We study the arithmetic of certain del Pezzo surfaces of degree 2. We produce examples of Brauer-Manin obstruction to the Hasse principle, coming from 2- and 4-torsion elements in the Brauer group.

A line bundle over a complex projective variety is called big and 1-ample if a large multiple of it is generated by global sections and a morphism induced by the evaluation of the spanning sections is generically finite and has at most 1-dimensional fibers. A vector bundle is called big and 1-ample if the relative hyperplane line bundle over its projectivisation is big and 1-ample.

The main theorem of the present paper asserts that any complex projective manifold of dimension 4 or more, whose tangent bundle is big and 1-ample, is equal either to a projective space or to a smooth quadric. Since big and 1-ample bundles are ‘almost’ ample, the present result is yet another extension of the celebrated Mori paper ‘Projective manifolds with ample tangent bundles’ (Ann. of Math. 110 (1979) 593–606).

The proof of the theorem applies results about contractions of complex symplectic manifolds and of manifolds whose tangent bundles are numerically effective. In the appendix we re-prove these results.

The techniques and concepts we present are flags of regular schemes and their persistence under blow-up, the Gauss–Bruhat decomposition of the group of formal automorphisms of affine space, and coordinate-free initial ideals. All three are used to construct and study invariants for resolution of singularities.

We construct explicit examples of elementary extremal contractions, both birational and of fiber type, from smooth projective $n$-dimensional varieties, with $n \geq 4$, onto smooth projective varieties, arising from classical projective geometry and defined over sufficiently small fields, not necessarily algebraically closed.

The examples considered come from particular special homaloidal and subhomaloidal linear systems, which are usually degenerations of general phenomena classically investigated by Bordiga, Severi, Todd, Room, Fano, Semple and Tyrrell and more recently by Ein and Shepherd-Barron.

The first series of examples is associated to particular codimension 2 determinantal smooth subvarieties of $\mathbf{P}^{m}$, with $3 \leq m \leq 5$. We get another series of examples by considering special cubic hypersurfaces through some surfaces in $\mathbf{P}^5$, or some 3-folds in $\mathbf{P}^7$ having one apparent double point. The last examples come from an intriguing birational elementary extremal contraction in dimension 6, studied by Semple and Tyrrell and fully described in the last section of the paper.

The cohomology ring of the moduli space of stable holomorphic vector bundles of rank $n$ and degree $d$ over a Riemann surface of genus $g > 1$ has a standard set of generators when $n$ and $d$ are coprime. When $n = 2$ the relations between these generators are well understood, and in particular a conjecture of Mumford, that a certain set of relations is a complete set, is known to be true. In this article generalisations are given of Mumford's relations to the cases when $n > 2$ and also when the bundles are parabolic bundles, and these are shown to form complete sets of relations.

Bertini's theorem on variable singular points may fail in characteristic $p$; there are algebraic fibrations that are pathological in the sense that all fibres are non-smooth though the total space admits only a finite number of singular points. We prove that if an algebraic fibration by plane projective quartic curves in characteristic 7 is pathological then, after an eventual cyclic base extension of degree 3, it is, up to birational equivalence, obtained by a base extension from the pencil of quartic curves cut out by the equations $ zx^3 + xy^3 + t yz^3 = 0$.

More generally, we classify pathological fibrations by canonical curves of arithmetic genus $g = \frac{p - 1}{2}$ in the projective space of dimension $\frac{p - 3}{2}$. We prove that the gap sequences of the smooth Weierstrass points of the fibres have the property that a positive integer $\ell$ is a gap if and only if $2g + 1 - \ell$ is a non-gap. In analogy to the Kodaira–Néron classification of special fibres of minimal fibrations by elliptic curves, we construct minimal proper regular models of pathological fibrations, determine the structure of the bad fibres, and study the global geometry of the total spaces.

In this paper, we investigate the structure of Ariki–Koike algebras and their Specht modules using Gröbner–Shirshov basis theory and combinatorics of Young tableaux. For a multipartition $\lambda$, we find a presentation of the Specht module $S^{\lambda}$ given by generators and relations, and determine its Gröbner–Shirshov pair. As a consequence, we obtain a linear basis of $S^{\lambda}$ consisting of standard monomials with respect to the Gröbner–Shirshov pair. We show that this monomial basis can be canonically identified with the set of cozy tableaux of shape $\lambda$.

Starting from a Whitney decomposition of a symmetric cone $\Omega$, analogous to the dyadic partition $[2^j, 2^{ j + 1})$ of the positive real line, in this paper we develop an adapted Littlewood–Paley theory for functions with spectrum in $\Omega$. In particular, we define a natural class of Besov spaces of such functions, $B^{p, q}_\nu$, where the role of the usual derivation is now played by the generalized wave operator of the cone $\Delta(\frac{\partial}{\partial x})$. We show that $B^{p, q}_\nu$ consists precisely of the distributional boundary values of holomorphic functions in the Bergman space $A^{p, q}_\nu (T_\Omega)$, at least in a 'good range' of indices $1 \leq q < q_{\nu, p}$. We obtain the sharp $q_{\nu, p}$ when $p \leq 2$, and conjecture a critical index for $p > 2$. Moreover, we show the equivalence of this problem with the boundedness of Bergman projectors $P_\nu \colon L^{p, q}_\nu \to A^{p, q}_\nu$, for which our result implies a positive answer when $q_{\nu, p}' < q < q_{\nu, p}$. This extends, to general cones, previous work of the authors on the light-cone.

Finally, we focus on light-cones and introduce new necessary and sufficient conditions for our conjecture to hold in terms of inequalities related to the cone multiplier problem. In particular, using recent work by Laba and Wolff, we establish the validity of our conjecture for light-cones when $p$ is sufficiently large.

Let $F$ be an algebraically closed field of characteristic $p$, and let $H$ be a finite group. A natural and important problem in representation theory is to classify the pairs $(G, D)$, where $G$ is a subgroup of $H$, and $D$ is an irreducible $FH$-module that remains irreducible when restricted to $G$. For example, if $H$ is an almost simple group or a central extension of an almost simple group, then one encounters this problem in the attempt to determine the maximal subgroups of the finite classical groups. We solve the above problem for Schur's double covers $H = {\widehat S}_n$ and ${\widehat A}_n$ of the symmetric and alternating groups, and $G$ a Young-type subgroup. The answer is given in terms of the combinatorics of restricted, $p$-strict partitions; such partitions are used to parameterize the irreducible spin representations of ${\widehat S}_n$ and ${\widehat A}_n$. We exploit the connections, recently developed by Ariki, Grojnowski, Brundan, Kleshchev, and others, between modular branching rules, crystal graphs of affine Kac–Moody algebras, and affine Hecke algebras and related objects.

In this paper we describe a tensor space representation of the blob algebra over a ring allowing base change to every interesting (that is, non-semisimple) specialisation. We show that in quasihereditary specialisations this passes to a full tilting module. We consider the nature of the resultant Ringel dual algebra.

A finite permutation group is said to be innately transitive if it contains a transitive minimal normal subgroup. In this paper, we give a characterisation and structure theorem for the finite innately transitive groups, as well as describing those innately transitive groups which preserve a product decomposition. The class of innately transitive groups contains all primitive and quasiprimitive groups.

The summatory function of the Möbius function is denoted $M(x)$. In this article we deduce conditional results concerning $M(x)$ assuming the Riemann hypothesis and a conjecture of Gonek and Hejhal on the negative moments of the Riemann zeta function. Assuming these conjectures, we show that $M(x)$, when appropriately normalized, possesses a limiting distribution, and also that a strong form of the weak Mertens conjecture is true. Finally, we speculate on the lower order of $M(x)$ by studying the constructed distribution function.

Let $G$ be a locally compact group, and let $L^1 (G)$ be the Banach algebra which is the group algebra of $G$. We consider a variety of Banach left $L^1 (G)$ -modules over $L^1 (G)$ , and seek to determine conditions on $G$ that determine when these modules are either projective or injective or flat in the category. The answers typically involve $G$ being compact or discrete or amenable. For example, in the case where $G$ is discrete and $1 < p < \infty$, we find that the module $\ell^p (G)$ is injective whenever $G$ is amenable, and that, if it is amenable, then $G$ is ‘pseudo-amenable’, a property very close to that of amenability.

We estimate the $L^2$-norm of the $s$-dimensional Riesz transforms on some Cantor sets in ${\mathbb R}^d$. Towards this end, we show that the Riesz transforms truncated at different scales behave in a quasiorthogonal way. As an application, we obtain some precise numerical estimates for the Lipschitz harmonic capacity of these sets.

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.

This paper is devoted to the well-posedness of abstract Cauchy problems for quasi-linear evolution equations. The notion of Hadamard well-posedness is considered, and a new type of stability condition is introduced from the viewpoint of the theory of finite difference approximations. The result obtained here generalizes not only some results on abstract Cauchy problems closely related with the theory of integrated semigroups or regularized semigroups but also the Kato theorem on quasi-linear evolution equations. An application to some quasi-linear partial differential equation of weakly hyperbolic type is also given.

We analyze the decomposition rank (a notion of covering dimension for nuclear C*-algebras introduced by E. Kirchberg and the author) of subhomogeneous C*-algebras. In particular, we show that a subhomogeneous C*-algebra has decomposition rank $n$ if and only if it is recursive subhomogeneous of topological dimension $n$, and that $n$ is determined by the primitive ideal space.

As an application, we use recent results of Q. Lin and N. C. Phillips to show the following. Let $A$ be the crossed product C*-algebra coming from a compact smooth manifold and a minimal diffeomorphism. Then the decomposition rank of $A$ is dominated by the covering dimension of the underlying manifold.

We develop the Fredholm theory for Toeplitz operators, with symbols in the C*-algebra $D = [SO, SAP]_{n, n}$ generated by all slowly oscillating (SO) and semi-almost periodic (SAP) $n\times n$ matrix functions, on the Hardy spaces $H^p_n$ (with $1 < p < \infty$) over the upper half-plane. Using limit operator techniques, we get necessary Fredholm conditions for any operator in the Banach algebra ${\rm alg}(S, D)$ of singular integral operators with coefficients in $D$ on the space $[L^p (\mathbb{R})]_n$. Applying the Allan–Douglas local principle and the theory of Toeplitz operators with SAP matrix symbols, we also establish Fredholm criteria for Toeplitz operators with matrix symbols $g \in D$ on the space $H^p_n$. An index formula for Fredholm Toeplitz operators with matrix symbols in $D$ is obtained on the basis of an appropriate approximation of slowly oscillating components of the symbols.

For quantized irreducible flag manifolds the locally finite part of the dual coalgebra is shown to coincide with a natural quotient coalgebra $\overline{U}$ of $U_q ( \mathfrak{g} )$. On the way the coradical filtration of $\overline{U}$ is determined. A graded version of the duality between $\overline{U}$ and the quantized coordinate ring is established. This leads to a natural construction of several examples of quantized vector spaces.

As an application, covariant first order differential calculi on quantized irreducible flag manifolds are classified.

Let $G$ be a locally compact group, $\mathrm{L}^p (G)$ be the usual $\mathrm{L}^p$-space for $1 \leq p \leq \infty$, and $\mathrm{A} (G)$ be the Fourier algebra of $G$. Our goal is to study, in a new abstract context, the completely bounded multipliers of $\mathrm{A} (G)$, which we denote $\mathrm{M_{cb}A} (G)$. We show that $\mathrm{M_{cb}A} (G)$ can be characterised as the ‘invariant part’ of the space of (completely) bounded normal $\mathrm{L}^\infty (G)$-bimodule maps on $\mathcal{B}(\mathrm{L}^2 (G))$, the space of bounded operators on $\mathrm{L}^2 (G)$. In doing this we develop a function-theoretic description of the normal $\mathrm{L}^\infty (X, \mu)$-bimodule maps on $\mathcal{B} (\mathrm{L}^2 (X, \mu))$, which we denote by $\mathrm{V}^\infty (X, \mu)$, and name the {\it measurable Schur multipliers} of $(X, \mu)$. Our approach leads to many new results, some of which generalise results hitherto known only for certain classes of groups. Those results which we develop here are a uniform approach to obtaining the functorial properties of $\mathrm{M_{cb} A} (G)$, and a concrete description of a standard predual of $\mathrm{M_{cb} A} (G)$.