126. CJM 2003 (vol 55 pp. 1100)
 Khesin, Boris; Rosly, Alexei

Polar Homology
For complex projective manifolds we introduce polar homology
groups, which are holomorphic analogues of the homology groups in
topology. The polar $k$chains are subvarieties of complex
dimension $k$ with meromorphic forms on them, while the boundary
operator is defined by taking the polar divisor and the Poincar\'e
residue on it. One can also define the corresponding analogues for the
intersection and linking numbers of complex submanifolds, which have the
properties similar to those of the corresponding topological notions.
Keywords:Poincar\' e residue, holomorphic linking Categories:14C10, 14F10, 58A14 

127. CJM 2003 (vol 55 pp. 897)
 Archinard, Natália

Hypergeometric Abelian Varieties
In this paper, we construct abelian varieties associated to Gauss' and
AppellLauricella hypergeometric series.
Abelian varieties of this kind and the algebraic curves we define
to construct them were considered by several authors in settings
ranging from monodromy groups (Deligne, Mostow), exceptional sets
(Cohen, Wolfart, W\"ustholz), modular embeddings (Cohen, Wolfart) to
CMtype (Cohen, Shiga, Wolfart) and modularity (Darmon).
Our contribution is to provide a complete, explicit and selfcontained
geometric construction.
Categories:11, 14 

128. CJM 2003 (vol 55 pp. 766)
 Kerler, Thomas

Homology TQFT's and the AlexanderReidemeister Invariant of 3Manifolds via Hopf Algebras and Skein Theory
We develop an explicit skeintheoretical algorithm to compute the
Alexander polynomial of a 3manifold from a surgery presentation
employing the methods used in the construction of quantum invariants
of 3manifolds. As a prerequisite we establish and prove a rather
unexpected equivalence between the topological quantum field theory
constructed by Frohman and Nicas using the homology of
$U(1)$representation varieties on the one side and the
combinatorially constructed Hennings TQFT based on the quasitriangular
Hopf algebra $\mathcal{N} = \mathbb{Z}/2 \ltimes \bigwedge^*
\mathbb{R}^2$ on the other side. We find that both TQFT's are $\SL
(2,\mathbb{R})$equivariant functors and, as such, are isomorphic.
The $\SL (2,\mathbb{R})$action in the Hennings construction comes
from the natural action on $\mathcal{N}$ and in the case of the
FrohmanNicas theory from the HardLefschetz decomposition of the
$U(1)$moduli spaces given that they are naturally K\"ahler. The
irreducible components of this TQFT, corresponding to simple
representations of $\SL(2,\mathbb{Z})$ and $\Sp(2g,\mathbb{Z})$, thus
yield a large family of homological TQFT's by taking sums and products.
We give several examples of TQFT's and invariants that appear to fit
into this family, such as Milnor and Reidemeister Torsion,
SeibergWitten theories, Casson type theories for homology circles
{\it \`a la} Donaldson, higher rank gauge theories following Frohman
and Nicas, and the $\mathbb{Z}/p\mathbb{Z}$ reductions of
ReshetikhinTuraev theories over the cyclotomic integers $\mathbb{Z}
[\zeta_p]$. We also conjecture that the Hennings TQFT for
quantum$\mathfrak{sl}_2$ is the product of the ReshetikhinTuraev
TQFT and such a homological TQFT.
Categories:57R56, 14D20, 16W30, 17B37, 18D35, 57M27 

129. CJM 2003 (vol 55 pp. 839)
 Lee, Min Ho

Cohomology of Complex Torus Bundles Associated to Cocycles
Equivariant holomorphic maps of Hermitian symmetric domains into
Siegel upper half spaces can be used to construct families of
abelian varieties parametrized by locally symmetric spaces, which
can be regarded as complex torus bundles over the parameter spaces.
We extend the construction of such torus bundles using 2cocycles of
discrete subgroups of the semisimple Lie groups associated to the
given symmetric domains and investigate some of their properties.
In particular, we determine their cohomology along the fibers.
Categories:14K10, 14D06, 14F99 

130. CJM 2003 (vol 55 pp. 693)
 Borne, Niels

Une formule de RiemannRoch Ã©quivariante pour les courbes
Soit $G$ un groupe fini agissant sur une courbe alg\'ebrique
projective et lisse $X$ sur un corps alg\'ebriquement clos $k$. Dans
cet article, on donne une formule de RiemannRoch pour la
caract\'eristique d'Euler \'equivariante d'un $G$faisceau inversible
$\mathcal{L}$, \`a valeurs dans l'anneau $R_k (G)$ des caract\`eres du
groupe $G$. La formule donn\'ee a un bon comportement fonctoriel en
ce sens qu'elle rel\`eve la formule classique le long du morphisme
$\dim \colon R_k (G) \to \mathbb{Z}$, et est valable m\^eme pour une
action sauvage. En guise d'application, on montre comment calculer
explicitement le caract\`ere de l'espace des sections globales d'une
large classe de $G$faisceaux inversibles, en s'attardant sur le cas
particulier d\'elicat du faisceau des diff\'erentielles sur la courbe.
Keywords:group actions on varieties or schemes,, RiemannRoch theorems Categories:14L30, 14C40 

131. CJM 2003 (vol 55 pp. 649)
132. CJM 2003 (vol 55 pp. 609)
 Moraru, Ruxandra

Integrable Systems Associated to a Hopf Surface
A Hopf surface is the quotient of the complex surface $\mathbb{C}^2
\setminus \{0\}$ by an infinite cyclic group of dilations of
$\mathbb{C}^2$. In this paper, we study the moduli spaces
$\mathcal{M}^n$ of stable $\SL (2,\mathbb{C})$bundles on a Hopf
surface $\mathcal{H}$, from the point of view of symplectic geometry.
An important point is that the surface $\mathcal{H}$ is an elliptic
fibration, which implies that a vector bundle on $\mathcal{H}$ can be
considered as a family of vector bundles over an elliptic curve. We
define a map $G \colon \mathcal{M}^n \rightarrow \mathbb{P}^{2n+1}$
that associates to every bundle on $\mathcal{H}$ a divisor, called the
graph of the bundle, which encodes the isomorphism class of the bundle
over each elliptic curve. We then prove that the map $G$ is an
algebraically completely integrable Hamiltonian system, with respect
to a given Poisson structure on $\mathcal{M}^n$. We also give an
explicit description of the fibres of the integrable system. This
example is interesting for several reasons; in particular, since the
Hopf surface is not K\"ahler, it is an elliptic fibration that does
not admit a section.
Categories:14J60, 14D21, 14H70, 14J27 

133. CJM 2003 (vol 55 pp. 561)
134. CJM 2003 (vol 55 pp. 533)
 Edo, Eric

Automorphismes modÃ©rÃ©s de l'espace affine
Le probl\`eme de JungNagata ({\it cf.}\ [J], [N]) consiste \`a savoir
s'il existe des automorphismes de $k[x,y,z]$ qui ne sont pas
mod\'er\'es. Nous proposons une approche nouvelle de cette question,
fond\'ee sur l'utilisation de la th\'eorie des automates et du
polygone de Newton. Cette approche permet notamment de g\'en\'eraliser
de fa\c con significative les r\'esultats de [A].
The JungNagata's problem ({\it cf.}\ [J], [N]) asks if there exists
nontame (or wild) automorphisms of $k[x,y,z]$. We give a new way to
attack this question, based on the automata theory and the Newton
polygon. This new approch allows us to generalize significantly the
results of [A].
Keywords:tame automorphisms, automata, Newton polygon Category:14R10 

135. CJM 2003 (vol 55 pp. 331)
 Savitt, David

The Maximum Number of Points on a Curve of Genus $4$ over $\mathbb{F}_8$ is $25$
We prove that the maximum number of rational points on a smooth,
geometrically irreducible genus 4 curve over the field of 8 elements
is 25. The body of the paper shows that 27 points is not possible by
combining techniques from algebraic geometry with a computer
verification. The appendix shows that 26 points is not possible by
examining the zeta functions.
Categories:11G20, 14H25 

136. CJM 2003 (vol 55 pp. 248)
 Dhillon, Ajneet

A Generalized Torelli Theorem
Given a smooth projective curve $C$ of positive genus $g$, Torelli's
theorem asserts that the pair $\bigl( J(C),W^{g1} \bigr)$ determines
$C$. We show that the theorem is true with $W^{g1}$ replaced by
$W^d$ for each $d$ in the range $1\le d\le g1$.
Category:14H99 

137. CJM 2003 (vol 55 pp. 157)
 Shimada, Ichiro

Zariski Hyperplane Section Theorem for Grassmannian Varieties
Let $\phi \colon X\to M$ be a morphism from a smooth irreducible
complex quasiprojective variety $X$ to a Grassmannian variety $M$
such that the image is of dimension $\ge 2$. Let $D$ be a reduced
hypersurface in $M$, and $\gamma$ a general linear automorphism of
$M$. We show that, under a certain differentialgeometric condition
on $\phi(X)$ and $D$, the fundamental group $\pi_1 \bigl( (\gamma
\circ \phi)^{1} (M\setminus D) \bigr)$ is isomorphic to a central
extension of $\pi_1 (M\setminus D) \times \pi_1 (X)$ by the cokernel
of $\pi_2 (\phi) \colon \pi_2 (X) \to \pi_2 (M)$.
Categories:14F35, 14M15 

138. CJM 2003 (vol 55 pp. 133)
 Shimada, Ichiro

On the Zariskivan Kampen Theorem
Let $f \colon E\to B$ be a dominant morphism, where $E$ and $B$ are
smooth irreducible complex quasiprojective varieties, and let $F_b$
be the general fiber of $f$. We present conditions under which the
homomorphism $\pi_1 (F_b)\to \pi_1 (E)$ induced by the inclusion is
injective.
Category:14F35 

139. CJM 2002 (vol 54 pp. 1319)
 Yekutieli, Amnon

The Continuous Hochschild Cochain Complex of a Scheme
Let $X$ be a separated finite type scheme over a noetherian base ring
$\mathbb{K}$. There is a complex $\widehat{\mathcal{C}}^{\cdot} (X)$
of topological $\mathcal{O}_X$modules, called the complete Hochschild
chain complex of $X$. To any $\mathcal{O}_X$module
$\mathcal{M}$not necessarily quasicoherentwe assign the complex
$\mathcal{H}om^{\cont}_{\mathcal{O}_X} \bigl(
\widehat{\mathcal{C}}^{\cdot} (X), \mathcal{M} \bigr)$ of continuous
Hochschild cochains with values in $\mathcal{M}$. Our first main
result is that when $X$ is smooth over $\mathbb{K}$ there is a
functorial isomorphism
$$
\mathcal{H}om^{\cont}_{\mathcal{O}_X} \bigl(
\widehat{\mathcal{C}}^{\cdot} (X), \mathcal{M} \bigr) \cong \R
\mathcal{H}om_{\mathcal{O}_{X^2}} (\mathcal{O}_X, \mathcal{M})
$$
in the derived category $\mathsf{D} (\Mod \mathcal{O}_{X^2})$, where
$X^2 := X \times_{\mathbb{K}} X$.
The second main result is that if $X$ is smooth of relative dimension
$n$ and $n!$ is invertible in $\mathbb{K}$, then the standard maps
$\pi \colon \widehat{\mathcal{C}}^{q} (X) \to \Omega^q_{X/
\mathbb{K}}$ induce a quasiisomorphism
$$
\mathcal{H}om_{\mathcal{O}_X} \Bigl( \bigoplus_q \Omega^q_{X/
\mathbb{K}} [q], \mathcal{M} \Bigr) \to
\mathcal{H}om^{\cont}_{\mathcal{O}_X} \bigl(
\widehat{\mathcal{C}}^{\cdot} (X), \mathcal{M} \bigr).
$$
When $\mathcal{M} = \mathcal{O}_X$ this is the quasiisomorphism
underlying the Kontsevich Formality Theorem.
Combining the two results above we deduce a decomposition of the
global Hochschild cohomology
$$
\Ext^i_{\mathcal{O}_{X^2}} (\mathcal{O}_X, \mathcal{M}) \cong
\bigoplus_q \H^{iq} \Bigl( X, \bigl( \bigwedge^q_{\mathcal{O}_X}
\mathcal{T}_{X/\mathbb{K}} \bigr) \otimes_{\mathcal{O}_X} \mathcal{M}
\Bigr),
$$
where $\mathcal{T}_{X/\mathbb{K}}$ is the relative tangent sheaf.
Keywords:Hochschild cohomology, schemes, derived categories Categories:16E40, 14F10, 18G10, 13H10 

140. CJM 2002 (vol 54 pp. 554)
 Hausen, Jürgen

Equivariant Embeddings into Smooth Toric Varieties
We characterize embeddability of algebraic varieties into smooth toric
varieties and prevarieties. Our embedding results hold also in an
equivariant context and thus generalize a wellknown embedding theorem
of Sumihiro on quasiprojective $G$varieties. The main idea is to
reduce the embedding problem to the affine case. This is done by
constructing equivariant affine conoids, a tool which extends the
concept of an equivariant affine cone over a projective $G$variety to
a more general framework.
Categories:14E25, 14C20, 14L30, 14M25 

141. CJM 2002 (vol 54 pp. 595)
 Nahlus, Nazih

Lie Algebras of ProAffine Algebraic Groups
We extend the basic theory of Lie algebras of affine algebraic groups
to the case of proaffine algebraic groups over an algebraically
closed field $K$ of characteristic 0. However, some modifications
are needed in some extensions. So we introduce the prodiscrete
topology on the Lie algebra $\mathcal{L}(G)$ of the proaffine
algebraic group $G$ over $K$, which is discrete in the
finitedimensional case and linearly compact in general. As an
example, if $L$ is any sub Lie algebra of $\mathcal{L}(G)$, we show
that the closure of $[L,L]$ in $\mathcal{L}(G)$ is algebraic in
$\mathcal{L}(G)$.
We also discuss the Hopf algebra of representative functions $H(L)$ of
a residually finite dimensional Lie algebra $L$. As an example, we
show that if $L$ is a sub Lie algebra of $\mathcal{L}(G)$ and $G$
is connected, then the canonical Hopf algebra morphism from $K[G]$
into $H(L)$ is injective if and only if $L$ is algebraically dense
in $\mathcal{L}(G)$.
Categories:14L, 16W, 17B45 

142. CJM 2002 (vol 54 pp. 352)
 Haines, Thomas J.

On Connected Components of Shimura Varieties
We study the cohomology of connected components of Shimura varieties
$S_{K^p}$ coming from the group $\GSp_{2g}$, by an approach modeled on
the stabilization of the twisted trace formula, due to Kottwitz and
Shelstad. More precisely, for each character $\olomega$ on
the group of connected components of $S_{K^p}$ we define an operator
$L(\omega)$ on the cohomology groups with compact supports $H^i_c
(S_{K^p}, \olbbQ_\ell)$, and then we prove that the virtual
trace of the composition of $L(\omega)$ with a Hecke operator $f$ away
from $p$ and a sufficiently high power of a geometric Frobenius
$\Phi^r_p$, can be expressed as a sum of $\omega${\em weighted}
(twisted) orbital integrals (where $\omega${\em weighted} means that
the orbital integrals and twisted orbital integrals occuring here each
have a weighting factor coming from the character $\olomega$).
As the crucial step, we define and study a new invariant $\alpha_1
(\gamma_0; \gamma, \delta)$ which is a refinement of the invariant
$\alpha (\gamma_0; \gamma, \delta)$ defined by Kottwitz. This is done
by using a theorem of Reimann and Zink.
Categories:14G35, 11F70 

143. CJM 2002 (vol 54 pp. 55)
 Ban, Chunsheng; McEwan, Lee J.; Némethi, András

On the Milnor Fiber of a Quasiordinary Surface Singularity
We verify a generalization of (3.3) from \cite{Le} proving
that the homotopy type of the Milnor fiber of a reduced
hypersurface singularity depends only on the embedded
topological type of the singularity. In particular, using
\cite{Za,Li1,Oh1,Gau} for irreducible quasiordinary germs,
it depends only on the normalized distinguished pairs of the
singularity. The main result of the paper provides an explicit
formula for the Eulercharacteristic of the Milnor fiber in the
surface case.
Categories:14B05, 14E15, 32S55 

144. CJM 2001 (vol 53 pp. 1309)
145. CJM 2001 (vol 53 pp. 923)
 Geramita, Anthony V.; Harima, Tadahito; Shin, Yong Su

Decompositions of the Hilbert Function of a Set of Points in $\P^n$
Let $\H$ be the Hilbert function of some set of distinct points
in $\P^n$ and let $\alpha = \alpha (\H)$ be the least degree
of a hypersurface of $\P^n$ containing these points. Write $\alpha
= d_s + d_{s1} + \cdots + d_1$ (where $d_i > 0$). We canonically
decompose $\H$ into $s$ other Hilbert functions $\H
\leftrightarrow (\H_s^\prime, \dots, \H_1^\prime)$ and show
how to find sets of distinct points $\Y_s, \dots, \Y_1$,
lying on reduced hypersurfaces of degrees $d_s, \dots, d_1$
(respectively) such that the Hilbert function of $\Y_i$ is
$\H_i^\prime$ and the Hilbert function of $\Y = \bigcup_{i=1}^s
\Y_i$ is $\H$. Some extremal properties of this canonical
decomposition are also explored.
Categories:13D40, 14M10 

146. CJM 2001 (vol 53 pp. 834)
 Veys, Willem

Zeta Functions and `Kontsevich Invariants' on Singular Varieties
Let $X$ be a nonsingular algebraic variety in characteristic zero. To
an effective divisor on $X$ Kontsevich has associated a certain
motivic integral, living in a completion of the Grothendieck ring of
algebraic varieties. He used this invariant to show that birational
(smooth, projective) CalabiYau varieties have the same Hodge
numbers. Then Denef and Loeser introduced the invariant {\it motivic
(Igusa) zeta function}, associated to a regular function on $X$, which
specializes to both the classical $p$adic Igusa zeta function and the
topological zeta function, and also to Kontsevich's invariant.
This paper treats a generalization to singular varieties. Batyrev
already considered such a `Kontsevich invariant' for log terminal
varieties (on the level of Hodge polynomials of varieties instead of
in the Grothendieck ring), and previously we introduced a motivic zeta
function on normal surface germs. Here on any $\bbQ$Gorenstein
variety $X$ we associate a motivic zeta function and a `Kontsevich
invariant' to effective $\bbQ$Cartier divisors on $X$ whose support
contains the singular locus of~$X$.
Keywords:singularity invariant, topological zeta function, motivic zeta function Categories:14B05, 14E15, 32S50, 32S45 

147. CJM 2001 (vol 53 pp. 73)
 Fukui, Toshizumi; Paunescu, Laurentiu

Stratification Theory from the Weighted Point of View
In this paper, we investigate stratification theory in terms of the
defining equations of strata and maps (without tube systems), offering
a concrete approach to show that some given family is topologically
trivial. In this approach, we consider a weighted version of
$(w)$regularity condition and Kuo's ratio test condition.
Categories:32B99, 14P25, 32Cxx, 58A35 

148. CJM 2001 (vol 53 pp. 3)
 Bell, J. P.

The Equivariant Grothendieck Groups of the RussellKoras Threefolds
The RussellKoras contractible threefolds are the smooth affine threefolds
having a hyperbolic $\mathbb{C}^*$action with quotient isomorphic to the
corresponding quotient of the linear action on the tangent space at the
unique fixed point. Koras and Russell gave a concrete description of all such
threefolds and determined many interesting properties they possess.
We use this description and these properties to compute the equivariant
Grothendieck groups of these threefolds. In addition, we give certain
equivariant invariants of these rings.
Categories:14J30, 19L47 

149. CJM 2000 (vol 52 pp. 1235)
 Hurtubise, J. C.; Jeffrey, L. C.

Representations with Weighted Frames and Framed Parabolic Bundles
There is a wellknown correspondence (due to Mehta and Seshadri in
the unitary case, and extended by Bhosle and Ramanathan to other
groups), between the symplectic variety $M_h$ of representations of
the fundamental group of a punctured Riemann surface into a compact
connected Lie group~$G$, with fixed conjugacy classes $h$ at the
punctures, and a complex variety ${\cal M}_h$ of holomorphic bundles
on the unpunctured surface with a parabolic structure at the puncture
points. For $G = \SU(2)$, we build a symplectic variety $P$ of pairs
(representations of the fundamental group into $G$, ``weighted frame''
at the puncture points), and a corresponding complex variety ${\cal
P}$ of moduli of ``framed parabolic bundles'', which encompass
respectively all of the spaces $M_h$, ${\cal M}_h$, in the sense that
one can obtain $M_h$ from $P$ by symplectic reduction, and ${\cal
M}_h$ from ${\cal P}$ by a complex quotient. This allows us to
explain certain features of the toric geometry of the $\SU(2)$ moduli
spaces discussed by Jeffrey and Weitsman, by giving the actual toric
variety associated with their integrable system.
Categories:58F05, 14D20 

150. CJM 2000 (vol 52 pp. 1149)
 Ban, Chunsheng; McEwan, Lee J.

Canonical Resolution of a Quasiordinary Surface Singularity
We describe the embedded resolution of an irreducible quasiordinary
surface singularity $(V,p)$ which results from applying the canonical
resolution of BierstoneMilman to $(V,p)$. We show that this process
depends solely on the characteristic pairs of $(V,p)$, as predicted
by Lipman. We describe the process explicitly enough that a resolution
graph for $f$ could in principle be obtained by computer using only
the characteristic pairs.
Keywords:canonical resolution, quasiordinary singularity Categories:14B05, 14J17, 32S05, 32S25 
