))$ by studying the threefold $W\subset \mathbb{P}(d/2,1,1,1)$ given by $y^2+x^d+f=0$. It turns out that in the case that $C$ has only ADE singularities the Alexander polynomial of $C$ determines the group of Weil Divisors on $W$ modulo $\mathbb{Q}$-Cartier divisors on $W$. One can use this to find a series of subschemes $J_i$ of the singular locus of $C$, such that the number of highest degree syzygies of $J_i$ has a geometric interpretation. We end by giving some higher dimensional examples.

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S. Kudla (University of Toronto)
Modular generating functions for arithmetic cycles: a survey

In this talk I will give a survey of some recent results on the relations between the Fourier coefficients of modular forms and the classes of certain cycles in arithmetic Chow groupsShimura varieties. When the generating series for such cycle classes are modular forms, they may be viewed as an exotic type of theta function. The behavior of such forms under natural geometric operations,such as pullback to subvarieties, is of particular interest. I will describe several examples and discuss some open problems.

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A. Kumar (MIT, USA)
Elliptic fibrations on Kummer surfaces

I will describe computations regarding elliptic fibrations on Kummer surfaces, and some applications, such as explicit algebraic families of K3 surfaces with Shioda-Inose structure.

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C. Liedtke (Stanford University, USA)
Rational Curves on K3 Surfaces

We show that projective K3 surfaces with odd Picard rank contain infinitely many rational curves. Our proof extends the Bogomolov-Hassett-Tschinkel approach, i.e., uses moduli spaces of stable maps and reduction to positive characteristic. This is joint work with Jun Li.

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H. Movasati (IMPA, Brazil)
Eisenstein type series for mirror quintic Calabi-Yau varieties

In this talk we introduce an ordinary differential equation associated to the one parameter family of Calabi-Yau varieties which is mirror dual to the universal family of smooth quintic three folds. It is satisfied by seven functions written in the $q$-expansion form and the Yukawa coupling turns out to be rational in these functions. We prove that these functions are algebraically independent over the field of complex numbers, and hence, the algebra generated by such functions can be interpreted as the theory of quasi-modular forms attached to the one parameter family of Calabi-Yau varieties.Our result is a reformulation and realization of a problem of Griffiths around seventies on the existence of automorphic functions for the moduli of polarized Hodge structures. It is a generalization of the Ramanujan differential equation satisfied by three Eisenstein series.

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S. Mukai (RIMS, Japan)
Enriques surfaces and root systems

There are many interesting families of Enriques surfaces which are characterized by the presence of (negative definite) root sublattices ADE's in their twisted Picard lattices. In this talk I will discuss two such families (a) Enriques surfaces with many M-semi-symplectic automorphisms and (d) Enriques surfaces of Lieberman type related with the joint work with H. Ohashi, and another kind of family of (e) Enriques surfaces of type $E_7$.

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V. Nikulin (University of Liverpool, UK, and Steklov Mathematical
Institute, Moscow, Russia)
Elliptic fibrations on K3 surfaces

We discuss, how many elliptic fibrations and elliptic fibrations with infinite automorphism groups an algebraic K3 surface over an algebraically closed field can have. As examples of applications of the same ideas, we also consider K3 surfaces with exotic structures: with finite number of Enriques involutions, and with naturally arithmetic automorphism groups. See details in arXiv:1010.3904.
Lecture Notes
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K. O'Grady (Sapienza Universita' di Roma)
Moduli and periods of double EPW-sextics

We analyze the GIT-quotient of the parameter space for (double covers of) EPW-sextics i.e. the symplectic grassmannian of lagrangian subspaces of the third wedge-product of a $6$-dimensional complex vector-space (equipped with the symplectic form defined by wedge product on $3$-vectors) modulo the natural action of $PGL(6)$. Our goal is to analyze the period map for the GIT-quotient, thus we aim to establish a dictionary between (semi)stability conditions and properties of the periods. We are inspired by the works of C.Voisin and R.Laza on cubic 4-folds.

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K. Oguiso
(Osaka University, Japan)
Group of automorphisms of Wehler type on Calabi-Yau manifolds and compact hyperkaehler manifolds

Wehler pointed out, without proof, that a K3 surface defined by polynomial of multi-degree $(2,2,2)$ in the product of three projective lines admits a biholomorphic group action of the free product of three cyclic groups of order two. I would like to first explain one proof of his result and in which aspects his example is interesting. Then I would like to give a "fake" generalization for Calabi-Yau manifolds and explain why it is fake. Finally I would like to give a right generalization for Calabi-Yau manifolds of any even dimensions and compact hyperk\"ahler manifolds of any degree.

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H. Ohashi (Nagoya University, Japan)
On automorphisms of Enriques surfaces

We will discuss a possible extension to Enriques surfaces of an outstanding result of Mukai about the automorphism groups of K3 surfaces. We define the notion of Mathieu-semi-symplectic actions on Enriques surfaces and classify them. The maximal groups will be characterized in terms of the small Mathieu group $M_{12}$. This is a joint work with S. Mukai.

Lecture Notes

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G. Pearlstein (Michigan State University, USA)
Jumps in the Archimedean Height

We answer a question of Richard Hain regarding the asymptotic behavior of the archimedean heights and explain its connection to the Hodge conjecture via the work of Griffiths and Green.

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J.-C. Rohde (Universitaet Hamburg, Germany)
Shimura varieties and Calabi-Yau manifolds versus Mirror Symmetry

There are examples of Calabi-Yau $3$-manifolds $X$, which cannot be a fiber of a maximal family of Calabi-Yau $3$-manifolds with maximally unipotent monodromy. This contradicts the assumptions of the mirror symmetry conjecture. All known examples of this kind can be constructed by quotients of products of K3 surfaces $S$ and elliptic curves by an automorphism of order 3 or 4. Moreover the associated period domain of a maximal family with a fiber isomorphic to $X$ is a complex ball containing a dense set of complex multiplication points. In some examples the K3 surfaces S used for the construction of $X$ can also be used to construct pairs of subfamilies of pairs of mirror families with dense sets of complex multiplication fibers.

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A. Sarti (University of Poitiers, France)
The BHCR-mirror symmetry for K3 surfaces

The aim of this talk is to apply the construction of mirror pairs of Berglund and H\"ubsch to K3 surfaces with non symplectic involution and to investigate a recent result of Chiodo and Ruan. They apply the construction to pairs $(X,G)$ where $X$ is a Calabi Yau manifold of dimension at least three, given as the zero set of a non degenerate potential in some weighted projective space, and $G$ is a finite group acting on the manifold.\\For this reason we call the symmetry the {\it BHCR-mirror symmetry. In the talk I will show that this symmetry coincides with the mirror symmetry for lattice polarized K3 surfaces described by Dolgachev.\\This is a joint work with Michela Artebani and Samuel Boissi\`ere.

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C. Schnell (IPMU, Japan)
Derived equivalences and the fundamental group

I will describe an example (constructed by Gross and Popescu) of a simply connected Calabi-Yau threefold $X$, with a free action by the group $G = Z/5Z x Z/5Z$, for which $X$ and $X/G$ are derived equivalent. This shows that being simply connected is not a derived invariant.

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C. Schoen (Duke University, USA)
Desingularized fiber products of elliptic surfaces

The varieties of the title are sufficiently complex to exhibit many of the phenomena which arise when one studies smooth projective threefolds, but are often significantly simpler to work with than general threefolds because of the well understood elliptic surfaces from which they are built. So far these varieties have contributed to our understanding of algebraic cycles, modularity of Galois representations, phenomena peculiar to postive characteristic, superstring theory, Brauer groups, Calabi-Yau threefolds, and families of Kummer surfaces. Many open problems remain.

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S. Schroeer (University of Duesseldorf, Germany)
Enriques manifolds

Enriques manifolds are complex spaces whose universal coverings are hyperkahler manifolds. We give several examples, construct period domains, and establish a local Torelli theorem. The theory applies to various situations related to punctual Hilbert schemes, moduli spaces of stable sheaves, and Mukai flops. This is a joint work of K. Oguiso.

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A. Thompson (Oxford University, UK)
Degenerations of K3 surfaces of degree two

We consider semistable degenerations of K3 surfaces of degree two, with the aim of explicitly studying the geometric behaviour at the boundary of the moduli space of such surfaces. We begin by showing that results of the minimal model program may be used to bring these degenerations into a uniquely determined normal form: the relative log canonical model. We then proceed to describe a result that explicitly classifies the central fibres that may appear in this relative log canonical model, as complete intersections in certain weighted projective spaces.

Lecture Notes

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D. van Straten (Universitaet Mainz, Germany)
CY-period expansions

The local power series expansion of period-functions have strong integrality properties. Such expansions can be used effectively to find Picard–Fuchs equations
in situations, where the traditional “Dwork–Griffiths–Method” is not available or cumbersome to use. We give examples how to use “conifold expansions” to obtain the Picard–Fuchs equations for some one-parameter families of Calabi–Yau 3-folds.
(Work in progress, joint with S. Cynk).

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U. Whitcher (Harvey Mudd College, USA)
Picard-Fuchs equations for lattice-polarized K3 surfaces

The moduli spaces of K3 surfaces polarized by the lattices $H\oplus E_8\oplus E_8$ and $H\oplus E_8 \oplus E_7$ are related to moduli spaces of polarized abelian surfaces. We use Picard-Fuchs equations for the lattice-polarized K3 surfaces to explore this correspondence and characterize subloci of the moduli spaces of particular interest.

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K.-I. Yoshikawa (Kyoto University, Japan)
On the value of Borcherds $\Phi$-function

It is well known that the Petersson norm of Jacobi Delta-function is expressed as the product of the discriminant of cubic curve and the $L_2$ norm of appropriately normalized $1$-form on the curve. We give a generalization this fact to Enriques surfaces and Borcherds $\Phi$-function.

Slides

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J.-D. Yu (National Taiwan University, Taiwan)
On Dwork congruences

The Dwork congruences refer to a system of congruences among the coefficients of periods of certain Calabi-Yau pencils. They are used to derive the unit root formula for the zeta functions of the reductions of the fibers. Examples include certain hypergeometric series proved by Dwork himself via ad hoc methods. Here we give a geometric interpretation of these congruences.

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Y. Zarhin
(Pennsylvania State University, USA)
Hodge groups

We discuss computations of Hodge groups of certain superelliptic jacobians (based on joint papers with Jiangwei Xue).

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### Contributed Speaker Abstracts:

M.J. Bertin (Université Paris, 6)
Elliptic fibrations on the modular surface associated to $\Gamma_1(8)$

This is a joint work with Odile Lecacheux. Using Nishiyama's method, we determine all the elliptic fibrations with section on the elliptic surface $$X+\frac {1}{X}+Y+\frac {1}{Y}+Z+\frac {1}{Z}=2.$$ This $K3$-surface, of discriminant $-8$, is explained to be the modular surface associated to the modular group $\Gamma_1(8)$.We illustrate the method with examples and show how to get, for a given fibration, the rank and torsion of the Mordell-Weil group.Moreover, from a Weierstrass equation of an elliptic fibration, we explain one of the various ways to obtain a Weierstrass equation of another fibration.

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Yasuhiro Goto (Hokkaido University of Education Hakodate)
On K3 surfaces with involution

K3 surfaces with involution are classified by Nikulin's invariants. We calculate these invariants for K3 surfaces defined in weighted projective $3$-spaces by Delsarte-type equations.

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L. H. Halle (University of Oslo, Norway)
Motivic zeta functions for degenerations of Calabi-Yau varieties

I will discuss a global version of Denef and Loeser's motivic zeta functions. More precisely, to any Calabi-Yau variety $X$ defined over a discretely valued field $K$, I will define a formal power series $Z_X(T)$ with coefficients in a certain localized Grothendieck ring of varieties over the residue field $k$ of $K$. The series $Z_X(T)$ has properties analogous to Denef and Loeser's zeta function, in particular one can formulate a global version of the motivic monodromy conjecture. I will present a few cases where this conjecture has been proved. This is joint work with Johannes Nicaise.

Lecture Notes

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S. Sijsling (IMPA, Brazil)
Calculating arithmetic Picard-Fuchs equations

We consider second-order Picard-Fuchs equations that are obtained by uniformizing certain genus 1 Shimura curves. These equations are distinguished by having a particularly beautiful monodromy group, generated by two elements and commensurable with the group of units of a quaternion order. They describe the periods of certain families of fake elliptic curves that are as yet hard to write down.We explore the methods for determining these equations explicitly, and discuss the
open questions that remain.

Lecture Notes