Expand all Collapse all | Results 151 - 159 of 159 |
151. CJM 1998 (vol 50 pp. 863)
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 |
152. CJM 1998 (vol 50 pp. 829)
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 |
153. CJM 1998 (vol 50 pp. 581)
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 |
154. CJM 1998 (vol 50 pp. 525)
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 |
155. CJM 1998 (vol 50 pp. 378)
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 |
156. CJM 1997 (vol 49 pp. 1281)
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 |
157. CJM 1997 (vol 49 pp. 675)
Some adjunction-theoretic properties of codimension two non-singular subvarities of quadrics We make precise the structure of the first two reduction morphisms
associated with codimension two non-singular subvarieties
of non-singular quadrics $\Q^n$, $n\geq 5$.
We give a coarse classification of the same class of subvarieties
when they are assumed not to be of log-general-type.}
Keywords:Adjunction Theory, classification, codimension two, conic bundles,, low codimension, non log-general-type, quadric, reduction, special, variety. Categories:14C05, 14E05, 14E25, 14E30, 14E35, 14J10 |
158. CJM 1997 (vol 49 pp. 749)
Twisted Hasse-Weil $L$-functions and the rank of Mordell-Weil groups Following a method outlined by Greenberg, root
number computations give a conjectural lower bound for the ranks of
certain Mordell-Weil groups of elliptic curves. More specifically,
for $\PQ_{n}$ a \pgl{{\bf Z}/p^{n}{\bf Z}}-extension of ${\bf Q}$ and
$E$ an elliptic curve over {\bf Q}, define the motive $E \otimes
\rho$, where $\rho$ is any irreducible representation of
$\Gal (\PQ_{n}/{\bf Q})$. Under some restrictions, the root number in
the conjectural functional equation for the $L$-function of $E
\otimes \rho$ is easily computible, and a `$-1$' implies, by the
Birch and Swinnerton-Dyer conjecture, that $\rho$ is found in
$E(\PQ_{n}) \otimes {\bf C}$. Summing the dimensions of such $\rho$
gives a conjectural lower bound of
$$
p^{2n} - p^{2n - 1} - p - 1
$$
for the rank of $E(\PQ_{n})$.
Categories:11G05, 14G10 |
159. CJM 1997 (vol 49 pp. 417)
Characteristic cycles in Hermitian symmetric spaces
We give explicit combinatorial expresssions for the characteristic
cycles associated to certain canonical sheaves on Schubert varieties
$X$ in the classical Hermitian symmetric spaces: namely the
intersection homology sheaves $IH_X$ and the constant sheaves $\Bbb
C_X$. The three main cases of interest are the Hermitian symmetric
spaces for groups of type $A_n$ (the standard Grassmannian), $C_n$
(the Lagrangian Grassmannian) and $D_n$. In particular we find that
$CC(IH_X)$ is irreducible for all Schubert varieties $X$ if and only
if the associated Dynkin diagram is simply laced. The result for
Schubert varieties in the standard Grassmannian had been established
earlier by Bressler, Finkelberg and Lunts, while the computations in
the $C_n$ and $D_n$ cases are new.
Our approach is to compute $CC(\Bbb C_X)$ by a direct geometric
method, then to use the combinatorics of the Kazhdan-Lusztig
polynomials (simplified for Hermitian symmetric spaces) to compute
$CC(IH_X)$. The geometric method is based on the fundamental formula
$$CC(\Bbb C_X) = \lim_{r\downarrow 0} CC(\Bbb C_{X_r}),$$ where the
$X_r \downarrow X$ constitute a family of tubes around the variety
$X$. This formula leads at once to an expression for the coefficients
of $CC(\Bbb C_X)$ as the degrees of certain singular maps between
spheres.
Categories:14M15, 22E47, 53C65 |