arXiv subjects
Publications and source records attributed to Donu Arapura.
We study the first step of the weight filtration on the cohomology of a proper complex algebraic variety, which we call the combinatorial part. We obtain a natural upper bound on its size, which gives rather strong information about the topology of rational singularities.
The purpose of this note is prove that the mixed Hodge structure constructed by the author in math.AG/0301140 [The Leray spectral sequence is motivic, Invent. 2005] for geometric variations of Hodge structure coincides with the structure coming from M. Saito's mixed Hodge module theory. This note also contains a small erratum for the earlier paper.
Let SU_X(n,L) be the moduli space of rank n semistable vector bundles with fixed determinant L on a smooth projective genus g>1 curve X. Let SU_X^s(n,L) denote the open subset parameterizing stable bundles. We show that for small i, the mixed Hodge structure on H^i(SU_X^s(n, L), Q) is independent of the degree of L, and hence pure of weight i. Moreover any simple factors is, up to Tate twisting, isomorphic to a summand of a tensor power of H^1(X,Q). A more precise statement for i = 3, yields a Torelli theorem complementing earlier work of several authors. This is a replacement of our preprint Intermediate Jacobians of Moduli spaces which contained a gap.
Given a complex algebraic variety X, we define a natural number called the motivic dimension which measures the amount of transcendental (co)homology of X. It is zero precisely when all the (co)homolgy is spanned by algebraic cycles. Most of this paper is concerned with giving estimates on this number, along with examples where it is small. As an application, we check or recheck the Hodge conjectue in a number of examples: uniruled fourfolds, rationally connected fivefolds, fourfolds fibred by surfaces with p_g=0, Hilbert schemes of a small number points on surfaces with p_g=0, and generic hypersurfaces.
An abelian category of relative pure motives is constructed along the lines of André (over a field of characteristic 0). An algebraic stack is shown to possess a motive in this sense. This motive is studied for the moduli stack of G-bundles over the universal curve for appropriate G, and a motivic Atiyah-Bott formula is obtained. By taking Hodge realization, a description of the corresponding variations of Hodge structures over M_g is given. In particular, it follows that these lie in the tensor category generated by the variation of Hodge structure associated to the curve itself. Similar results are also given for low degree variations of Hodge structures associated to the moduli space of stable G-bundles.
This is a sequel to the paper "Frobenius amplitude and strong vanishing theorems for vector bundles" (math.AG/0202129). We introduce a more elementary variant of the notion of F-amplitude from the earlier paper which we call amplitude. This provides another measure of positivity of a vector bundle which is related to a number of preexisting positivity notions such as k-ampleness or q-convexity. We use this to refine the estimates of F-amplitude from the first paper, and to deduce some further vanishing theorems as a consequence. We also give some new proofs of some known vanishing theorems for Abelian and toric varieties by analogous methods. For technical reasons, we need to develop a theory of partial Castelnuovo-Mumford regularity which provides a rough measure of the cohomological complexity of a sheaf. Since this material may have independent interest, it is contained in a section which can be read on its own.
Given two smooth projective varieties X and Y over a field, we say that X motivates Y if the (suitably defined) motive of Y is contained in the category generated from X by taking sums, summands and products. This notion has appeared implicitly in many places, but it seems useful to make it explicit. Some techniques are given for checking this condition, but in a nutshell it involves building a correspondence which "dominates" Y. Among the more interesting examples dealt with are moduli spaces. We show that in number of cases moduli spaces of sheaves over curves or surfaces are motivated by underly curve or surface. This allows us to check (or recheck) the (generalized) Hodge and Lefschetz standard conjectures for some of these examples. The dominating correspondence in these examples is built from the universal sheaf, and can, in some instances, be realized as a kind of Fourier-Mukai transform.
It is shown that if the generalized Hodge conjecture, or some weaker form of it, holds for a Calabi-Yau variety then it holds for any Calabi-Yau variety birationally equivalent to it. The key idea is to construct suitable homomorphisms between the Grothendieck group of varieties and the Grothendieck group of the exact category of filtered Hodge structures. This along with motivic integration yields the first statement.
The Hodge conjecture is shown to hold for rationally connected fivefolds, or more generally for fivefolds for which the base of the maximal rationally connected fibration is at most 3 dimensional.
There is a natural descending filtration on the singular cohomology of a complex smooth projective variety called the coniveau filtration. The generalized Hodge conjecture would imply, rather trivially, that the coniveau filtration is compatible with pushforwards, pullbacks and products. The purpose of this paper is to prove this statement unconditionally. This completes the argument in math.AG/0102070.
Our goal is to prove that the Leray spectral sequence associated to a map of algebraic varieties is motivic in the following sense: If the singular cohomology groups of the category of quasiprojective varieties defined over a subfield of C can be canonically endowed with some additional structure, such as its mixed Hodge structure or its motive (in Nori's sense or as a product of realizations). Then the Leray spectral sequence for a projective map in this category is automatically compatible with this structure. As byproduct, we get a relatively cheap construction of a mixed Hodge structure on the cohomology of variation of Hodge structure of geometric origin.
Compact Kähler solvmanifolds are classified up to biholomorphism. A proof of a conjecture Benson and Gordon, that completely solvable compact Kähler solvmanifolds are tori is deduced from this. The main ingredient in the proof is a restriction theorem for polycyclic Kähler groups proved by Nori and the author.
The goal is to verify the Hodge conjecture (and some related conjectures) for certain moduli spaces. It is shown that the (generalized) Hodge conjecture holds for the projective moduli spaces of vector bundles over an abelian or K3 surface or a curve if it holds for all powers of the surface or curve. In the latter case, the curve can be replaced by its Jacobian. As a consequence the generalized Hodge conjecture holds for these moduli spaces when the curve is very general. A similar result is proved for the Hilbert schemes of points over an arbitrary surface.
The primary goal of this paper is to systematically exploit the method of Deligne-Illusie to obtain Kodaira type vanishing theorems for vector bundles and more generally coherent sheaves on algebraic varieties. The key idea is to introduce a number which provides a cohomological measure of the positivity of a coherent sheaf called the Frobenius or F-amplitude. As the terminology indicates, the definition is most natural over a field of positive characteristic, however it can be forced into characteristic 0 by standard tricks. The F-amplitude enters into the statement of the basic vanishing theorem, and this leads to the problem of calculating, or at least estimating, this number. Most of the work in this paper is devoted to doing this various situations. The key result in this direction is a bound on the F-amplitude of an ample vector bundle. When combined with the vanishing theorem, this has some reasonably down to earth corollaries (both old and new).
In this short note, we determine the Kodaira dimension and some of the plurigenera of (a desingularization of) a symmetric power of a smooth projective variety. We use it to obtain bounds on the genus of curve passing through a fixed number of general points.
This paper contains a Kawamata-Viehweg-Kollár type vanishing theorem for vector bundles. In order to formulate and prove this cleanly, we introduce a class of sheaves that automatically satisfies a vanishing theorem. This is obtained by closing up the subclass of sheaves of adjoint type under extensions, direct images, and taking direct summands.
This is primarily an expository piece and the first sentence of the introduction pretty much sums it up: This article is aimed at people who already know what mixed Hodge structures are and what they are good for, but who are not sure how to construct them.
This is a long overdue write up of the following: If the fundamental group of a normal complex algebraic variety (respectively Zariski open subset of a compact Kähler manifold) is a solvable group of matrices over Q (respectively polycyclic group), then it is virtually nilpotent. This should be interpreted as saying that a large class of solvable groups will not occur as fundamental groups of such spaces. The essential strategy is to first "complete" these groups to algebraic groups (generalizing constructions of Malcev and Mostow), and then check that the identity component is unipotent. This is done by Galois theoretic methods in the algebraic case, and is reduced to homological properties of one dimensional characters in the analytic case.