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126. CJM 2003 (vol 55 pp. 157)

 Zariski Hyperplane Section Theorem for Grassmannian Varieties Let $\phi \colon X\to M$ be a morphism from a smooth irreducible complex quasi-projective 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 differential-geometric 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

127. CJM 2003 (vol 55 pp. 133)

 On the Zariski-van Kampen Theorem Let $f \colon E\to B$ be a dominant morphism, where $E$ and $B$ are smooth irreducible complex quasi-projective 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

128. 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 quasi-coherent---we 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 quasi-isomorphism $$\mathcal{H}om_{\mathcal{O}_X} \Bigl( \bigoplus_q \Omega^q_{X/ \mathbb{K}} [q], \mathcal{M} \Bigr) \to \mathcal{H}om^{\cont}_{\mathcal{O}_X} \bigl( \widehat{\mathcal{C}}^{\cdot} (X), \mathcal{M} \bigr).$$ When $\mathcal{M} = \mathcal{O}_X$ this is the quasi-isomorphism underlying the Kontsevich Formality Theorem. Combining the two results above we deduce a decomposition of the global Hochschild cohomology $$\Ext^i_{\mathcal{O}_{X^2}} (\mathcal{O}_X, \mathcal{M}) \cong \bigoplus_q \H^{i-q} \Bigl( X, \bigl( \bigwedge^q_{\mathcal{O}_X} \mathcal{T}_{X/\mathbb{K}} \bigr) \otimes_{\mathcal{O}_X} \mathcal{M} \Bigr),$$ where $\mathcal{T}_{X/\mathbb{K}}$ is the relative tangent sheaf. Keywords:Hochschild cohomology, schemes, derived categoriesCategories:16E40, 14F10, 18G10, 13H10

129. 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 well-known 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

130. CJM 2002 (vol 54 pp. 595)

Nahlus, Nazih
 Lie Algebras of Pro-Affine Algebraic Groups We extend the basic theory of Lie algebras of affine algebraic groups to the case of pro-affine algebraic groups over an algebraically closed field $K$ of characteristic 0. However, some modifications are needed in some extensions. So we introduce the pro-discrete topology on the Lie algebra $\mathcal{L}(G)$ of the pro-affine algebraic group $G$ over $K$, which is discrete in the finite-dimensional 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

131. 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

132. CJM 2002 (vol 54 pp. 55)

Ban, Chunsheng; McEwan, Lee J.; Némethi, András
 On the Milnor Fiber of a Quasi-ordinary 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 quasi-ordinary germs, it depends only on the normalized distinguished pairs of the singularity. The main result of the paper provides an explicit formula for the Euler-characteristic of the Milnor fiber in the surface case. Categories:14B05, 14E15, 32S55

133. CJM 2001 (vol 53 pp. 1309)

Steer, Brian; Wren, Andrew
 The Donaldson-Hitchin-Kobayashi Correspondence for Parabolic Bundles over Orbifold Surfaces A theorem of Donaldson on the existence of Hermitian-Einstein metrics on stable holomorphic bundles over a compact K\"ahler surface is extended to bundles which are parabolic along an effective divisor with normal crossings. Orbifold methods, together with a suitable approximation theorem, are used following an approach successful for the case of Riemann surfaces. Categories:14J17, 57R57

134. 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 $\alpha = d_s + d_{s-1} + \cdots + d_1$ (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

135. 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) Calabi-Yau 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 functionCategories:14B05, 14E15, 32S50, 32S45

136. 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

137. 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

138. 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

139. 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 singularityCategories:14B05, 14J17, 32S05, 32S25

140. 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

141. 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

142. 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 subgroupCategories:14M15, 20G05

143. CJM 2000 (vol 52 pp. 348)

González Pérez, P. D.
 SingularitÃ©s quasi-ordinaires toriques et polyÃ¨dre 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

144. 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$. Keywords:Generators, syzygies, resolution, fat points, maximal rank, plane, Weyl groupCategories:13P10, 14C99, 13D02, 13H15 145. CJM 1999 (vol 51 pp. 1175) Lehrer, G. I.; Springer, T. A.  Reflection Subquotients of Unitary Reflection Groups Let$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$. A key tool is the investigation of extensions of invariant vector fields on the eigenspace, which leads to some results and questions concerning the geometry of intersections of invariant hypersurfaces. Categories:51F15, 20H15, 20G40, 20F55, 14C17 146. CJM 1999 (vol 51 pp. 1226) McKay, John  Semi-Affine Coxeter-Dynkin Graphs and$G \subseteq \SU_2(C)$The semi-affine Coxeter-Dynkin graph is introduced, generalizing both the affine and the finite types. Categories:20C99, 05C25, 14B05 147. CJM 1999 (vol 51 pp. 1123) Arnold, V. I.  First Steps of Local Contact Algebra We consider germs of mappings of a line to contact space and classify the first simple singularities up to the action of contactomorphisms in the target space and diffeomorphisms of the line. Even in these first cases there arises a new interesting interaction of local commutative algebra with contact structure. Keywords:contact manifolds, local contact algebra, Diracian, contactianCategories:53D10, 14B05 148. CJM 1999 (vol 51 pp. 1089) Vakil, Ravi  The Characteristic Numbers of Quartic Plane Curves The characteristic numbers of smooth plane quartics are computed using intersection theory on a component of the moduli space of stable maps. This completes the verification of Zeuthen's prediction of characteristic numbers of smooth plane curves. A short sketch of a computation of the characteristic numbers of plane cubics is also given as an illustration. Categories:14N10, 14D22 149. CJM 1999 (vol 51 pp. 936) David, Chantal; Kisilevsky, Hershy; Pappalardi, Francesco  Galois Representations with Non-Surjective Traces Let$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$is special, and is also treated in the paper. We also classify all those occurences. Category:14H52 150. CJM 1999 (vol 51 pp. 771) Flicker, Yuval Z.  Stable Bi-Period Summation Formula and Transfer Factors This paper starts by introducing a bi-periodic summation formula for automorphic forms on a group$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
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