We use twisted sheaves and their moduli spaces to study the Brauer group of a scheme. In particular, we (1) show how twisted methods can be efficiently used to re-prove the basic facts about the Brauer group and cohomological Brauer group (including Gabber’s theorem that they coincide for a separated union of two affine schemes), (2) give a new proof of de Jong’s period-index theorem for surfaces over algebraically closed fields, and (3) prove an analogous result for surfaces over finite fields. We also include a reduction of all period-index problems for Brauer groups of function fields over algebraically closed fields to characteristic zero, which (among other things) extends de Jong’s result to include classes of period divisible by the characteristic of the base field. Finally, we use the theory developed here to give counterexamples to a standard type of local-to-global conjecture for geometrically rational varieties over the function field of the projective plane.

Sikora has given results which confirm the analogy between number fields and 3-manifolds. However, he has given proofs of his results which are very different in the arithmetic and in the topological case. In this paper, we show how to provide a unified approach to the results in the two cases. For this we introduce an equivariant cohomology which satisfies a localization theorem. In particular, we obtain a satisfactory explanation for the coincidences between Sikora’s formulas which leads us to clarify and to extend the dictionary of arithmetic topology.

For p≥3 an odd prime and a nonnegative integer r≤p−2, we prove a conjecture of Breuil on lattices in semi-stable representations, that is, the anti-equivalence of categories between the category of strongly divisible lattices of weight r and the category of Galois stable -lattices in semi-stable p-adic Galois representations with Hodge–Tate weights in {0,…,r}.

In this paper, we use the perspective of linear series, and in particular results following from the degeneration tools of limit linear series, to give a number of new results on the existence and non-existence of tamely branched covers of the projective line in positive characteristic. Our results are both in terms of ramification indices and the sharper invariant of monodromy cycles, and the first class of results are obtained by intrinsically algebraic and positive-characteristic arguments.

Let k be a non-Archimedean field, let X be a k-affinoid space and let f1,…,fn, with , be analytic functions over X. If X is irreducible, we prove that the analytic domain is still irreducible, provided that is small enough. Then, for a general X, we precisely describe how the geometric connected components of the spaces behave with regards to ε. Finally, we obtain a result concerning privileged neighbourhoods and adapt a theorem from complex analytic geometry about Noetherianity for germs of analytic functions.

A K3 category is by definition a Calabi–Yau category of dimension two. Geometrically K3 categories occur as bounded derived categories of (twisted) coherent sheaves on K3 or abelian surfaces. A K3 category is generic if there are no spherical objects (or just one up to shift). We study stability conditions on K3 categories as introduced by Bridgeland and prove his conjecture about the topology of the stability manifold and the autoequivalences group for generic twisted projective K3, abelian surfaces, and K3 surfaces with trivial Picard group.

We study the action of a real reductive group G on a real submanifold X of a Kähler manifold Z. We suppose that the action of G extends holomorphically to an action of the complexified group and that with respect to a compatible maximal compact subgroup U of the action on Z is Hamiltonian. There is a corresponding gradient map where is a Cartan decomposition of . We obtain a Morse-like function on X. Associated with critical points of are various sets of semistable points which we study in great detail. In particular, we have G-stable submanifolds Sβ of X which are called pre-strata. In cases where is proper, the pre-strata form a decomposition of X and in cases where X is compact they are the strata of a Morse-type stratification of X. Our results are generalizations of results of Kirwan obtained in the case where and X=Z is compact.

The generalized Weierstrass representation is used to analyze the asymptotic behavior of a constant mean curvature surface that arises locally from an ordinary differential equation (ODE) with a regular singularity. We prove that a holomorphic perturbation of an ODE that represents a Delaunay surface generates a constant mean curvature surface which has a properly immersed end that is asymptotically Delaunay. Furthermore, that end is embedded if the Delaunay surface is unduloidal.

We introduce the notion of an alternate product of Frobenius manifolds and we give, after Ciocan-Fontanine et al., an interpretation of the Frobenius manifold structure canonically attached to the quantum cohomology of G(r,n+1) in terms of alternate products. We also investigate the relationship with the alternate Thom–Sebastiani product of Laurent polynomials.

For every prime p and integer n≥3 we explicitly construct an abelian variety of dimension n such that for a suitable prime l the group of quasi-isogenies of  of l-power degree is canonically a dense subgroup of the nth Morava stabilizer group at p. We also give a variant of this result taking into account a polarization. This is motivated by the recent construction by Behrens and Lawson of topological automorphic forms which generalizes topological modular forms. For this, we prove some arithmetic results of independent interest: a result about approximation of local units in maximal orders of global skew fields which also gives a precise solution to the problem of extending automorphisms of the p-divisible group of a simple abelian variety over a finite field to quasi-isogenies of the abelian variety of degree divisible by as few primes as possible.