Calabi-Yau Varieties and Mirror Symmetry
Org: James Lewis (Alberta) and Noriko Yui (Queen's)
- REZA AKHTAR, Miami University, Oxford, Ohio 45056, USA
Chow-Kunneth and Lefschetz decompositions for quotients of
Let A be an abelian variety and G a finite group acting on A.
The quotient variety A/G is not necessarily an abelian variety or
even smooth. We discuss the motive of A/G, in particular how one
can construct explicit Chow-Kunneth and Lefschetz decompositions for
this motive in terms of the corresponding decompositions for the
motive of A.
This is joint work with Roy Joshua.
- MARIE JOSÉ BERTIN, University Paris 6, Institut de Mathématiques, 175 Rue du
Chevaleret, 75013 Paris, France
Mahler's measure and L-series of K3 hypersurfaces
The Mahler measure of a multivariate polynomial has been introduced by
Mahler in 1962 to measure the size of factors of polynomials.
Later in 1996, Deninger guessed a link between the Mahler measure of a
certain polynomial defining an elliptic curve and the L-series of
I plan to show how these facts extend to K3-hypersurfaces. I give
several examples of singular K3-hypersurfaces for which the Mahler
measure is expressed in terms of the L-series of the variety plus
eventually a term related to the faces of the Newton polyhedron.
- VINCENT BOUCHARD, Perimeter Institute, 31 Caroline Street North, Waterloo,
Ontario, N2L 2Y5, Canada
On the landscape of standard-model bundles on non-simply
connected Calabi-Yau threefolds
Let X be a smooth Calabi-Yau threefold of Schoen's type,
i.e., a fiber product of two rational elliptic surfaces over
P1. We classify all pairs (X,G), where G is a finite
group acting freely on X, such that the quotient X/G is a
non-simply connected torus-fibered Calabi-Yau threefold. We also
systematize the construction of stable G-invariant vector bundles on
X with structure group SU(4) or SU(5). This work is
motivated by the search for vacua of heterotic string theory yielding
realistic four-dimensional physics, and the study of the landscape of
such heterotic vacua.
This is joint work with Ron Donagi.
- XI CHEN, University of Alberta, Edmonton, AB, T6G 2G1
Vojta's 1+epsilon Conjecture and Bergman metric
I'll talk about a proof of Vojta's 1+epsilon conjecture. One
interesting aspect of this proof is an application of Bergman metric.
- ADRIAN CLINGHER, University of Missouri in St. Louis
On a special class of lattice polarized K3 surfaces
Using standard Hodge theory, one can associate to any K3 surface which
is polarized by the rank 17 even lattice H+E8+E7 an abelian surface
A(C) obtained as the jacobian of a genus-two curve. I will discuss
a geometric method that leads to an explicit description of the
This is joint work with Charles Doran (University of Washington).
- CHUCK DORAN, University of Washington, Seattle, WA, USA
On Stokes Matrices of Calabi-Yau Hypersurfaces
We consider Laplace transforms of the Picard-Fuchs differential
equations of Calabi-Yau hypersurfaces and calculate their Stokes
matrices. We also introduce two different types of Laplace transforms
of Gel'fand-Kapranov-Zelevinski hypergeometric systems.
This is joint work with Shinobu Hosono (University of Tokyo).
- YASUHIRO GOTO, Hokkaido University of Education, 1-2 Hachiman-cho,
Hakodate, 040-8567 Japan
Formal groups of Calabi-Yau threefolds in positive
Calabi-Yau varieties (over an algebraically closed field of positive
characteristic) are associated with formal groups of dimension one and
they are classified by the height. The height of the formal group of
a K3 surface is bounded by 10 when it is finite. The existence of
such a bound for Calabi-Yau threefolds is unknown. In this talk, we
compute the height of formal groups of various Calabi-Yau threefolds
arising from weighted Fermat/Delsarte threefolds and analyze its
- SHENGDA HU, Université de Montréal, Département de Math. et de
Stat., CP 6128, succ. Centre-Ville, Montreal, QC, H3C 3J7,
T-duality with H-flux from generalized Kähler
The construction of reduction by Poisson Lie group action in Poisson
geometry can be extended to generalized complex geometry. A
generalized Kähler manifold has two compatible generalized complex
structures. We present a construction in generalized Kähler geometry
via Poisson Lie reduction on either of the generalized complex
structure. The two quotient manifolds thus obtained are possible
candidates of a T-dual pair with H-flux.
- MATT KERR, University of Chicago (Math Dept.), 5734 S. University Ave
Algebraic K-theory and local mirror symmetry
We discuss an aspect of recent joint work with C. Doran, relating our
construction of families of cycles in higher K-theory to the mirror
map in local mirror symmetry, and from there to asymptotics of
- NAM-HOON LEE, Korea Institute for Advanced Study
Some attempt at constructing infinitely many families of
Some Calabi-Yau construction by smoothing normal crossings, that may
possibly lead to construction of infinitely many families of
Calabi-Yau manifolds, will be discussed. It depends on the existence
of a specific type of threefolds and some K3 surfaces.
As byproducts, we will also discuss some generalization of Enriques
- JAMES LEWIS, University of Alberta
The Abel-Jacobi Map for Higher Chow Groups, II
Let X be a projective algebraic manifold, and Y in X a normal
crossing divisor. We describe the regulator map (on the level of
complexes) from Bloch's higher Chow groups of X-Y into absolute
Hodge cohomology, and give a description of the corresponding
This is based on joint work with Matt Kerr.
- BONG H. LIAN, Brandeis University and NUS
Counting Fourier-Mukai Partners
I will discuss an explicit formula for counting Fourier-Mukai partners
of any given algebraic K3 surface.
This is joint work with Hosono, Oguiso and Yau.
- LING LONG, Iowa State University
On the Coefficients of Noncongruence Modular Forms
By a theorem of Belyi, it is known that meromorphic functions on
compact smooth orientable Riemann surfaces defined over the algebraic
closure of Q are modular functions for finite index
subgroups of the modular group. Predominately, most of these modular
functions are for noncongruence subgroups. It was observed by Atkin
and Swinnerton-Dyer that the Fourier coefficients having unbounded
denominators is a clear distinction between noncongruence and
congruence modular forms. However, it is unknown whether the
coefficients of a genuine noncongruence modular form (with algebraic
coefficients) will have unbounded denominators.
In this talk, we will discuss the unbounded denominator property
satisfied by the coefficients of several types of noncongruence
modular forms, including some meromorphic functions parameterizing
- STEVEN LU, Université du Québec à Montréal, Dépt. de Math.,
CP 8888 Succursale Centre-ville, Montréal, QC, H3C 3P8
On complex invariant metrics for some Calabi-Yau varieties
I will comment on some recent ideas of Claire Voisin on constructing
invariant measures so as to show that they vanish on some Calabi-Yau
varieties. The natural construction obtained is used to give the
vanishing of a corresponding invariant metric for these varieties.
- JOHN MCKAY, Concordia University
Towards an understanding of Monstrous Moonshine
We are working towards a better understanding of the geometry of the
monster simple group, ultimately leading to a reduction in the number
of sporadic finite simple groups. Some problems arising will be
- MATTHIAS SCHUETT, Harvard University
Classifying singular K3 surfaces
A complex K3 surface is called singular if it has Picard number 20.
Singular K3 surfaces behave in many ways like elliptic curves with
complex multiplication. For instance, they are defined over some
number field. After reviewing some classical results, as of
Shioda-Inose and Shafarevich, we will discuss the problem which
singular K3 surfaces actually can be defined over Q.
- ANDREY TODOROV, University of California at Santa Cruz
Regularized Determinants of CY Metrics, Borchards Products;
Applications to K3 Surfaces
In this talk we will prove that there exists a holomorphic section of
the relative dualizing sheaf over some finite cover of the moduli
space of polarized CY manifolds whose L2 norm is equal to the
regularized determinant of the CY metric of the Laplacian acting on
(0,1) forms. We will show that in case of M-polarized K3 surfaces
this section can be represented as Borcherds' product. Its
logarithmic derivative counts nonsingular rational curves on K3
surface when M is an unimodular lattice.
- JOHANNES WALCHER, Institute for Advanced Study, Einstein Drive, Princeton, NJ
Open Mirror Symmetry on the Quintic
The physics of string theory leads to interesting enumerative results
for curves in Calabi-Yau manifolds. The most recent example concerns
the number of holomorphic disks ending on the real Lagrangian in the
quintic three-fold. This prediction has now been proven and can also
be interpreted in the context of homological mirror symmetry.
The open mirror theorem is joint work with R. Pandharipande and
- SHING-TUNG YAU, Harvard University
Complex manifolds with torsion
My talk will be on the construction of solution to a supersymmetric
configuration for the compactification of Heterotic string model due
It consists of constructing nonKahler manifolds with hermitian metrics
coupled with Hermitian Yang Mills connections.
- JENG-DAW YU, Queen's University, Kingston, ON, Canada
Local structure on non-supersingular Newton strata of K3
Over finite characteristic, the moduli space of K3 surfaces has a
natural stratification according to the heights of the associated
formal Brauer groups. We show that the formal completion of a
non-supersingular stratum at a point has an extension structure by a
- NORIKO YUI, Queen's University
Motives, Mirror Symmetry and Modularity
We consider certain families of Calabi-Yau orbifolds and their mirror
partners constructed from Fermat hypersurfaces in weighted projective
spaces. We use Fermat motives to interpret the topological mirror
symmetry phenonemon. These Calabi-Yau orbifolds are defined over
Q, and we can discuss the modularity of the associated
We address the modularity question at motivic level. We give some
examples of modular Fermat motives. We then formulate a modularity
conjecture about rank 4 Fermat motives that there exist Siegel
modular forms on some congruence subgroups of Sp(4,Z).