let k be an algebraically closed field of positive characteristic p > 0 and $c \to {\mathbb p}^1_k$ a p-cyclic cover of the projective line ramified in exactly one point. we are interested in the p-sylow subgroups of the full automorphism group autkc. we prove that for curves c with genus 2 or higher, these groups are exactly the extensions of a p-cyclic group by an elementary abelian p-group. the main tool is an efficient algorithm to compute the p-sylow subgroups of autkc starting from an artin–schreier equation for the cover $c \to {\mathbb p}^1_k$. we also characterize curves c with genus $g_c\geq 2$ and a p-group action $g\subset \text{aut}_k c$ such that $2p/(p-1)<|g|/g_c$ and $4/(p-1)^2\leq |g|/g_c^2$. our methods rely on previous work by stichtenoth whose approach we have adopted.

We give a full description of the Chow ring of the complex Cayley plane $\mathbb{O}\mathbb{P}^2$. For this, we describe explicitly the most interesting of its Schubert varieties and compute their intersection products. Translating our results into the Borel presentation, i.e. in terms of Weyl group invariants, we are able to compute the degree of the variety of reductions Y8 introduced by the current authors in arXiv: math.AG/0306328.

We define invariants of the blow-Nash equivalence of Nash function germs, in a similar way to the motivic zeta functions of Denef and Loeser. As a key ingredient, we extend the virtual Betti numbers, which were known for real algebraic sets, to a generalized Euler characteristic for projective constructible arc-symmetric sets. Actually we prove more: the virtual Betti numbers are not only algebraic invariants, but also Nash invariants of arc-symmetric sets. Our zeta functions enable one to distinguish the blow-Nash equivalence classes of Brieskorn polynomials of two variables. We prove moreover that there are no moduli for the blow-Nash equivalence in the case of an algebraic family with isolated singularities.

We give a purely algebraic proof of the hypersurface case of the Toric Residue Mirror Conjecture recently proposed by Batyrev and Materov.

We characterize the Fermat quartic K3 surface, among all K3 surfaces, by means of its finite group symmetries.

Let F be a non-Archimedean local field. Let $\mathcal{G}_n^{\rm et}(F)$ be the set of equivalence classes of irreducible, n-dimensional representations of the Weil group $\mathcal{W}_F$ of F which are essentially tame. Let $\mathcal{A}_n^{\rm et}(F)$ be the set of equivalence classes of irreducible, essentially tame, supercuspidal representations of GLn(F). The Langlands correspondence induces a canonical bijection $\mathcal{L}:\mathcal{G}_n^{\rm et}(F) \to \mathcal{A}_n^{\rm et}(F)$. We continue the programme of describing this map in terms of explicit descriptions of the sets $\mathcal{G}_n^{\rm et}(F)$ and $\mathcal{A}_n^{\rm et}(F)$. These descriptions are in terms of admissible pairs $(E/F, \xi)$, consisting of a tamely ramified field extension $E/F$ of degree n and a quasicharacter $\xi$ of $E^\times$ subject to certain technical conditions. If Pn(F) is the set of isomorphism classes of admissible pairs of degree n, we have explicit bijections $P_n(F) \cong \mathcal{G}_n^{\rm et}(F)$ and $P_n(F) \cong \mathcal{A}_n^{\rm et}(F)$. In an earlier paper we showed that, if $\sigma \in \mathcal{G}_n^{\rm et}(F)$ corresponds to an admissible pair $(E/F,\xi)$, then $\mathcal{L}(\sigma)$ corresponds to the admissible pair $(E/F,\mu\xi)$, for a certain tamely ramified character $\mu$ of $E^\times$. In this paper, we determine the character $\mu$ when $E/F$ is totally ramified.

We give a realization of Nakajima varieties and the action of the Weyl group on them using certain canonical structures of homological algebras and their natural generalization, which we develop in this paper. We consider in detail the case of an affine quiver, where we present a simple homological characterization of Nakajima varieties and its relation to moduli of sheaves on the projective plane.

we find a minimal differential graded (dg) operad whose generic representations in ${\mathbb r}^n$ are in one-to-one correspondence with formal germs of those endomorphisms of the tangent bundle to ${\mathbb r}^n$ which satisfy the nijenhuis integrability condition. this operad is of a surprisingly simple origin: it is the cobar construction on the quadratic operad of homologically trivial dg lie algebras. as a by-product we obtain a strong-homotopy generalization of this geometric structure and show its homotopy equivalence to the structure of contractible dg manifolds.

let g be an algebraic group, $\gamma$ an arithmetic lattice of g and $x=\gamma\backslash g$. if h is an algebraic subgroup of g such that $h\cap \gamma$ is a lattice of h, then $\gamma\backslash \gamma h\subset x$ is endowed with a canonical h-invariant probability measure $\mu_h$. using ratner's theory, we give general examples where $\mu_{h_n}$ converges weakly to $\mu_g$ if hn is a strict sequence of algebraic subgroups of g. if $\gamma$ is a congruence subgroup of g, we define another probability measure $\mu_h^a$ on x by using the adelic description of the quotient. we conjecture that $\mu_{h_n}^a$ always converges weakly to $\mu_g$ if hn is a strict sequence. using automorphic forms and l-functions, we describe the case $g=\textit{sl}(2,f)$ for a number field f and a sequence of tori hn. the relation with similar problems on shimura varieties is explained.

We combine the geometric realization of principal series representations on partially holomorphic cohomology spaces, with the Bott–Borel–Weil theorem for direct limits of compact Lie groups, obtaining limits of principal series representations for direct limits of real reductive Lie groups. We introduce the notion of weakly parabolic direct limits and relate it to the conditions that the limit representations are norm-preserving representations on a Banach space or unitary representations on a Hilbert space. We specialize the results to diagonal embedding direct limit groups. Finally we discuss the possibilities of extending the results to limits of tempered series other than the principal series.

We introduce an essentially new Grothendieck topology, the Weil-étale topology, on schemes over finite fields. The cohomology groups associated with this topology should behave better than the standard étale cohomology groups. In particular there is a very natural definition of an Euler characteristic and a plausible conjecture relating the Euler characteristic of Z to the value of the zeta-function at s = 0. This conjecture is proved in certain cases.

Let $\mathcal{E}$ be a rank-2 bundle over a smooth complex projective surface X. Whenever the subscheme of zeros of a global section of $\mathcal{E}$ is zero-dimensional, it gives a geometric realization of the second Chern class $c_2(\mathcal{E})$. Taking the incidence correspondence

\[\xymatrix{{\mathcal{Z} =\{(x,[e]) \in X\times \bf{P} (H^0(\mathcal{E}))\mid e(x)= 0\}} \ar[r]^>>>>>{p_2}& {\bf{P} (H^0(\mathcal{E}))}}\]

and considering the Zariski open subset $U_f \subset \bf{P} (H^0(\mathcal{E}))$ over which the morphism p2 is finite, we have

\[\mathcal{Z}_f = p^{-1}_2(U_f)\longrightarrow U_f,\]

a family of zero-dimensional subschemes of X of length $\deg (c_2(\mathcal{E}))$. This can be viewed as a distinguished geometric representative of $c_2(\mathcal{E})$.

We define a new invariant of $\mathcal{E}$ which can be viewed as a ‘lifting’ of $c_2(\mathcal{E})$ to Uf. This invariant is a sequence of sections of some coherent sheaves on Uf. These sheaves are built from the following cohomology cup-products:

\begin{align}\gamma_1 &: H^1 (\mathcal{E}^{\ast}) \otimes H^0 (\mathcal{E}) \longrightarrow H^1 (\mathcal{O}_X),\\ \gamma_2&: S^2 H^1 (\mathcal{E}^{\ast})\longrightarrow H^2 (\mathcal{O}_X (\det(\mathcal{E}^{\ast})) ).\end{align}

The main property of our invariant is as follows: either it determines the family $\mathcal{Z}_f \stackrel{p_2}{\longrightarrow} U_f$, or the vector bundle $\mathcal{E}$ is ‘special’. The speciality is expressed in terms of the special geometry of zero-loci of global sections of $\mathcal{E}$ and the special geometry of X.

The sequence of sections entering the definition of our invariant is obtained by starting with the one defined by the cup-product $\gamma_2$ and deriving others inductively by using a geometric interpretation of a part of the cup-product $\gamma_1$ together with the Grothendieck residue map. So one can view our invariant as $\gamma_2$ together with some kind of higher-order cohomology cup-products.

The emergence of these higher-order cohomology cup-products is explained conceptually as higher-order derivatives of a natural deformation of $\gamma_2$ associated to a certain ‘natural’ deformation of the complex structure on $\mathcal{E}$. The variety of these natural deformations of $\mathcal{E}$ has all the features of the classical Jacobian of curves: it carries a distinguished divisor which either determines the family $\mathcal{Z}_f \stackrel{p_2}{\longrightarrow} U_f$ or ‘sees’ that $\mathcal{E}$ is special in the aforementioned sense. An essentially new feature of this Jacobian of $\mathcal{E}$ is that it also carries a variation of Hodge-like structures which arises naturally from our invariant.

We show that the norm of a Bethe vector in the $sl_{r+1}$ Gaudin model is equal to the Hessian of the corresponding master function at the corresponding critical point. In particular the Bethe vectors corresponding to non-degenerate critical points are non-zero vectors. This result is a byproduct of functorial properties of Bethe vectors studied in this paper. As another byproduct of functoriality we show that the Bethe vectors form a basis in the tensor product of several copies of first and last fundamental $sl_{r+1}$-modules.

In 1987, Sabbah proved the existence of Bernstein–Sato polynomials associated with several analytic functions. The purpose of this paper is to give a more elementary and constructive proof of Sabbah's result based on the notion of the analytic Gröbner fan of a $\mathcal D$-module.

Let X be an irreducible projective variety of dimension n in a projective space and let x be a point of X. Denote by Curvesd(X, x) the space of curves of degree d lying on X and passing through x. We will show that the number of components of Curvesd(X, x) for any smooth point x outside a subvariety of codimension $\geq 2$ is bounded by a number depending only on n and d. An effective bound is given. A key ingredient of the proof is an argument from Ein, Küchle and Lazarsfeld's work on Seshadri numbers.

we interpret the five-parameter family of macdonald–koornwinder polynomials as vector valued spherical functions on quantum grassmannians.

A polar hypersurface P of a complex analytic hypersurface germ f = 0 can be investigated by analyzing the invariance of certain Newton polyhedra associated with the image of P, with respect to suitable coordinates, by certain morphisms appropriately associated with f. We develop this general principle of Teissier when f = 0 is a quasi-ordinary hyper-surface germ and P is the polar hypersurface associated with any quasi-ordinary projection of f = 0. We show a decomposition of P into bunches of branches which characterizes the embedded topological types of the irreducible components of f = 0. This decomposition is also characterized by some properties of the strict transform of P by the toric embedded resolution of f = 0 given by the second author. In the plane curve case this result provides a simple algebraic proof of a theorem of Lê et al.

We classify quadruples $(M, g, m, \tau)$ in which (M, g) is a compact Kähler manifold of complex dimension m > 2 and $\tau$ is a nonconstant function on M such that the conformally related metric $g/\tau^{2}$, defined wherever $\tau \ne 0$, is an Einstein metric. It turns out that M then is the total space of a holomorphic $\mathbb{C}{\rm P}^1$ bundle over a compact Kähler–Einstein manifold (N, h). The quadruples in question constitute four disjoint families: one, well known, with Kähler metrics g that are locally reducible; a second, discovered by Bérard Bergery (1982), and having $\tau \ne 0$ everywhere; a third one, related to the second by a form of analytic continuation, and analogous to some known Kähler surface metrics; and a fourth family, present only in odd complex dimensions $m \ge 9$. Our classification uses a moduli curve, which is a subset $\mathcal{C}$, depending on m, of an algebraic curve in $\mathbb{R}^2$. A point (u, v) in $\mathcal{C}$ is naturally associated with any $(M, g, m, \tau)$ having all of the above properties except for compactness of M, replaced by a weaker requirement of ‘vertical’ compactness. One may in turn reconstruct M, g and $\tau$ from (u, v) coupled with some other data, among them a Kähler–Einstein base (N, h) for the $\mathbb{C}{\rm P}^1$ bundle M. The points (u, v) arising in this way from $(M, g, m, \tau)$ with compactM form a countably infinite subset of \mathcal{C}$.

This work is concerned with type theory for reductive groups over a non Archimedean field. Given such a field F, and a division algebra D of finite dimension over its center F, we obtain results concerning the construction of simple types for the group GL(m, D), $m\geqslant1$. More precisely, for each simple stratum of the matrix algebra M(m, D), we produce a set of β-extensions in the sense of Bushnell and Kutzko.

This paper shows that a nontrivial uniform decay estimate for complete exponential sums modulo pr, determined by a polynomial map ${\bf P} = (P_1, P_2)$ follows from the existence of a ‘good P decomposition’ of ${\mathbb Z}_p^n$, a property that can be proved with geometric methods, and which was introduced in an earlier article by the present author.