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126. 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

127. CJM 2001 (vol 53 pp. 3)

Bell, J. P.
The Equivariant Grothendieck Groups of the Russell-Koras Threefolds
The Russell-Koras 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

128. CJM 2000 (vol 52 pp. 1235)

Hurtubise, J. C.; Jeffrey, L. C.
Representations with Weighted Frames and Framed Parabolic Bundles
There is a well-known 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

129. CJM 2000 (vol 52 pp. 1149)

Ban, Chunsheng; McEwan, Lee J.
Canonical Resolution of a Quasi-ordinary Surface Singularity
We describe the embedded resolution of an irreducible quasi-ordinary surface singularity $(V,p)$ which results from applying the canonical resolution of Bierstone-Milman 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, quasi-ordinary singularity
Categories:14B05, 14J17, 32S05, 32S25

130. CJM 2000 (vol 52 pp. 1018)

Reichstein, Zinovy; Youssin, Boris
Essential Dimensions of Algebraic Groups and a Resolution Theorem for $G$-Varieties
Let $G$ be an algebraic group and let $X$ be a generically free $G$-variety. We show that $X$ can be transformed, by a sequence of blowups with smooth $G$-equivariant centers, into a $G$-variety $X'$ with the following property the stabilizer of every point of $X'$ is isomorphic to a semidirect product $U \sdp A$ of a unipotent group $U$ and a diagonalizable group $A$. As an application of this result, we prove new lower bounds on essential dimensions of some algebraic groups. We also show that certain polynomials in one variable cannot be simplified by a Tschirnhaus transformation.

Categories:14L30, 14E15, 14E05, 12E05, 20G10

131. CJM 2000 (vol 52 pp. 982)

Lárusson, Finnur
Holomorphic Functions of Slow Growth on Nested Covering Spaces of Compact Manifolds
Let $Y$ be an infinite covering space of a projective manifold $M$ in $\P^N$ of dimension $n\geq 2$. Let $C$ be the intersection with $M$ of at most $n-1$ generic hypersurfaces of degree $d$ in $\mathbb{P}^N$. The preimage $X$ of $C$ in $Y$ is a connected submanifold. Let $\phi$ be the smoothed distance from a fixed point in $Y$ in a metric pulled up from $M$. Let $\O_\phi(X)$ be the Hilbert space of holomorphic functions $f$ on $X$ such that $f^2 e^{-\phi}$ is integrable on $X$, and define $\O_\phi(Y)$ similarly. Our main result is that (under more general hypotheses than described here) the restriction $\O_\phi(Y) \to \O_\phi(X)$ is an isomorphism for $d$ large enough. This yields new examples of Riemann surfaces and domains of holomorphy in $\C^n$ with corona. We consider the important special case when $Y$ is the unit ball $\B$ in $\C^n$, and show that for $d$ large enough, every bounded holomorphic function on $X$ extends to a unique function in the intersection of all the nontrivial weighted Bergman spaces on $\B$. Finally, assuming that the covering group is arithmetic, we establish three dichotomies concerning the extension of bounded holomorphic and harmonic functions from $X$ to $\B$.

Categories:32A10, 14E20, 30F99, 32M15

132. CJM 2000 (vol 52 pp. 265)

Brion, Michel; Helminck, Aloysius G.
On Orbit Closures of Symmetric Subgroups in Flag Varieties
We study $K$-orbits in $G/P$ where $G$ is a complex connected reductive group, $P \subseteq G$ is a parabolic subgroup, and $K \subseteq G$ is the fixed point subgroup of an involutive automorphism $\theta$. Generalizing work of Springer, we parametrize the (finite) orbit set $K \setminus G \slash P$ and we determine the isotropy groups. As a consequence, we describe the closed (resp. affine) orbits in terms of $\theta$-stable (resp. $\theta$-split) parabolic subgroups. We also describe the decomposition of any $(K,P)$-double coset in $G$ into $(K,B)$-double cosets, where $B \subseteq P$ is a Borel subgroup. Finally, for certain $K$-orbit closures $X \subseteq G/B$, and for any homogeneous line bundle $\mathcal{L}$ on $G/B$ having nonzero global sections, we show that the restriction map $\res_X \colon H^0 (G/B, \mathcal{L}) \to H^0 (X, \mathcal{L})$ is surjective and that $H^i (X, \mathcal{L}) = 0$ for $i \geq 1$. Moreover, we describe the $K$-module $H^0 (X, \mathcal{L})$. This gives information on the restriction to $K$ of the simple $G$-module $H^0 (G/B, \mathcal{L})$. Our construction is a geometric analogue of Vogan and Sepanski's approach to extremal $K$-types.

Keywords:flag variety, symmetric subgroup
Categories:14M15, 20G05

133. CJM 2000 (vol 52 pp. 348)

González Pérez, P. D.
Singularités quasi-ordinaires toriques et polyèdre de Newton du discriminant
Nous \'etudions les polyn\^omes $F \in \C \{S_\tau\} [Y] $ \`a coefficients dans l'anneau de germes de fonctions holomorphes au point sp\'ecial d'une vari\'et\'e torique affine. Nous g\'en\'eralisons \`a ce cas la param\'etrisation classique des singularit\'es quasi-ordinaires. Cela fait intervenir d'une part une g\'en\'eralization de l'algorithme de Newton-Puiseux, et d'autre part une relation entre le poly\`edre de Newton du discriminant de $F$ par rapport \`a $Y$ et celui de $F$ au moyen du polytope-fibre de Billera et Sturmfels~\cite{Sturmfels}. Cela nous permet enfin de calculer, sous des hypoth\`eses de non d\'eg\'en\'erescence, les sommets du poly\`edre de Newton du discriminant a partir de celui de $F$, et les coefficients correspondants \`a partir des coefficients des exposants de $F$ qui sont dans les ar\^etes de son poly\`edre de Newton.

Categories:14M25, 32S25

134. CJM 2000 (vol 52 pp. 123)

Harbourne, Brian
An Algorithm for Fat Points on $\mathbf{P}^2
Let $F$ be a divisor on the blow-up $X$ of $\pr^2$ at $r$ general points $p_1, \dots, p_r$ and let $L$ be the total transform of a line on $\pr^2$. An approach is presented for reducing the computation of the dimension of the cokernel of the natural map $\mu_F \colon \Gamma \bigl( \CO_X(F) \bigr) \otimes \Gamma \bigl( \CO_X(L) \bigr) \to \Gamma \bigl( \CO_X(F) \otimes \CO_X(L) \bigr)$ to the case that $F$ is ample. As an application, a formula for the dimension of the cokernel of $\mu_F$ is obtained when $r = 7$, completely solving the problem of determining the modules in minimal free resolutions of fat point subschemes\break $m_1 p_1 + \cdots + m_7 p_7 \subset \pr^2$. All results hold for an arbitrary algebraically closed ground field~$k$.

Keywords:Generators, syzygies, resolution, fat points, maximal rank, plane, Weyl group
Categories:13P10, 14C99, 13D02, 13H15

135. CJM 1999 (vol 51 pp. 1175)

Lehrer, G. I.; Springer, T. A.
Reflection Subquotients of Unitary Reflection Groups
Let $G$ be a finite group generated by (pseudo-) reflections in a complex vector space and let $g$ be any linear transformation which normalises $G$. In an earlier paper, the authors showed how to associate with any maximal eigenspace of an element of the coset $gG$, a subquotient of $G$ which acts as a reflection group on the eigenspace. In this work, we address the questions of irreducibility and the coexponents of this subquotient, as well as centralisers in $G$ of certain elements of the coset. A criterion is also given in terms of the invariant degrees of $G$ for an integer to be regular for $G$. A key tool is the investigation of extensions of invariant vector fields on the eigenspace, which leads to some results and questions concerning the geometry of intersections of invariant hypersurfaces.

Categories:51F15, 20H15, 20G40, 20F55, 14C17

136. CJM 1999 (vol 51 pp. 1226)

McKay, John
Semi-Affine Coxeter-Dynkin Graphs and $G \subseteq \SU_2(C)$
The semi-affine Coxeter-Dynkin graph is introduced, generalizing both the affine and the finite types.

Categories:20C99, 05C25, 14B05

137. CJM 1999 (vol 51 pp. 1123)

Arnold, V. I.
First Steps of Local Contact Algebra
We consider germs of mappings of a line to contact space and classify the first simple singularities up to the action of contactomorphisms in the target space and diffeomorphisms of the line. Even in these first cases there arises a new interesting interaction of local commutative algebra with contact structure.

Keywords:contact manifolds, local contact algebra, Diracian, contactian
Categories:53D10, 14B05

138. CJM 1999 (vol 51 pp. 1089)

Vakil, Ravi
The Characteristic Numbers of Quartic Plane Curves
The characteristic numbers of smooth plane quartics are computed using intersection theory on a component of the moduli space of stable maps. This completes the verification of Zeuthen's prediction of characteristic numbers of smooth plane curves. A short sketch of a computation of the characteristic numbers of plane cubics is also given as an illustration.

Categories:14N10, 14D22

139. CJM 1999 (vol 51 pp. 936)

David, Chantal; Kisilevsky, Hershy; Pappalardi, Francesco
Galois Representations with Non-Surjective Traces
Let $E$ be an elliptic curve over $\q$, and let $r$ be an integer. According to the Lang-Trotter conjecture, the number of primes $p$ such that $a_p(E) = r$ is either finite, or is asymptotic to $C_{E,r} {\sqrt{x}} / {\log{x}}$ where $C_{E,r}$ is a non-zero constant. A typical example of the former is the case of rational $\ell$-torsion, where $a_p(E) = r$ is impossible if $r \equiv 1 \pmod{\ell}$. We prove in this paper that, when $E$ has a rational $\ell$-isogeny and $\ell \neq 11$, the number of primes $p$ such that $a_p(E) \equiv r \pmod{\ell}$ is finite (for some $r$ modulo $\ell$) if and only if $E$ has rational $\ell$-torsion over the cyclotomic field $\q(\zeta_\ell)$. The case $\ell=11$ is special, and is also treated in the paper. We also classify all those occurences.

Category:14H52

140. CJM 1999 (vol 51 pp. 771)

Flicker, Yuval Z.
Stable Bi-Period Summation Formula and Transfer Factors
This paper starts by introducing a bi-periodic summation formula for automorphic forms on a group $G(E)$, with periods by a subgroup $G(F)$, where $E/F$ is a quadratic extension of number fields. The split case, where $E = F \oplus F$, is that of the standard trace formula. Then it introduces a notion of stable bi-conjugacy, and stabilizes the geometric side of the bi-period summation formula. Thus weighted sums in the stable bi-conjugacy class are expressed in terms of stable bi-orbital integrals. These stable integrals are on the same endoscopic groups $H$ which occur in the case of standard conjugacy. The spectral side of the bi-period summation formula involves periods, namely integrals over the group of $F$-adele points of $G$, of cusp forms on the group of $E$-adele points on the group $G$. Our stabilization suggests that such cusp forms---with non vanishing periods---and the resulting bi-period distributions associated to ``periodic'' automorphic forms, are related to analogous bi-period distributions associated to ``periodic'' automorphic forms on the endoscopic symmetric spaces $H(E)/H(F)$. This offers a sharpening of the theory of liftings, where periods play a key role. The stabilization depends on the ``fundamental lemma'', which conjectures that the unit elements of the Hecke algebras on $G$ and $H$ have matching orbital integrals. Even in stating this conjecture, one needs to introduce a ``transfer factor''. A generalization of the standard transfer factor to the bi-periodic case is introduced. The generalization depends on a new definition of the factors even in the standard case. Finally, the fundamental lemma is verified for $\SL(2)$.

Categories:11F72, 11F70, 14G27, 14L35

141. CJM 1999 (vol 51 pp. 616)

Panyushev, Dmitri I.
Parabolic Subgroups with Abelian Unipotent Radical as a Testing Site for Invariant Theory
Let $L$ be a simple algebraic group and $P$ a parabolic subgroup with Abelian unipotent radical $P^u$. Many familiar varieties (determinantal varieties, their symmetric and skew-symmetric analogues) arise as closures of $P$-orbits in $P^u$. We give a unified invariant-theoretic treatment of various properties of these orbit closures. We also describe the closures of the conormal bundles of these orbits as the irreducible components of some commuting variety and show that the polynomial algebra $k[P^u]$ is a free module over the algebra of covariants.

Categories:14L30, 13A50

142. CJM 1998 (vol 50 pp. 1209)

Fukuma, Yoshiaki
A lower bound for $K_X L$ of quasi-polarized surfaces $(X,L)$ with non-negative Kodaira dimension
Let $X$ be a smooth projective surface over the complex number field and let $L$ be a nef-big divisor on $X$. Here we consider the following conjecture; If the Kodaira dimension $\kappa(X)\geq 0$, then $K_{X}L\geq 2q(X)-4$, where $q(X)$ is the irregularity of $X$. In this paper, we prove that this conjecture is true if (1) the case in which $\kappa(X)=0$ or $1$, (2) the case in which $\kappa(X)=2$ and $h^{0}(L)\geq 2$, or (3) the case in which $\kappa(X)=2$, $X$ is minimal, $h^{0}(L)=1$, and $L$ satisfies some conditions.

Keywords:Quasi-polarized surface, sectional genus
Category:14C20

143. CJM 1998 (vol 50 pp. 1253)

López-Bautista, Pedro Ricardo; Villa-Salvador, Gabriel Daniel
Integral representation of $p$-class groups in ${\Bbb Z}_p$-extensions and the Jacobian variety
For an arbitrary finite Galois $p$-extension $L/K$ of $\zp$-cyclotomic number fields of $\CM$-type with Galois group $G = \Gal(L/K)$ such that the Iwasawa invariants $\mu_K^-$, $ \mu_L^-$ are zero, we obtain unconditionally and explicitly the Galois module structure of $\clases$, the minus part of the $p$-subgroup of the class group of $L$. For an arbitrary finite Galois $p$-extension $L/K$ of algebraic function fields of one variable over an algebraically closed field $k$ of characteristic $p$ as its exact field of constants with Galois group $G = \Gal(L/K)$ we obtain unconditionally and explicitly the Galois module structure of the $p$-torsion part of the Jacobian variety $J_L(p)$ associated to $L/k$.

Keywords:${\Bbb Z}_p$-extensions, Iwasawa's theory, class group, integral representation, fields of algebraic functions, Jacobian variety, Galois module structure
Categories:11R33, 11R23, 11R58, 14H40

144. CJM 1998 (vol 50 pp. 929)

Broer, Abraham
Decomposition varieties in semisimple Lie algebras
The notion of decompositon class in a semisimple Lie algebra is a common generalization of nilpotent orbits and the set of regular semisimple elements. We prove that the closure of a decomposition class has many properties in common with nilpotent varieties, \eg, its normalization has rational singularities. The famous Grothendieck simultaneous resolution is related to the decomposition class of regular semisimple elements. We study the properties of the analogous commutative diagrams associated to an arbitrary decomposition class.

Categories:14L30, 14M17, 15A30, 17B45

145. CJM 1998 (vol 50 pp. 863)

Yekutieli, Amnon
Smooth formal embeddings and the residue complex
Let $\pi\colon X \ar S$ be a finite type morphism of noetherian schemes. A {\it smooth formal embedding\/} of $X$ (over $S$) is a bijective closed immersion $X \subset \mfrak{X}$, where $\mfrak{X}$ is a noetherian formal scheme, formally smooth over $S$. An example of such an embedding is the formal completion $\mfrak{X} = Y_{/ X}$ where $X \subset Y$ is an algebraic embedding. Smooth formal embeddings can be used to calculate algebraic De~Rham (co)homology. Our main application is an explicit construction of the Grothendieck residue complex when $S$ is a regular scheme. By definition the residue complex is the Cousin complex of $\pi^{!} \mcal{O}_{S}$, as in \cite{RD}. We start with I-C.~Huang's theory of pseudofunctors on modules with $0$-dimensional support, which provides a graded sheaf $\bigoplus_{q} \mcal{K}^{q}_{\,X / S}$. We then use smooth formal embeddings to obtain the coboundary operator $\delta \colon\mcal{K}^{q}_{X / S} \ar \mcal{K}^{q + 1}_{\,X / S}$. We exhibit a canonical isomorphism between the complex $(\mcal{K}^{\bdot}_{\,X / S}, \delta)$ and the residue complex of \cite{RD}. When $\pi$ is equidimensional of dimension $n$ and generically smooth we show that $\mrm{H}^{-n} \mcal{K}^{\bdot}_{\,X/S}$ is canonically isomorphic to to the sheaf of regular differentials of Kunz-Waldi \cite{KW}. Another issue we discuss is Grothendieck Duality on a noetherian formal scheme $\mfrak{X}$. Our results on duality are used in the construction of $\mcal{K}^{\bdot}_{\,X / S}$.

Categories:14B20, 14F10, 14B15, 14F20

146. CJM 1998 (vol 50 pp. 829)

Putcha, Mohan S.
Conjugacy classes and nilpotent variety of a reductive monoid
We continue in this paper our study of conjugacy classes of a reductive monoid $M$. The main theorems establish a strong connection with the Bruhat-Renner decomposition of $M$. We use our results to decompose the variety $M_{\nil}$ of nilpotent elements of $M$ into irreducible components. We also identify a class of nilpotent elements that we call standard and prove that the number of conjugacy classes of standard nilpotent elements is always finite.

Categories:20G99, 20M10, 14M99, 20F55

147. CJM 1998 (vol 50 pp. 581)

Kamiyama, Yasuhiko
The homology of singular polygon spaces
Let $M_n$ be the variety of spatial polygons $P= (a_1, a_2, \dots, a_n)$ whose sides are vectors $a_i \in \text{\bf R}^3$ of length $\vert a_i \vert=1 \; (1 \leq i \leq n),$ up to motion in $\text{\bf R}^3.$ It is known that for odd $n$, $M_n$ is a smooth manifold, while for even $n$, $M_n$ has cone-like singular points. For odd $n$, the rational homology of $M_n$ was determined by Kirwan and Klyachko [6], [9]. The purpose of this paper is to determine the rational homology of $M_n$ for even $n$. For even $n$, let ${\tilde M}_n$ be the manifold obtained from $M_n$ by the resolution of the singularities. Then we also determine the integral homology of ${\tilde M}_n$.

Keywords:singular polygon space, homology
Categories:14D20, 57N65

148. CJM 1998 (vol 50 pp. 525)

Brockman, William; Haiman, Mark
Nilpotent orbit varieties and the atomic decomposition of the $q$-Kostka polynomials
We study the coordinate rings~$k[\Cmubar\cap\hbox{\Frakvii t}]$ of scheme-theoretic intersections of nilpotent orbit closures with the diagonal matrices. Here $\mu'$ gives the Jordan block structure of the nilpotent matrix. de Concini and Procesi~\cite{deConcini&Procesi} proved a conjecture of Kraft~\cite{Kraft} that these rings are isomorphic to the cohomology rings of the varieties constructed by Springer~\cite{Springer76,Springer78}. The famous $q$-Kostka polynomial~$\Klmt(q)$ is the Hilbert series for the multiplicity of the irreducible symmetric group representation indexed by~$\lambda$ in the ring $k[\Cmubar\cap\hbox{\Frakvii t}]$. \LS~\cite{L&S:Plaxique,Lascoux} gave combinatorially a decomposition of~$\Klmt(q)$ as a sum of ``atomic'' polynomials with non-negative integer coefficients, and Lascoux proposed a corresponding decomposition in the cohomology model. Our work provides a geometric interpretation of the atomic decomposition. The Frobenius-splitting results of Mehta and van der Kallen~\cite{Mehta&vanderKallen} imply a direct-sum decomposition of the ideals of nilpotent orbit closures, arising from the inclusions of the corresponding sets. We carry out the restriction to the diagonal using a recent theorem of Broer~\cite{Broer}. This gives a direct-sum decomposition of the ideals yielding the $k[\Cmubar\cap \hbox{\Frakvii t}]$, and a new proof of the atomic decomposition of the $q$-Kostka polynomials.

Keywords:$q$-Kostka polynomials, atomic decomposition, nilpotent conjugacy classes, nilpotent orbit varieties
Categories:05E10, 14M99, 20G05, 05E15

149. CJM 1998 (vol 50 pp. 378)

Kurth, Alexandre
Equivariant polynomial automorphism of $\Theta$-representations
We show that every equivariant polynomial automorphism of a $\Theta$-repre\-sen\-ta\-tion and of the reduction of an irreducible $\Theta$-representation is a multiple of the identity.

Categories:14L30, 14L27

150. CJM 1997 (vol 49 pp. 1281)

Sottile, Frank
Pieri's formula via explicit rational equivalence
Pieri's formula describes the intersection product of a Schubert cycle by a special Schubert cycle on a Grassmannian. We present a new geometric proof, exhibiting an explicit chain of rational equivalences from a suitable sum of distinct Schubert cycles to the intersection of a Schubert cycle with a special Schubert cycle. The geometry of these rational equivalences indicates a link to a combinatorial proof of Pieri's formula using Schensted insertion.

Keywords:Pieri's formula, rational equivalence, Grassmannian, Schensted insertion
Categories:14M15, 05E10
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