We determine the intersection theory on the Igusa compactification and the second Voronoi compactification of ${\cal A}_4$.

Let v be a henselian valuation of a field K. In this paper it is proved that any finite extension (K′, v′) of (K, v) is tame if and only if there exists α ≠ 0 in K′ such that v′(α) = v(TrK′/K(α)) using elementary results of valuation theory. A special case of this result, when the characteristic of the residue field of v is p > 0 and (K′, v′)/(K, v) is an extension of degree p, was proved in 1990 by J. P. Tignol (J. Reine Angew. Math. 404 (1990) 1–38).

The author proves in this paper that every profinite group G with polynomial subgroup growth is boundedly generated; that is, it is a product of finitely many procyclic subgroups. This answers a question of P. Zalesskii. By contrast, if G is a boundedly generated group, then the subgroup growth of G is at most nclogn . As a byproduct, a short, elementary proof demonstrates that Aut(Fr) (for r [ges ] 2) and many other related groups are not boundedly generated.

We show that any gauge-equivalence class of solutions (∇, Φ) of the monopole equation F∇ = *d∇Φ, defined on an open set Ω of S3 (equipped with its standard metric), induces canonically a holomorphic function on the complex 2-dimensional space of all geodesics on S3 entirely contained in Ω.

Let X be an infinite-dimensional Banach space, and let A be a Cp Lipschitz bounded starlike body (for instance the unit ball of a smooth norm). We prove that:

(1) the boundary ∂A is Cp Lipschitz contractible;

(2) there is a Cp Lipschitz retraction from A onto ∂A;

(3) there is a Cp Lipschitz map T : A → A with no approximate fixed points.

This paper presents an algebraic approach to Mislin's theorem that control of $p$-fusion is equivalent to inducing a mod-$p$ cohomology isomorphism. There are consequences for the cohomology of $p$-permutation modules.

Let $C$ be a germ at $O \in {\mathbb R}^2$ of a real analytic plane curve, and $C^{\mathbb C}$ its complexification; let $C_t \subset B_{\varepsilon}$ be a fiber of a real smooth deformation of $C$ in the ball $B_{\varepsilon} = B(O, \varepsilon)$. The following inequality is proved between the integrals of real curvature $k$ of $C_t$ and those of Gaussian curvature $K$ of $C_{t}^{\mathbb C}$: $$ 2 \lim_{\varepsilon, t \rightarrow 0} \int_{C_t^\varepsilon} \vert k \vert \leq \lim_{\varepsilon, t \rightarrow 0} \int_{C_t^{\varepsilon {\mathbb C}}} \vert K \vert.$$ The sharpness of this inequality is proved in the case where $C$ is a real irreducible germ. Similar results are proved for an affine algebraic curve $C \subset {\mathbb R}^2$ of degree $d$.

In 1980, Robertson, Tweddle and Yeomans solved the metrizable BCE problem in certain special cases, for example, in the normable case under assumption of the Continuum Hypothesis (CH). A complete positive solution was recently given by the author and L. M. Sánchez Ruiz under an assumption, weaker than CH, that scales exist. A better partial solution for the general BCE problem obviates scales or other extra-ZFC assumptions, and entirely lays to rest the metrizable BCE problem. The proof stimulates thought about how the least infinite-dimensionality of normed barrelled spaces impinges on the separable quotient problem for Banach spaces.

It is shown that a function whose critical locus is an isolated complete intersection singularity of arbitrary dimension, and that has finite codimension (in the sense of R. Pellikaan, Proc. London Math. Soc. (3) 57 (1998) 357–382) with respect to the ideal defining the isolated complete intersection singularity, can be approximated by a function whose critical locus is a finite number of Morse points together with the Milnor fibre of the isolated complete intersection singularity, having there well-known types of singularities.

A flip can be thought of as a diagram X− → X ← X+ of complex threefolds satisfying some conditions. One often thinks of a flip as being in two parts: the first part, X− → X, is the given, while the second, X ← X+, is the unknown. I calculate cohomological properties of the canonical classes, K− = KX− and so on, and in particular properties of the function

formula here

In the case of the toric flips of Danilov [3] and Reid [7], this function can be expressed in terms of lattice points of multiplies of a polyhedron giving sharpness for the more general result.

This paper concerns systems of r homogeneous diagonal equations of degree k in s variables, with integer coefficients. Subject to a suitable non-singularity condition, it is shown that the expected asymptotic formula holds for the number of such systems inside a box [−P,P]s, provided only that s > (3r+1)2k−2. By way of comparison, classical methods based on the use of Hua's lemma would establish a similar conclusion, provided instead that s > r2k.

Requirements for boundary conditions are found which imply sharp by order estimates in L∞-metrics of eigenfunctions of the ordinary differential operator

formula here

It is shown that the inequalities

formula here

for the eigenfunctions {yλ} yield certain restrictions on the spectrum.

The projective tensor product $\ell_2\,{\hat{\otimes}}\,X$ of $\ell_2$ with any Banach space $X$ sits inside the space ${\rm Rad}(X)$ of all almost unconditionally summable sequences in $X$. If $X$ is of cotype 2 and $u: X \longrightarrow Y$ is 2-summing, then $u$ takes ${\rm Rad}(X)$ into $\ell_2\,{\hat{\otimes}}\,Y$. Consequently, if $X$ is of cotype 2, then every operator from $X$ to $\ell_2$ is 1-summing if and only if $\ell_1\,{\check{\otimes}}\,X \subseteq \ell_2\,{\hat{\otimes}}\,X$. In this case, each 2-summing operator from $\ell_2$ to $X$ is nuclear, and $X$ does not have non-trivial type provided that ${\rm dim}\,X = \infty$.

The determination of the generalized Springer correspondence for disconnected reductive groups is completed here by being achieved in the case of disconnected groups of types $D_4$ in characteristic 3 and $E_6$ in characteristic 2. It is expected that this correspondence will provide an important ingredient for the computation of character values of finite groups of Lie type. Some evidence is provided in support of the latter statement.

Sufficient conditions are given on a Banach space $X$ which ensure that $\ell_{\infty}$ embeds in ${\mathcal L}\, (X)$, the space of all bounded linear operators on $X$. A basic sequence $(e_n)$ is said to be quasisubsymmetric if for any two increasing sequences $(k_n)$ and $(\ell_n)$ of positive integers with $k_n \leq \ell_n$ for all $n$, $(e_{k_n})$ dominates $(e_{\ell_n})$. If a Banach space $X$ has a seminormalized quasisubsymmetric basis then $\ell_{\infty}$ embeds in ${\mathcal L}\, (X)$.

1. Introduction

Let k be an algebraically closed field of any characteristic. A toric variety over k is a normal variety X containing the algebraic group T=(k*)n as an open dense subset, with a group action T × X→X extending the group law of T.

On any smooth variety X over a field k, we can define the sheaf of differential operators [Dscr], which carries a natural structure as an [Oscr]X-bisubalgebra of Endk([Oscr]X). A [Dscr]-module on X is a sheaf [Fscr] of abelian groups having a structure as a left [Dscr]-module, such that [Fscr] is quasi-coherent as an [Oscr]X-module. A smooth variety X is called [Dscr]-affine if for every [Dscr]-module [Fscr] we have

[bull] [Fscr] is generated by global sections over [Dscr],

[bull] Hi(X, [Fscr])=0, i>0.

Beilinson and Bernstein have shown [1] that every flag variety over a field of characteristic zero is [Dscr]-affine, from which they deduced a conjecture of Kazhdan and Lusztig. In fact, flag varieties are the only known examples of [Dscr]-affine projective varieties. In this paper we prove that the [Dscr]-affinity of a smooth complete toric variety implies that it is a product of projective spaces. Part of the method will be to translate a proof of the non [Dscr]-affinity of a 2-dimensional Schubert variety, given by Haastert in [4], into the language of toric varieties.

Let [Ascr] be a complex algebra (with unit e), X a left [Ascr] module and a = (a1, . . ., am) a commuting tuple of elements in [Ascr]. Following Taylor [8], we have to consider the Koszul complex K·(a, X) in order to define the joint spectrum of the tuple a.

This paper is based on a talk given at a joint meeting of the London Mathematical Society and the British Society for the History of Mathematics held in Oxford on 13 May 1995.

For any nonnegative self-adjoint operators $A$ and $B$ in a separable Hilbert space, the Trotter-type formula $[;(e^{i2tA/n}+e^{i2tB/n})/2]^n$ is shown to converge strongly in the norm closure of $\dom(A^{1/2})\cap\dom(B^{1/2})$ for some subsequence and for almost every $t\in\mathbb{R}$. This result extends to the degenerate case, and to Kato-functions following the method of T. Kato (see ‘Trotter's product formula for an arbitrary pair of self-adjoint contraction semigroup’, Topics in functional analysis (ed. M. Kac, Academic Press, New York, 1978) 185–195). Moreover, the restrictions on the convergence can be removed by considering functions other than the exponential.

Extending a theorem of S. Mori [6], we characterize two-dimensional factorial algebras which are graded by a finitely generated abelian group of rank one and which are affine over an algebraically closed field. The result is related to the classification of weighted projective lines [2, 5] and to the topic of canonical algebras [7].