Height pairing and nearby cycles
We prove that, as was conjectured by Spencer Bloch, the Hodge period of some limit Hodge structures equals the height pairing of algebraic cycles on the resolution of singularities of the singular fiber.
math.AG↗
arXiv subjects
Alexander Beilinson
Publications and source records attributed to Alexander Beilinson.
Constructible sheaves are holonomic
We show that for any constructible sheaf F on a smooth algebraic variety X over a field of arbitrary characteristic its singular support SS(F) is equidimensional of dimension dim X. Here SS(F) is the minimal closed subset of the cotangent bundle of X such that every (local) function on X with df(X) disjoint from SS(F) is locally acyclic relative to F.
math.AG↗
I.M.Gelfand and his seminar -- a presence
These are reminiscences of I.M.Gelfand's mathematical seminar of 1970s-1980s. The essay will appear in the March 2016 issue of Notices of the AMS.
math.HO↗
Topological polylogarithms and $p$-adic interpolation of $L$-values of totally real fields
We develop the topological polylogarithm which provides an integral version of Nori's Eisenstein cohomology classes for $GL_n(\mathbf{Z})$ and yields classes with values in an Iwasawa algebra. This implies directly the integrality properties of special values of $L$-functions of totally real fields and a construction of the associated $p$-adic $L$-function. Using a result of Graf, we also apply this to prove some integrality and $p$-adic interpolation results for the Eisenstein cohomology of Hilbert modular varieties.
math.NT↗
Relative continuous K-theory and cyclic homology
We show that for an associative algebra A and its ideal I such that the I-adic topology on A coincides with the p-adic topology, the relative continuous K-theory pro-spectrum "lim"K(A_i, IA_i), where A_i :=A/p^i A, is naturally isogenous to the cyclic chain pro-complex "lim"CC(A_i) (subject to minor conditions on A). This identification is a continuous version of the classical Goodwillie isomorphism. The work comes from an attempt to understand the article of Bloch, Esnault, and Kerz "p-adic deformations of algebraic cycle classes".
math.AG↗
On the crystalline period map
The paper contains a proof of the Fontaine-Jannsen conjecture based on a crystalline version of the p-adic Poincar'e lemma (different proofs were found earlier by Faltings, Niziol and Tsuji).
math.AG↗
p-adic periods and derived de Rham cohomology
We construct p-adic period map using derived de Rham cohomology of Illusie and give a simple proof of Fontaine's C_{dR} conjecture.
math.AG↗
On a theorem of Kisin
The note provides a simple proof of Kisin's theorem about the restriction of crystalline representations to certain subgroup of the Galois group.
math.NT↗
A remark on primitive cycles and the Radon transform
We show that the use of Brylinski's Radon transform elucidates some points of the Green-Griffiths approach to the Hodge conjecture.
math.AG↗
A corollary of the b-function lemma
Let $X$ be an algebraic variety, $f$ a regular function, $j:U\subset X$ the complement to the locus of vanishing of $f$, and $M$ a holonomic D-module on $U$. Consider the $D_U[s]$-module $M\otimes "f^s"$. The goal of this note is to describe all $D_X[s]$-submodules $N\subset j_*(M\otimes "f^s")$ such that $j^*(N)\simeq M\otimes "f^s"$.
math.AG↗
Remarks on Grothendieck's standard conjectures
We show that Grothendieck's standard conjectures are implied by either of two other motivic conjectures: (a) by that of the existence of the motivic t-structure, and (b) by (a weak form of) Suslin's Lawson homology conjecture.
math.AG↗
$ε$-Factors for the Period Determinants of Curves
The article provides a factorization formalism for determinants of the period matrices for D-modules on curves. Unlike previous approach due to Bloch, Deligne, and Esnault, it does not use Fourier transform.
math.AG↗
Topological $ε$-factors
The article describes a purely topological counterpart of the $ε$-factorization of constants in the functional equations (which is a key ingredient in the interplay between L-functions and classical automorphic forms). We consider the determinant of the cohomology of a constructible sheaf F on a real analytic manifold X (or a bit more precise object, which is $RΓ(X,F)$ seen as a homotopy point of the K-theory spectrum), and show that it can be "computed" by means of a "spectral" version of the Dubson-Kashiwara formula, which yields, in particular, the $ε$-factorization format. This picture may lead to a better understanding of a recent work of Bloch-Deligne-Esnault on the determinant of the period matrix.
math.AG↗
$ε$-factors for Gauss-Manin determinants
This is the last version of AG/0111277. Here the old abstract: We define $ε$-factors in the de Rham setting and calculate the determinant of the Gauß-Manin connection for a family of (affine) curves and a vector bundle equipped with a flat connection.
math.AG↗
Torelli theorem via Fourier-Mukai transform
We show that the Fourier transform on the Jacobian of a curve interchanges "$δ$ functions" at the curve and the theta divisor. The Torelli theorem is an immediate consequence.
math.AG↗
Wall-Crossing functors and D-modules
We study Translation functors and Wall-Crossing functors on infinite dimensional representations of a complex semisimple Lie algebra using D-modules. This functorial machinery is then used to prove the Endomorphism-theorem and the Structure-theorem, two important results established earlier by W. Soergel in a totally different way. Other applications to the category O of Bernstein-Gelfand-Gelfand are given, and some conjectural relationships between Koszul duality, Verdier duality and convolution functors are discussed. A geometric interpretation of tilting modules is given.
alg-geom↗