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 $L/k$.

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.

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.

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.

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 $\mcal{K}^{\bdot}_{\,X / S}$.

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.

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 ${\tilde M}_n$.

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.

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.

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.}

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 $E(\PQ_{n})$.

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.