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.
____________________________
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.
Lecture Notes 1
Lecture
Notes 2
____________________________
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.
____________________________
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
____________________________
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.
____________________________
N. Yui (Queen's University, Canada)
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.
*************************************************************
Invited Speaker Abstracts:
M. Artebani (Universidad de Concepcion, Chile)
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.
____________________________
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
____________________________
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.
____________________________
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.
____________________________
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.
____________________________
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.
____________________________
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.
____________________________
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.
____________________________
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
____________________________
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.
____________________________
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.
____________________________
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.
____________________________
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.
____________________________
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.
____________________________
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.
____________________________
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.
____________________________
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 $E_7$.
____________________________
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
___________________________
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.
____________________________
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.
____________________________
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
____________________________
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.
____________________________
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.
____________________________
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.
____________________________
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.
____________________________
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.
____________________________
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.
____________________________
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
____________________________
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).
____________________________
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.
____________________________
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
____________________________
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.
____________________________
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).
*************************************************************
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.
____________________________
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.
____________________________
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
____________________________
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
Back to top
