This is the fifth in a series of papers constructing explicit examples of special Lagrangian submanifolds in ${\mathbb C}^m$. A submanifold of ${\mathbb C}^m$ is ruled if it is fibred by a family of real straight lines in ${\mathbb C}^m$. This paper studies ruled special Lagrangian 3-folds in ${\mathbb C}^3$, giving both general theory and families of examples. Our results are related to previous work of Harvey and Lawson, Borisenko, and Bryant. Special Lagrangian cones in ${\mathbb C}^3$ are automatically ruled, and each ruled special Lagrangian 3-fold is asymptotic to a unique special Lagrangian cone. We study the family of ruled special Lagrangian 3-folds N asymptotic to a fixed special Lagrangian cone N0. We find that this depends on solving a linear equation, so that the family of such N has the structure of a vector space. We also show that the intersection $\Sigma$ of N0 with the unit sphere ${\mathcal S}^5$ in ${\mathbb C}^3$ is a Riemann surface, and construct a ruled special Lagrangian 3-fold N asymptotic to N0 for each holomorphic vector field w on $\Sigma$. As corollaries of this we write down two large families of explicit special Lagrangian 3-folds in ${\mathbb C}^3$ depending on a holomorphic function on $\mathbb C$, which include many new examples of singularities of special Lagrangian 3-folds. We also show that each special Lagrangian T2-cone N0 can be extended to a 2-parameter family of ruled special Lagrangian 3-folds asymptotic to N0, and diffeomorphic to $T^2\times{\mathbb R}$.

2000 Mathematical Subject Classification: 53C38, 53D12.

Let $X / T$ be a one parameter family of canonical 3-folds and let $D$ be a Weil divisor on it flat over $T$. We study the problem of when the $D_t$-minimal models of $X_t$ form a family and we obtain conditions for this to happen. As an application of this we classify terminal divisorial contractions $E \subset Y \leftarrow C \subset X$ contracting an irreducible surface $E$ onto the smooth curve $C$, in the case when the general section of $X$ through $C$ is a $D_5$ DuVal singularity.

This article addresses the question of which non-empty, compact, proper subsets $E$ of the extended complex plane $\widehat{\mathbb C}$ have the feature that, for some $K$ in $[1,\infty)$, the family of$K$-quasiconformal self-mappings of $\widehat{\mathbb C}$ which leave $E$ invariant acts transitively on the set $E\times E^c$, where $E^c$ is the complement of $E$ in $\widehat{\mathbb C}$. The main result in the paper asserts that the class of sets with this property comprises all one- and two-point subsets of $\widehat{\mathbb C}$, all quasicircles in $\widehat{\mathbb C}$ and all images of the Cantor ternary set under quasiconformal self-mappings of $\widehat{\mathbb C}$. It is shown that the third category includes the limit set of any non-cyclic, finitely generated Schottky group.

We study regularity properties of a positive measure in euclidean space, such as being absolutely continuous with respect to certain Hausdorff measures, in terms of their dyadic doubling properties. Applications of the main results to the distortion of homeomorphisms of the real line and to the regularity of the harmonic measure for some degenerate elliptic operators are given.

We find Zariski pairs of sextics with simple singularities having maximal total Milnornumber. We also relate them to the existence of distinct Mordell--Weil groups of extremal elliptic $K3$ surfaces with a fixed set of semistable singular fibres.

1991 Mathematics Subject Classification: 14F45, 14F27, 14F28.

Let $F: {\Bbb C}^n \to {\Bbb C}^n$ be a holomorphic map, $F^k$ be the $k$th iterate of $F$, and $p \in {\Bbb C}^n$ be a periodic point of $F$ of period $k$. That is, $F^k(p) = p$, but for any positive integer $j$ with $j < k$, $F^j(p) \ne p$. If $p$ is hyperbolic, namely if $DF^k(p)$ has no eigenvalue of modulus $1$, then it is well known that the dynamical behaviour of $F$ is stable near the periodic orbit $\Gamma = \{p, F(p),\ldots, F^{k-1}(p)\}$. But if $\Gamma$ is not hyperbolic, the dynamical behaviour of $F$ near $\Gamma$ may be very complicated and unstable. In this case, a very interesting bifurcational phenomenon may occur even though $\Gamma$ may be the only periodic orbit in some neighbourhood of $\Gamma$: for given $M \in {\Bbb N}\setminus \{1\}$, there may exist a $C^r$-arc $\{F_t: t\in [0,1]\}$ (where $r \in {\Bbb N}$ or $r = \infty$) in the space ${\Bbb H}({\Bbb C}^n)$ of holomorphic maps from ${\Bbb C}^n$ into ${\Bbb C}^n$, such that $F_0 =F$ and, for $t \in (0,1]$, $F_t$ has an $Mk$-periodic orbit $\Gamma_t$ with $d(\Gamma_t, \Gamma) = \sup_{p \in \Gamma_t}\inf_{q \in \Gamma} \|p - q\| \to 0$ as $t \to 0$. The period thus increases by a factor $M$ under a $C^r$-small perturbation! If such an $F_t$ does exist, then $\Gamma$, as well as $p$, is said to be {\em $M$-tupling bifurcational.} This definition is independent of~$r$.

For the above $F$, there may exist a $C^r$-arc $F^*_t$ in ${\Bbb H}({\Bbb C}^n)$, with $t \in [0,1]$, such that $F^*_0 = F$ and, for $t \in (0,1]$, $F^*_t$ has two distinct $k$-periodic orbits $\Gamma_{t,1}$ and $\Gamma_{t,2}$ with $d(\Gamma_{t,i}, \Gamma) \to 0$ as $t \to 0$ for $i = 1,2$. If such an $F^*_t$ does exist, then $\Gamma$, as well as $p$, is said to be {\em $1$-tupling bifurcational.}

In recent decades, there have been many papers and remarkable results which deal with period doubling bifurcations of periodic orbits of parametrized maps. L. Block and D. Hart pointed out that period $M$-tupling bifurcations cannot occur for $M > 2$ in the 1-dimensional case. There are examples showing that for any $M\in {\Bbb N}$, period $M$-tupling bifurcations can occur in higher-dimensional cases.

An $M$-tupling bifurcational periodic orbit as defined here acts as a critical orbit which leads to period $M$-tupling bifurcations in some parametrized maps. The main result of this paper is the following.

{\sc Theorem.} {\em Let $k \in {\Bbb N}$ and $M \in {\Bbb N}$, and let $F: {\Bbb C}^2 \to {\Bbb C}^2$ be a holomorphic map with $k$-periodic point $p$. Then $p$ is $M$-tupling bifurcational if and only if $DF^k(p)$ has a non-zero periodic point of period $M$.}

1991 Mathematics Subject Classification: 32H50, 58F14.

In previous joint work with Cappell and Shaneson, we have established an Atiyah–Lusztig–Meyer-type multiplicative characteristic class formula for the twisted signature and, more generally, the twisted $L$-class, of a stratified Witt space. The present paper shows that these formulae hold even when the stratified space does not satisfy the Witt condition. It constitutes one of the first applications of signature homology.

Denote by $\phi_{\beta,p}$ the normal state on${\cal B}( \ell ^2 ({\Bbb N}))$ with list of eigenvalues $\lambda_{\beta ,p,k} =p^{-\beta k} (1-p^{-\beta} )$, where $k \in {\Bbb N}$, $\beta \in (0,1]$and $p$ is a prime number. For each $\beta $ and each subset ${\cal S}$ of the set ${\cal P}$ of all prime numbers, denote by $M_{\beta ,{\cal S}}$ the ITPFI factor (infinite tensor product of factors of type I)$$ \bigotimes\limits_{ p\in {\cal S}} \big( {\cal B}(\ell ^2 ({\Bbb N} )) ,\phi_{\beta,p} \big).$$ We prove that for any $\beta \in (0,1]$ and $\lambda \in [ 0,1]$, there exists ${\cal S}={\cal S}_{\beta,\lambda}$, asubset of ${\cal P}$, such that $M_{\beta, {\cal S}}$ isof type III$_\lambda$. Moreover, $M_{\beta ,{\cal P}}$ is the Araki--Woods factor $R_\infty$ for any $\beta \in (0,1]$. For any $\beta \in (0.5,1]$ and ${\cal S} \subset {\cal P}$, we show that $M_{\beta ,{\cal S}}$ is an ITPFI$_2$ factor. We study the class of Connes' T-groups of type III factors of the form $M_{\beta,{\cal S}}$, where ${\cal S} \subset {\cal P}$, which we call $\beta$-representable subgroups of ${\Bbb R}$ and show that it is rich, proving for each $\beta \in (0,1]$ that if $H$ is a countable subgroup of ${\Bbb R}$and $\Sigma$ a countable subset of ${\Bbb R} \setminus H$, there exists a $\beta$-representable group $\Gamma$ which contains $H$ and does not intersect $\Sigma$.Moreover, we prove that any $\beta_0$-representable group is $\beta$-representable if $1\geq \beta_0 \geq \max (\beta ,0.07)$ and that the class of $1$-representable groups is closed under the natural action of ${\Bbb R}^*_+$on the subgroups of ${\Bbb R}$. Our results show in particular that all hyperfinite type III$_\lambda$ factors, with $\lambda \in (0,1]$, and a large class of type III$_0$ ITPFI factors are isomorphic to group von Neumann algebras associated with some restricted adelic productsof affine motion groups of $p$-adic lines.

1991 Mathematics Subject Classification: 46L35, 11N05.

In this paper, we consider Kawanaka's generalized Gelfand-Greav characters associated with the various unipotent classes of a finite group of Lie type $G(q)$ (where $q$ is a power of a prime~$p$). Under the assumption that $p$ is large enough, Lusztig [{\em Advances in Math.} 94 (1992) 139-179] has expressed these characters in terms of characteristic functions of certain character sheaves on $G$, where the resulting formulae contain as unknown quantities certain fourth roots of unity. It is one purpose of this paper to determine these roots of unity explicitly, in the case where the centre of $G$ is connected. We then use these results to study the restriction of character sheaves to the unipotent variety of~$G$. Our motivation was to find a more conceptual approach to the properties listed by Lusztig at the end of the introduction of [{\em J. Algebra} 104 (1986) 146-194].

The aim of this paper is to provide an analysis of non-linear approximation in the $L_p$-norm $p = d / (d - 1)$ of functions of bounded variation on $\mathbb{R}^d$ with $d > 1$ by polynomials in the Haar system. The exponent $p$ is the natural exponent as it is the correct exponent in the Sobolev inequality. The approximation schemes that we discuss in this paper are mostly related to Haar thresholding and $m$-term approximation. These problems for $d = 2$ are studied in detail in a paper by Cohen, DeVore, Petrushev and Xu. The main aim of this paper is to extend their results to the case $d \geq 2$.

We obtain the optimal order of the $m$-term Haar approximation and prove the stability of Haar thresholding in the BV-norm. As one of the main tools, we establish the boundedness of certain averaging projections in BV.

The main goal of this paper is to establish new uniform estimates on the size of sublevel sets of plurisubharmonic functions (called plurisubharmonic lemniscates) in terms of Hausdorff–Riesz measures and capacities of certain orders.

We first prove a new uniform version of Skoda's integrability theorem for a given class of plurisubharmonic functions in terms of Borel measures of Hausdorff–Riesz type of certain orders with a precise estimate of the integrability exponent in terms of Lelong numbers of the class and the order of the measures.

Then we present several applications of this result. We first deduce uniform estimates on the size of plurisubharmonic lemniscates associated to functions from some important classes of plurisubharmonic functions in terms of Hausdorff–Riesz measures.

We also derive a new comparison inequality between certain Hausdorff–Riesz capacities and the pluricomplex logarithmic capacity for borelean sets of a fixed bounded domain in $\mathbb{C}^n$ or more generally in an affine algebraic manifold.

Furthermore, using results from classical potential theory, we finally deduce from this comparison inequality new estimates of the size of polynomial lemniscates in terms of Hausdorff contents in the spirit of the famous lemma of H. Cartan.

By studying the braid group action on Milnor's construction of the 1-sphere, we show that the general higher homotopy group of the 3-sphere is the fixed set of the pure braid group action on certain combinatorially described groups. This establishes a relation between the braid groups and the homotopy groups of the sphere.

2000Mathematical Subject Classification: 20F36, 55P35, 55Q05, 55Q40, 55U10.

The main part of the paper finds necessary conditions for solubility of a pencil of curves of genus 1, each of which is a 2-covering of an elliptic curve with at least one 2-division point. As in previous work, these are proved subject to Schinzel's Hypothesis and to the finiteness of the Tate-\u{S}afarevi\u{c} group of elliptic curves defined over a number field. It thus generalizes earlier work of Colliot-Thélène, Skorobogatov and the second author.

The final section gives necessary conditions (though of a rather ugly nature) for the solubility of a Del Pezzo surface of degree 4.

2000 Mathematical Subject Classification: 11D25.

We correct the statements of Theorem 1 and Corollary 2 of the above-mentioned paper and provide a new argument for Corollary 2. We also update some parts of the paper in the light of recent major developments in the theory of bihermitian structures, due to M. Gualtieri and N. Hitchin.

It is known that a Markov map $T$ of the unit interval preserves a measure $\mu$, say, equivalent to Lebesgue measure, and that almost every point of the interval has a forward orbit under $T$ that is uniformly distributed with respect to $\mu$. In the opposite direction the main result of this paper states that there is a set of points having Hausdorff dimension $1$ whose forward orbits are in a certain sense very far from being so distributed.

1991 Mathematics Subject Classification: 58F08, 28A80.

We construct a new class of representations of the canonical commutation relations, which generalizes previously known classes. We perturb the infinitesimal generator of the initial Fock representation (in other words, the free quantum field) by a function of the field which is square-integrable with respect to the associated Gaussian measure. We characterize perturbations which lead to representations of the canonical commutation relations. We provide conditions entailing the irreducibility of such representations, show explicitly that our class of representations subsumes previously studied classes, and give necessary and sufficient conditions for our representations to be unitarily equivalent, or just quasi-equivalent, with Fock, coherent or quasifree representations. 1991 Mathematics Subject Classification: 81S05, 46L60, 81T05.

This paper proves that a triangulated graph is an outline triple system of even index if and only if it is an amalgamated triple system of even index.

2000 Mathematical Subject Classification: primary 05B07; secondary 05C99.

In this paper we study the problem of $L^p$-independence of the spectrum of second-order elliptic operators with measurable coefficients. A new technique is developed to treat the problem in cases when the kernel of the corresponding semigroup does not satisfy upper bounds of Gaussian type and the semigroup itself exists only in a certain interval of the $L^p$-scale. This allows the authors to treat second-order elliptic operators of divergence type with singular coefficients in the main part and singular lower-order terms. A criterion of $L^p$-independence obtained in the paper applies also to certain classes of higher-order elliptic operators. The problem of stability of the essential spectrum is also studied. It is shown, under some mild conditions on the coefficients of the elliptic operators, that its essential spectrum is the same as the one of the Laplacian. 1991 Mathematics Subject Classification: 35P99, 35B25, 35J15.

The aim of this work is to expand Bushnell and Kutzko's theory of $G$-covers [Proc. London Math. Soc. 77 (1998) 582–634] up to a full description of the generalized principal series of the $p$-adic group ${\rm Sp}_4(F)$, with $p$ odd.

We start with a Levi component $M$ of a maximal parabolic subgroup $P$ of $G = {\rm Sp}_4(F)$ and an explicit type $(J_M, \tau_M)$ for the inertial class $S$ in $M$ of a supercuspidal representation of $M$. We compute the Hecke algebra of a $G$-cover $(J, \tau)$ of $(J_M, \tau_M)$ constructed in our previous work [Ann. Inst. Fourier 49 (1999) 1805–1851]: it is a convolution algebra on a Coxeter group (namely, the affine Weyl group of either $U(1,1)(F)$, in the case of the Siegel parabolic, or ${\rm SL}_2(F)$), described explicitly by generators and relations.

From this and Bushnell and Kutzko's work we derive the structure of the parabolically induced representations ${\rm ind}_P^G \pi$, for $\pi$ in $S$, and we find their discrete series subrepresentations if any, thus recovering, through the theory of $G$-covers, results previously obtained by Shahidi using different methods.

The paper is written in French.

2000 Mathematical Subject Classification: 22E50, 11F70.

Given a generalized octagon which satisfies certain geometric conditions, we show that it must be isomorphic to the Ree--Tits octagon arisingnaturally from a Ree group of type $^2F_4$ over a perfect field of characteristic 2, in which the Frobenius automorphism has a square root.

1991 Mathematics Subject Classification: 51E12.