The rigid syntomic ring spectrum

The aim of this paper is to show that Besser syntomic cohomology is representable by a rational ring spectrum in the motivic homotopical sense. In fact, extending previous constructions, we exhibit a simple representability criterion and we apply it to several cohomologies in order to get our central result. This theorem gives new results for syntomic cohomology such as h-descent and the compatibility of cycle classes with Gysin morphisms. Along the way, we prove that motivic ring spectra induces a complete Bloch-Ogus cohomological formalism and even more. Finally, following a general motivic homotopical philosophy, we exhibit a natural notion of syntomic coefficients.Comment: Final version to appear in the Journal de l'institut des Math\'ematiques de Jussieu. Many typos have been corrected and the exposition has been improved according to the suggestions of the referees: we thank them a lot

A geometric model for Hochschild homology of Soergel bimodules

An important step in the calculation of the triply graded link homology theory of Khovanov and Rozansky is the determination of the Hochschild homology of Soergel bimodules for SL(n). We present a geometric model for this Hochschild homology for any simple group G, as equivariant intersection homology of B x B-orbit closures in G. We show that, in type A these orbit closures are equivariantly formal for the conjugation T-action. We use this fact to show that in the case where the corresponding orbit closure is smooth, this Hochschild homology is an exterior algebra over a polynomial ring on generators whose degree is explicitly determined by the geometry of the orbit closure, and describe its Hilbert series, proving a conjecture of Jacob Rasmussen.Comment: 19 pages, no figure

Karoubi's relative Chern character, the rigid syntomic regulator, and the Bloch-Kato exponential map

We construct a variant of Karoubi's relative Chern character for smooth, separated schemes over the ring of integers in a p-adic field and prove a comparison with the rigid syntomic regulator. For smooth projective schemes we further relate the relative Chern character to the etale p-adic regulator via the Bloch-Kato exponential map. This reproves a result of Huber and Kings for the spectrum of the ring of integers and generalizes it to all smooth projective schemes as above.Comment: v1:33 pages; v2:major revision (28 pages); v3:minor changes; v4:minor changes following suggestions by a refere

Moduli spaces for Bondal quivers

Given a sufficiently nice collection of sheaves on an algebraic variety V, Bondal explained how to build a quiver Q along with an ideal of relations in the path algebra of Q such that the derived category of representations of Q subject to these relations is equivalent to the derived category of coherent sheaves on V. We consider the case in which these sheaves are all locally free and study the moduli spaces of semistable representations of our quiver with relations for various stability conditions. We show that V can often be recovered as a connected component of such a moduli space and we describe the line bundle induced by a GIT construction of the moduli space in terms of the input data. In certain special cases, we interpret our results in the language of topological string theory.Comment: 17 pages, major revisio

Homological mirror symmetry for the quintic 3-fold

We prove homological mirror symmetry for the quintic Calabi-Yau 3-fold. The proof follows that for the quartic surface by Seidel (arXiv:math/0310414) closely, and uses a result of Sheridan (arXiv:1012.3238). In contrast to Sheridan's approach (arXiv:1111.0632), our proof gives the compatibility of homological mirror symmetry for the projective space and its Calabi-Yau hypersurface.Comment: 29 pages, 6 figures. v2: revised following the suggestions of the referee

2-Gerbes bound by complexes of gr-stacks, and cohomology

We define 2-gerbes bound by complexes of braided group-like stacks. We prove a classification result in terms of hypercohomology groups with values in abelian crossed squares and cones of morphisms of complexes of length 3. We give an application to the geometric construction of certain elements in Hermitian Deligne cohomology groups.Comment: 70 pages, latex+amsmath+xypi

BPS states of curves in Calabi--Yau 3--folds

The Gopakumar-Vafa conjecture is defined and studied for the local geometry of a curve in a Calabi-Yau 3-fold. The integrality predicted in Gromov-Witten theory by the Gopakumar-Vafa BPS count is verified in a natural series of cases in this local geometry. The method involves Gromov-Witten computations, Mobius inversion, and a combinatorial analysis of the numbers of etale covers of a curve.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper9.abs.html Version 3 is GT version 2 and has corrections to eq (2) on p 295, to 1st eq in Prop 2.1 and the tables on p 39

Mixed Artin-Tate motives over number rings

This paper studies Artin-Tate motives over number rings. As a subcategory of geometric motives, the triangulated category of Artin-Tate motives DATM(S) is generated by motives of schemes that are finite over the base S. After establishing stability of these subcategories under pullback and pushforward along open and closed immersions, a motivic t-structure is constructed. Exactness properties of these functors familiar from perverse sheaves are shown to hold in this context. The cohomological dimension of mixed Artin-Tate motives is two, and there is an equivalence of the triangulated category of Artin-Tate motives with the derived category of mixed Artin-Tate motives. Update in second version: a functorial and strict weight filtration for mixed Artin-Tate motives is established. Moreover, two minor corrections have been performed: first, the category of Artin-Tate motives is now defined to be the triangulated category generated by direct factors of $f_* \mathbf 1(n)$---as opposed to the thick category generated by these generators. Secondly, the exactness of $f^*$ for a finite map is only stated for etale maps $f$

Derived splinters in positive characteristic

This paper introduces the notion of a derived splinter. Roughly speaking, a scheme is a derived splinter if it splits off from the coherent cohomology of any proper cover. Over a field of characteristic 0, this condition characterises rational singularities by a result of Kov\'acs. Our main theorem asserts that over a field of characteristic p, derived splinters are the same as (underived) splinters, i.e., as schemes that split off from any finite cover. Using this result, we answer some questions of Karen Smith concerning extending Serre/Kodaira type vanishing results beyond the class of ample line bundles in positive characteristic; these are purely projective geometric statements independent of singularity considerations. In fact, we can prove "up to finite cover" analogues in characteristic p of many vanishing theorems one knows in characteristic 0. All these results fit naturally in the study of F-singularities, and are motivated by a desire to understand the direct summand conjectureComment: 22 pages, comments welcome

Lagrangian homology spheres in (A_m) Milnor fibres via C^*-equivariant A_infinity modules

We establish restrictions on Lagrangian embeddings of rational homology spheres into certain open symplectic manifolds, namely the (A_m) Milnor fibres of odd complex dimension. This relies on general considerations about equivariant objects in module categories (which may be applicable in other situations as well), as well as results of Ishii-Uehara and Ishii-Ueda-Uehara concerning the derived categories of coherent sheaves on the resolutions of (A_m) surface singularities.Comment: version 2: better results, simpler proofs; version 3: title changed as requested by referee, other very minor modification