We study the moduli spaces of polarised irreducible symplectic manifolds. By a comparison with locally symmetric varieties of orthogonal type of dimension 20, we show that the moduli space of polarised deformation K3[2] manifolds with polarisation of degree 2d and split type is of general type if d≥12.

Let K be a complete discrete valuation field of mixed characteristic (0,p) with a perfect residue field k. For a semi-stable scheme over the ring of integers OK of K or, more generally, for a log smooth scheme of semi-stable type over k, we define nearby cycles as a single 𝒟-module endowed with a monodromy ∂logt, whose cohomology should give the log crystalline cohomology. We also explicitly describe the monodromy filtration of the 𝒟-module with respect to the endomorphism ∂logt, and construct a weight spectral sequence for the cohomology of the nearby cycles.

Let V be a vector space over a p-adic field F, of finite dimension, let q be a non-degenerate quadratic form over V and let D be a non-isotropic line in V. Denote by W the hyperplane orthogonal to D, and by G and H the special orthogonal groups of V and W. Let π, respectively σ, be an irreducible admissible representation of G(F) , respectively H(F) . The representation σ appears as a quotient of the restriction of π to H(F)with a certain multiplicity m(π,σ) . We know that m(π,σ)≤1 . We assume that π is supercuspidal. Then we prove a formula that computes m(π,σ)as an integral of functions deduced from the characters of π and σ. Let Π, respectively Σ, be an L-packet of tempered irreducible representations of G(F) , respectively H(F) . Here we use the sophisticated notion of L-packet due to Vogan and we assume some usual conjectural properties of those packets. A weak form of the local Gross–Prasad conjecture says that there exists a unique pair (π,σ)∈Π×Σ such that m(π,σ)=1 . Assuming that the elements of Π are supercuspidal, we prove this assertion.

In a previous paper, the potential automorphy of certain Galois representations to GLn for n even was established, following the work of Harris, Shepherd–Barron and Taylor and using the lifting theorems of Clozel, Harris and Taylor. In this paper, we extend those results to n=3 and n=5, and conditionally to all other odd n. The key additional tools necessary are results which give the automorphy or potential automorphy of symmetric powers of elliptic curves, most notably those of Gelbert, Jacquet, Kim, Shahidi and Harris.

For a number field K and a finite abelian group G, we determine the probabilities of various local completions of a random G-extension of K when extensions are ordered by conductor. In particular, for a fixed prime ℘ of K, we determine the probability that ℘ splits into r primes in a random G-extension of K that is unramified at ℘. We find that these probabilities are nicely behaved and mostly independent. This is in analogy to Chebotarev’s density theorem, which gives the probability that in a fixed extension a random prime of K splits into r primes in the extension. We also give the asymptotics for the number of G-extensions with bounded conductor. In fact, we give a class of extension invariants, including conductor, for which we obtain the same counting and probabilistic results. In contrast, we prove that neither the analogy with the Chebotarev probabilities nor the independence of probabilities holds when extensions are ordered by discriminant.

Let X be a smooth projective variety of dimension n in Pr, and let π:X→Pn+c be a general linear projection, with c>0. In this paper we bound the scheme-theoretic complexity of the fibers of π. In his famous work on stable mappings, Mather extended the classical results by showing that the number of distinct points in the fiber is bounded by B:=n/c+1, and that, when n is not too large, the degree of the fiber (taking the scheme structure into account) is also bounded by B. A result of Lazarsfeld shows that this fails dramatically for n≫0. We describe a new invariant of the scheme-theoretic fiber that agrees with the degree in many cases and is always bounded by B. We deduce, for example, that if we write a fiber as the disjoint union of schemes Y′ and Y′′ such that Y′ is the union of the locally complete intersection components of Y, then deg Y′+deg Y′′red≤B. Our method also gives a sharp bound on the subvariety of Pr swept out by the l-secant lines of X for any positive integer l, and we discuss a corresponding bound for highly secant linear spaces of higher dimension. These results extend Ran’s ‘dimension +2 secant lemma’.

In this article we introduce and prove a ℤ/p meta-abelian form of the birationalp-adic section conjecture for curves. This is a much stronger result than the usual p-adic birational section conjecture for curves, and makes an effective p-adic section conjecture for curves quite plausible.

We give a construction of the moduli space of stable maps to the classifying stack Bμr of a cyclic group by a sequence of rth root constructions on . We prove a closed formula for the total Chern class of μr-eigenspaces of the Hodge bundle, and thus of the obstruction bundle of the genus-zero Gromov–Witten theory of stacks of the form [ℂN/μr]. We deduce linear recursions for genus-zero Gromov–Witten invariants.

Using the middle convolution functor MCχ introduced by N. Katz, we prove the existence of rigid local systems whose monodromy is dense in the simple algebraic group G2. We derive the existence of motives for motivated cycles which have a motivic Galois group of type G2. Granting Grothendieck’s standard conjectures, the existence of motives with motivic Galois group of type G2 can be deduced, giving a partial answer to a question of Serre.

We study Cox rings of K3 surfaces. A first result is that a K3 surface has a finitely generated Cox ring if and only if its effective cone is rational polyhedral. Moreover, we investigate degrees of generators and relations for Cox rings of K3 surfaces of Picard number two, and explicitly compute the Cox rings of generic K3 surfaces with a non-symplectic involution that have Picard number 2 to 5 or occur as double covers of del Pezzo surfaces.

In this paper, we define Swan conductors for unit-root overconvergent F-isocrystals using the theory of arithmetic 𝒟-modules due to Berthelot. Our Swan conductors are compared with the Swan conductors for ℓ-adic sheaves constructed by Kato and Saito using a geometric method. As an application, we prove the integrality of Swan conductors in the sense of Kato and Saito under the ‘resolution of singularities’ assumption.

Katz’s middle convolution algorithm provides a description of rigid connections on ℙ1 with regular singularities. We extend the algorithm by adding the Fourier transform to it. The extended algorithm provides a description of rigid connections with arbitrary singularities.

We introduce in this paper a hypercohomology version of the resonance varieties and obtain some relations to the characteristic varieties of rank one local systems on a smooth quasi-projective complex variety M. A logarithmic resonance variety is also considered and, as an application, we determine the first characteristic variety of the configuration space of n distinct labeled points on an elliptic curve. Finally, for a logarithmic 1-form α on M we investigate the relation between the resonance degree of α and the codimension of the zero set of α on a good compactification of M. This question was inspired by the recent work by Cohen, Denham, Falk and Varchenko.

Using the Gille–Merkurjev norm principle we compute in a uniform way the image of the degree map for quadrics (Springer’s theorem), for twisted forms of maximal orthogonal Grassmannians (a theorem of Bayer-Fluckiger and Lenstra), and for E6- (a theorem of Rost) and E7-varieties.

This paper studies affine Deligne–Lusztig varieties in the affine flag manifold of a split group. Among other things, it proves emptiness for certain of these varieties, relates some of them to those for Levi subgroups, and extends previous conjectures concerning their dimensions. We generalize the superset method, an algorithmic approach to the questions of non-emptiness and dimension. Our non-emptiness results apply equally well to the p-adic context and therefore relate to moduli of p-divisible groups and Shimura varieties with Iwahori level structure.

For a closed d-dimensional subvariety X of an abelian variety A and a canonically metrized line bundle L on A, Chambert-Loir has introduced measures c1(L∣X)∧d on the Berkovich analytic space associated to A with respect to the discrete valuation of the ground field. In this paper, we give an explicit description of these canonical measures in terms of convex geometry. We use a generalization of the tropicalization related to the Raynaud extension of A and Mumford’s construction. The results have applications to the equidistribution of small points.

An almost-direct product of free groups is an iterated semidirect product of finitely generated free groups in which the action of the constituent free groups on the homology of one another is trivial. We determine the structure of the cohomology ring of such a group. This is used to analyze the topological complexity of the associated Eilenberg–MacLane space.