We discuss some general properties of $\mathrm {TR}$ and its $K(1)$-localization. We prove that after $K(1)$-localization, $\mathrm {TR}$ of $H\mathbb {Z}$-algebras is a truncating invariant in the Land–Tamme sense, and deduce $h$-descent results. We show that for regular rings in mixed characteristic, $\mathrm {TR}$ is asymptotically $K(1)$-local, extending results of Hesselholt and Madsen. As an application of these methods and recent advances in the theory of cyclotomic spectra, we construct an analog of Thomason's spectral sequence relating $K(1)$-local $K$-theory and étale cohomology for $K(1)$-local $\mathrm {TR}$.

We characterize the Fermat quartic K3 surface, among all K3 surfaces, by means of its finite group symmetries.

Let Mμ0 denote S2×S2 endowed with a split symplectic form normalized so that μ≥1 and σ(S2)=1. Given a symplectic embedding of the standard ball of capacity c∈(0,1) into Mμ0, consider the corresponding symplectic blow-up . In this paper, we study the homotopy type of the symplectomorphism group and that of the space of unparametrized symplectic embeddings of Bc into Mμ0. Writing ℓ for the largest integer strictly smaller than μ, and λ∈(0,1] for the difference μ−ℓ, we show that the symplectomorphism group of a blow-up of ‘small’ capacity c<λ is homotopically equivalent to the stabilizer of a point in Symp(Mμ0), while that of a blow-up of ‘large’ capacity c≥λ is homotopically equivalent to the stabilizer of a point in the symplectomorphism group of a non-trivial bundle obtained by blowing down . It follows that, for c<λ, the space is homotopy equivalent to S2 ×S2, while, for c≥λ, it is not homotopy equivalent to any finite CW-complex. A similar result holds for symplectic ruled manifolds diffeomorphic to . By contrast, we show that the embedding spaces and , if non-empty, are always homotopy equivalent to the spaces of ordered configurations and . Our method relies on the theory of pseudo-holomorphic curves in 4 -manifolds, on the computation of Gromov invariants in rational 4 -manifolds, and on the inflation technique of Lalonde and McDuff.

We will introduce a regularization for $p$-adic multiple zeta values and show that the generalized double shuffle relations hold. This settles a question raised by Deligne, given as a project in the Arizona Winter School 2002. Our approach is to use the theory of Coleman functions on the moduli space of genus zero curves with marked points and its compactification. The main ingredients are the analytic continuation of Coleman functions to the normal bundle of divisors at infinity and definition of a special tangential base point on the moduli space.

The well-known fiber dimension theorem in algebraic geometry says that for every morphism f:X→Y of integral schemes of finite type the dimension of every fiber of f is at least dim X−dim Y. This has recently been generalized by Brosnan, Reichstein and Vistoli to certain morphisms of algebraic stacks f:𝒳→𝒴, where the usual dimension is replaced by essential dimension. We will prove a general version for morphisms of categories fibered in groupoids. Moreover, we will prove a variant of this theorem, where essential dimension and canonical dimension are linked. These results let us relate essential dimension to canonical dimension of algebraic groups. In particular, using the recent computation of the essential dimension of algebraic tori by MacDonald, Meyer, Reichstein and the author, we establish a lower bound on the canonical dimension of algebraic tori.

We give a criterion to check if, given a prime power pr with r > 1, the only rational points of the modular curve X0+ (pr) are trivial (i.e. cusps or points furnished by complex multiplication). We then prove that this criterion is verified if p satisfies explicit congruences. This applies in particular to the modular curves Xsplit (p), which intervene in the problem of Serre concerning uniform surjectivity of Galois representations associated to division points of elliptic curves.

Colmez has given a recipe to associate a smooth modular representation Ω(W) of the Borel subgroup of GL2(Qp) to a -representation W of by using Fontaine’s theory of (φ,Γ)-modules. We compute Ω(W) explicitly and we prove that if W is irreducible and dim (W)=2, then Ω(W) is the restriction to the Borel subgroup of GL2(Qp) of the supersingular representation associated to W by Breuil’s correspondence.

In this article, we prove that a free divisor in a three-dimensional complex manifold must be Euler homogeneous in a strong sense if the cohomology of its complement is the hypercohomology of its logarithmic differential forms. Calderón-Moreno et al. conjectured this implication in all dimensions and proved it in dimension two. We prove a theorem that describes in all dimensions a special minimal system of generators for the module of formal logarithmic vector fields. This formal structure theorem is closely related to the formal decomposition of a vector field by Kyoji Saito and is used in the proof of the above result. Another consequence of the formal structure theorem is that the truncated Lie algebras of logarithmic vector fields up to dimension three are solvable. We give an example that this may fail in higher dimensions.

We give an elementary proof of the function field (or geometric) case of the Mordell–Lang conjecture in characteristic zero.

We prove that the Witten–Reshetikhin–Turaev (WRT) SO(3) invariant of an arbitrary 3-manifold M is always an algebraic integer. Moreover, we give a rational surgery formula for the unified invariant dominating WRT SO(3) invariants of rational homology 3-spheres at roots of unity of order co-prime with the torsion. As an application, we compute the unified invariant for Seifert fibered spaces and for Dehn surgeries on twist knots. We show that this invariant separates Seifert fibered integral homology spaces and can be used to detect the unknot.

Kollár's conjecture, that log canonical singularities are Du Bois, is proved in the case of Cohen–Macaulay 3-folds. This in turn is used to derive Kodaira vanishing for this class of varieties. Finally it is proved that small deformations of Du Bois singularities are again Du Bois.

We study linear functions on fibrations whose central fibre is a linear free divisor. We analyse the Gauß–Manin system associated to these functions, and prove the existence of a primitive and homogenous form. As a consequence, we show that the base space of the semi-universal unfolding of such a function carries a Frobenius manifold structure.

Let f:X→S be a projective morphism of Noetherian schemes. We assume f purely of relative dimension d and finite Tor-dimensional. We associate to d+1 invertible sheaves $\cal L$1,$\ldots$,$\cal L$d+1 on X a line bundle IX/S($\cal L$1,$\ldots$,$\Cal L$d+1) on S depending additively on the $\cal L$i, commuting to ‘good’ base changes and which represents the integral along the fibres of f of the product of the first Chern classes of the $\cal L$i. If d=0, IX/S($\cal L$) is the norm $\cal N$X/S($\cal L$).

The Sarkisov program studies birational maps between varieties that are end products of the Minimal Model Program (MMP) on nonsingular uniruled varieties. If $X$ and $Y$ are terminal $ \mathbb{Q} $-factorial projective varieties endowed with a structure of Mori fibre space, a birational map $f: X\dashrightarrow Y$ is the composition of a finite number of elementary Sarkisov links. This decomposition is in general not unique: two such define a relation in the Sarkisov program. I define elementary relations, and show they generate relations in the Sarkisov program. Roughly speaking, elementary relations are the relations among the end products of suitable relative MMPs of $Z$ over $W$ with $\rho (Z/ W)= 3$.

This paper extends to quasi-projective varieties earlier work by the author and H. Blaine Lawson concerning spaces of algebraic cocycles on projective varieties. The topological monoid $C_r(Y)(U)$ of effective cocycles on a normal, quasi-projective variety $U$ with values in a projective variety $Y$ consists of algebraic cycles on $U×Y$ equi-dimensional of relative dimension r over $U$. A careful choice of topology enables the establishment of various good properties: the definition is essentially algebraic, the group completion $Z_r(Y)(U)$ has “sensible” homotopy groups, the construction is contravariant with respect to $U$, convariant with respect to $Y$, and there is a natural “quality map” to the topological group of cycles on $U×Y$. The fundamental theorem presented here is the extension of Friedlander-Lawson duality to this context: the duality map $Z_r(Y)(U)$ to $Z_r+m(U × Y)$ is a homotopy equivalence provided that both $U$ and $Y$ are smooth (where $m=dim U$). Various application are given, especially the determination of the homotopy types of certain topological groups of algeb raic morphisms.

Let G be a real reductive Lie group, K its compact subgroup. Let A be the algebra of G-invariant real-analytic functions on T*(G/K) (with respect to the Poisson bracket) and let C be the center of A. Denote by 2ε(G,K) the maximal number of functionally independent functions from A\C. We prove that ε(G,K) is equal to the codimension δ(G,K) of maximal dimension orbits of the Borel subgroup B⊂G$^{\Bbb C}$ in the complex algebraic variety G$^{\Bbb C}$/K$^{\Bbb C}$. Moreover, if δ(G,K)=1, then all G-invariant Hamiltonian systems on T*(G/K) are integrable in the class of the integrals generated by the symmetry group G. We also discuss related questions in the geometry of the Borel group action.

We prove the existence of extremal Sasakian structures occurring on a countably infinite number of distinct contact structures on ${T}^{2} \times {S}^{3} $ and certain related 5-manifolds. These structures occur in bouquets and exhaust the Sasaki cones in all except one case in which there are no extremal metrics.

We prove arithmetic Hilbert–Samuel type theorems for semi-positive singular hermitian line bundles of finite height. This includes the log-singular metrics of Burgos–Kramer–Kühn. The results apply in particular to line bundles of modular forms on some non-compact Shimura varieties. As an example, we treat the case of Hilbert modular surfaces, establishing an arithmetic analogue of the classical result expressing the dimensions of spaces of cusp forms in terms of special values of Dedekind zeta functions.

A general conjecture is stated on the cone of automorphic vector bundles admitting nonzero global sections on schemes endowed with a smooth, surjective morphism to a stack of $G$-zips of connected Hodge type; such schemes should include all Hodge-type Shimura varieties with hyperspecial level. We prove our conjecture for groups of type $A_{1}^{n}$, $C_{2}$, and $\mathbf{F}_{p}$-split groups of type $A_{2}$ (this includes all Hilbert–Blumenthal varieties and should also apply to Siegel modular $3$-folds and Picard modular surfaces). An example is given to show that our conjecture can fail for zip data not of connected Hodge type.

We prove that exact functors between the categories of perfect complexes supported on projective schemes are of Fourier–Mukai type if the functor satisfies a condition weaker than being fully faithful. We also get generalizations of the results in the literature in the case without support conditions. Some applications are discussed and, along the way, we prove that the category of perfect supported complexes has a strongly unique enhancement.