CYCLES, MOTIVES AND SHIMURA VARIETIES
TITLES AND ABSTRACTS
- C.S. Rajan
Title: Spectrum and Arithmetic
Abstract: We will discuss the relationship between spectrum and
arithmetic especially in the context of locally symmetric spaces. In
this context we raise some conjectures and give some partial evidence
that the arithmetic and the spectrum of such spaces should mutually
determine each other.
- C. Soule
Title: Motivic weight complexes for arithmetic varieties
Abstract:
Joint work with Henri Gillet. If k is a field of characteristic
zero and $X$ a reduced variety over $k$, one can attach
to $X$ a virtual Chow motive over $k$ in such a way
that, for any closed subvariety $Y$ in $X$, the motive of $X$
is the sum of the motives of $Y$ and $X-Y$. This resut was also obtained
by Guillen and Navarro. Its proof uses Hironaka's theorem.
In this talk, I shall discuss how De Jong's theorem can be
used to extend the result to varieties over a field
of arbitrary characteristic or over the ring of integers
in a number field.
- A. Nair
Title: On the motive of a Shimura variety.
- Manjul Bhargava
Title: Sums of Squares and the "290-Theorem"
Abstract: The famous "Four Squares Theorem" of Lagrange asserts
that any positive
integer can be expressed as the sum of four square numbers. That is, the
quadratic form $a^2 + b^2 + c^2 + d^2$ ``represents'' all (positive)
integers. When does a general quadratic form represent all integers?
When does it represent all odd integers? When does it represent all primes?
We show how all these questions turn out to have very simple and surprising
answers. In particular, we describe the recent work (joint with Jonathan
Hanke, Duke University) in proving Conway's "290-Conjecture".
- P. Griffiths
Title: Singularities of admissable normal functions
Abstract: pdf file
- V. Kumar Murty
Title: Tate cycles on Abelian varieties.
Abstract: For A an Abelian variety defined over a number field K,
we discuss some results about the relationship between the space of Tate
cycles on A and on the reduction modulo a prime v of K. For a simple
Abelian variety of CM type (defined over a number field), we show that
for a set of primes v of density 1, reduction mod v does not increase
the space of Tate cycles.
- Spencer Bloch
Title: limiting mixed Hodge structures and renormalization in
physics.
Abstract: Not given
- M.Asakura
Title: Indecomposable $K_1$ of elliptic surface and nonvanishing
of the p-adic regulator
Abstract: This is a joint work with K. Sato.
Let X be an elliptic surface defined over a number field $F$.
Let $D_i$ be the multiplicative singular fibers.
Put $U=X-\cup D_i$.
In this talk we show a computable method to obtain an upper bound of the
rank
of the Galois fixed part
$H^2(U\times_F \bar{F},Q_p(2))^{G_F}$ of the p-adic etale cohomology
group.
This is an application of Kato's explicit reciprocity law and the theory
of
syntomic cohomology.
Our result can be applied to the construction of indecomposable elements
of
$K_1(X)$ whose p-adic regulators do not vanish.
- E. Ghate
Title: The splitting question for ordinary modular Galois
representations
Abstract: The global p-adic Galois representation attached to a
cusp form is irreducible, but the local representation at p is reducible,
if p is an ordinary prime.
A fundamental question, asked by Greenberg, is whether the
local representation is semi-simple. In this talk we show that
this is so exactly when the underlying form has CM. We work under
some technical hypothesis (e.g. p is odd). This is recent joint
work with Hida, and builds on earlier work with Vatsal.
- Kartik Prasanna
Title: Algebraic cycles and exotic Heegner points
Abstract: I will describe a new (conjectural) construction of
rational points on CM elliptic curves using cycles on higher
dimensional varieties and the Abel-Jacobi map. This will be a report on
joint work in progress with Massimo Bertolini and Henri Darmon.
- H. Esnault
Title: Some properties of the splittings of the Galois group
into the arithmetic fundamental group of hyberbolic curves.
Abstract: This is a report on some work in progress with Oliver
Wittenberg; partly based on joint work with Ph\`ung H\^o Hai.
- A. Hogadi
Title: Arithmetic of Rationally Connected Varieties
Abstract:ationally connected varieties are a special class of
varieties in which any two points can be joined by a rational curve.
After a survey of known arithmetic results about these varieties, I will
describe a recent joint work with Chenyang Xu on arithmetic of singular
degenerations of such varieties.
- J.P.Murre
Title: The Picard Motive revisited
Abstract: (joint work with S.I.Kimura)
Around 1990 the lecturer constructed for every smooth, projective
variety X a Chow motive closely related to the Picard variety of X.
In the construction enter, by its very nature, a number of choices,
but it was conjectured that the motive itself is unique up to
isomorphism.
In the lecture we show now -and this is joint work with Shun-Ichi
Kimura- that indeed this " Picard Motive" is independent of the
choices
in the construction, i.e. it is by its intrinsic properties uniquely
determined up to a natural isomorphism .
- Jaya Iyer
Title: Regulators of canonical extensions
Abstract: In this talk, we will discuss the question of
Cheeger-Simons on the torsion property of the Chern Simons classes of
flat connections. This is related to Bloch's conjecture on triviality
of the Chern classes of flat bundles in the rational Deligne
cohomology. We discuss a recent joint work with C. Simpson, extending
this conjecture to canonical extensions with unipotent monodromy around
an irreducible smooth divisor.
- James Lewis
Title: Residues of Higher Chow Cycles
Abstract: Let $U/{\bf C}$ be a smooth quasiprojective
variety of dimension $d$,
${\rm CH}^{r}(U,m)$ Bloch's higher Chow group, and ${\rm cl}_{r,m} :
{\rm CH}^{r}(U,m)\otimes {\bf Q} \to \hom_{\rm MHS}$
$\big({\bf Q}(0), H^{2r-m}(U,{\bf Q}(r))\big)$ the cycle class map.
Beilinson once
conjectured ${\rm cl}_{r,m}$ to be surjective.
It was Jannsen however, who first
found a counterexample in the case $m=1$.
In this talk we discuss the map
${\rm cl}_{r,m}$ in more detail, and propose an amended version
of Beilinson's conjecture, generalizing the classical Hodge conjecture.
This is based on joint works with Rob de Jeu, and with
Su-Jeong Kang.
- A. Yafaev
Title: Manin-Mumford and Andre-Oort conjectures : a unified
approach.
Abstract: We present a unified approach to the Manin-Mumford and
the
Andre-Oort conjectures based on the combination of Galois-theoretic
and ergodic-theoretic
techniques.
- T. Terasoma
Title: DG category, Bar complex and their applications.
Abstract:For a DG algebra $A^{\bullet}$,we define a DG category
CA
associated to $A^{\bullet}$.In
DG category, we can define a notion of complexes, called DG complexes.
The category KC of DG complexes is naturally equipped with a structure
of
DG category. If A is connected, we can define a full sub category
$K^0CA$
of
KCA. The homotopy category of $K^0 CA$ is equivalent to (1)the category
of A
connection up to Gauge equivalence, and (2)the DG cageory of $H^0(B(A,
\in))²$)
comodule. We illustrate two examples as applications.
First application is to construct a coalgebra which classifies the
variation
of mixed Tate structure structure over a smooth variety X. The
associated
DGA is not graded commutative but is graded commutative up to homotpy.
For example it will be a dual of group algebra of Mumford-Tate group if
X
is a point.
Next application is to construct a ${\bf F}_p$ completion of etale
fundamental
group of a affine ${\bf F}_p$-scheme. We use Artin-Schreier DGA, which
is non-
commutative, but commutative up to homotopy.
- A. Rosenschon
Title: The modified Ceresa cycle modulo l
Abstract: We give examples of smooth projective varieties over
p-adic fields whose Chow group of codimension 2 cycles modulo certain
primes is infinite.
- C. Schoen
Title: Calabi-Yau threefolds with third Betti number zero.}
Abstract: We discuss examples and open problems concerning
smooth projective threefolds whose canonical sheaf is isomorphic
to the structure sheaf and for which the third l-adic cohomology
has rank zero. Varieties of this
description only occur when the characteristic of the base field
is positive. Examples have been constructed by Horikado,
Schroeer, Ekedahl, and the speaker using a variety of different
techniques.
- R. Sreekantan
Title: An analogue of the Hodge-D-conjecture
Abstract: In this talk we will discuss a non-Archimedean analogue
of the Hodge-D-conjecture regarding the surjectivity of the regulator
map.
- C. Weibel
Title: The Bloch-Kato Conjecture
Asbtract: About ten years ago, Rost and Voevodsky announced a
program to prove the Bloch-Kato conjecture, and thereby the
Quillen-Lichtenbaum conjectures. Rost's part is being written up.
Assuming that, we provide apatch to Voevodsky's part which finishes the
proof of this conjecture.
- A. Miller
Title: Chow-Motives of some mixed Shimura varieties.
Abstract: I will describe Chow-Kuenneth decompositions of
universal families over some unitary Shimura surfaces.
- Marc Levine
Title: Algebraic cobordism and the algebraic Thom complex
Abstract: Let $k$ be a field of characteristic zero. Together
with F. Morel, we have defined the algebraic cobordism $\Omega_*(X)$ of
a quasi-projective $k$-scheme $X$, simultaneously generalizing the
Chow groups $CH_*(X)$ and the Grothendieck group of coherent sheaves
$G_0(X)$; the theory $\Omega_*$ is the universal oriented Borel-
Moore homology theory on quasi-projective $k$-schemes. The algebraic
Thom complex $MGL$ in the Morel-Voevodsky motivic stable homotopy
category over $k$ gives rise to the universal oriented bi-graded
cohomology theory on smooth $k$-schemes. Relying on work of Panin and
Mocanasu, we show how to extend such a theory to an ``oriented
duality theory"; this yields in particular the Borel-Moore homology
$MGL'_{*,*}(X)$ for a quasi-projective $k$-scheme $X$. The universal
property of $\Omega_*$ gives a natural transformation $\vartheta:
\Omega_*\to MGL'_{2*,*}$; our main result is that $\vartheta(X)$ is
an isomorphism for all quasi-projective $k$-schemes $X$.
- K.H. Paranjape
Title: Integral modular forms and Calabi-Yau varieties
Abstract: The talk will present joint work with Dinakar
Ramakrishnan on the relationship between modular forms and Calabi-Yau
varieties. We explore the possibility that for every integral modular
form f there is a Calabi-Yau variety which contains the motive of f. To
substantiate this we have constructed a number of examples.
- A. Neeman
Title: Grothendieck duality via the homotopy category of flat
modules.
Abstract:
In the last couple of years we have learned a new way of looking at
dualizing complexes. I will discuss results due to Iyengar, Krause,
Jorgensen, myself and Murfet (a total of maybe six or seven articles, by
combinations of the authors cited).
- Kenichiro Kimura
Title: On the second Abel-Jacobi Map
Abstract: I will explain how ''Basic Lemma'' of Beilinson-Nori
can be applied to give a simple description of the complex $Rf_*Z_l$
for a variety $X$ with a structure morphism $f$. Then I will explain about
its application to give a simple a description of the second
Abel-Jacobi map.
- E. Looijenga
Title: A Baily-Borel package for certain period maps
Abstract: The undefined notion in the title packs
the essential ingredients of the Baily-Borel theory
(which are algebraic, geometric and topological
in character). This is available for some incomplete
locally symmetric varieties. The period maps for
K3 surfaces in a fixed projective space and for cubic
4-folds provide examples.
- A. Krishna
Title: Perfect complexes on Deligne-Mumford stacks and
applications
Abstract: We study the unbounded derived category of
quasi-coherent sheaves on certain Deligne-Mumford stacks.
As an application, we establish Thomason-Trobaugh
localization theorem for the K-theory of perfect complexes on
such stacks. We study Thomason's classification of thick tensor
triangulated categories of $D({perf}/X)$, and then identify the
spectrum of this category as defined by Balmer, with the moduli
scheme (if it exists) of the stack.
- D. Arapura
Title: A category of motivic sheaves.
Abstract: I want explain how to define a category
of motivic "sheaves" over any variety defined over a
subfield of C. This uses a modification of a construction
due to Nori. This category is abelian, and admits realizations
to the categories of sheaves for etale and classical topologies.
There is also a Hodge realization on a subcategory of
motivic local systems.
- T. Ito
Title: Generalized Hasse invariants and applications
Abstract:
The classical Hasse invariant is a modular form of weight p-1
in characteristic p which has a simple zero at each supersingular
point. In this talk, we will discuss a generalization of
the Hasse invariant to unitary Shimura varieties with
signature (1,n-1). We will also discuss some applications
to modular forms on unitary Shimura varieties
and l-adic Galois representations.
- Uwe Jannsen
Title: On embedded resolution of singularities for excellent
two-dimensional schemes
Abstract: This is a report on joint work in progress with Shuji
Saito and Vincent Cossart, on embedded resolution in the two-dimensional case.
There are two parts: First a resolution by blowing ups in regular
centers,
which automatically implies the embedded resolution, secondly the
additional
control of the position with respect to to a normal crossings divisor in
the embedded situation.
- V. Srinivas
Title: Zero cycles on affine varieties
Abstract:
In this talk, after motivating the study of Chow groups of 0-cycles on
affine varieties, I will survey results, conjectures and computations for these
Chow groups. I will report at the end on some very recent joint work
with Amalendu Krishna, towards verifying one of the conjectures in a
certain case.