On approximations of the de Rham complex and their cohomology

For a commutative algebra R, its de Rham cohomology is an important invariant of R. In the paper, an infinite chain of de Rham-like complexes is introduced where the first member of the chain is the de Rham complex. The complexes are called approximations of the de Rham complex. Their cohomologies are found for polynomial rings and algebras of power series over a field of characteristic zero.     

Rational points and cohomology of discriminant varieties

Let G be a finite unitary reflection group acting in a complex vector space . The discriminant varietyX_G of G is defined as the space of regular orbits of G on V. Classical examples include the varieties of complex polynomials of degree n with distinct (resp. non-zero distinct) roots. The normaliser of G in GL(V) acts on X_G; in this work we determine the action of on the cohomology of X_G. In the classical cases this amounts to computing the cohomology of X_G with certain local coefficient systems. Our methods are to compute equivariant weight polynomials by means of explicit counting of the rational points of certain varieties over finite fields, and then to exploit the weight purity of the relevant varieties. We obtain some power series identities as a byproduct.     

Vanishing and bases for cohomology of partially trivial local systems on hyperplane arrangements

In this paper we prove a vanishing theorem and construct bases for the cohomology of partially trivial local systems on complements of hyperplane arrangements. As a result, we obtain a non-resonance condition for partially trivial local systems.

     

Deligne's duality for de Rham realizations of 1‐motives

Let M be a 1‐motive over a base scheme S and M ′ its Cartier dual. We show the existence of a canonical duality between the de Rham realizations of M and M ′; this generalizes a result in [5]. Furthermore, we study universal extensions of 1‐motives and their relation with ?‐extensions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Minimal ‐crystals and isomorphism numbers of isosimple ‐crystals

In this paper we generalize minimal p‐divisible groups defined by Oort to minimal F‐crystals over algebraically closed fields of positive characteristic. We prove a structural theorem of minimal F‐crystals and give an explicit formula of the Frobenius endomorphism of the basic minimal F‐crystals that are the building blocks of the general minimal F‐crystals. We then use minimal F‐crystals to generalize minimal heights of p‐divisible groups and give an upper bound of the isomorphism numbers of F‐crystals, whose isogeny type are determined by simple F‐isocrystals, in terms of their ranks, Hodge slopes and Newton slopes.     

The Fischer decomposition for Hodge‐de Rham systems in Euclidean spaces

The classical Fischer decomposition of spinor‐valued polynomials is a key result on solutions of the Dirac equation in the Euclidean space . As is well‐known, it can be understood as an irreducible decomposition with respect to the so‐called L‐action of the Pin group Pin(m). But, in Clifford algebra valued polynomials, we can consider also the H‐action of Pin(m). In this paper, the corresponding Fischer decomposition for the H‐action is obtained. It turns out that, in this case, basic building blocks are the spaces of homogeneous solutions to the Hodge‐de Rham system. Moreover, it is shown that the Fischer decomposition for the H‐action can be viewed even as a refinement of the classical one. Copyright © 2011 John Wiley & Sons, Ltd.     

A class of non‐graded left‐symmetric algebraic structures on the Witt algebra

We classify the compatible left‐symmetric algebraic structures on the Witt algebra satisfying certain non‐graded conditions. It is unexpected that they are Novikov algebras. Furthermore, as applications, we study the induced non‐graded modules of the Witt algebra and the induced Lie algebras by Novikov‐Poisson algebras’ approach and Balinskii‐Novikov's construction.     

On a motivic interpretation of primitive,variable and fixed cohomology

If M is a smooth projective variety whose motive is Kimura finite‐dimensional and for which the standard Lefschetz Conjecture B holds, then the motive of M splits off a primitive motive whose cohomology is the primitive cohomology. Under the same hypotheses on M, let X be a smooth complete intersection of ample divisors within M. Then the motive of X is the sum of a variable and a fixed motive inducing the corresponding splitting in cohomology. I also give variants with group actions.     

Group theory and p‐adic valued models of swarm behaviour

The swarm behaviour can be controlled by different localizations of attractants (food pieces) and repellents (dangerous places), which, respectively, attract and repel the swarm propagation. If we assume that at each time step, the swarm can find out not more than p ?1 attractants ( ), then the swarm behaviour can be coded by p ‐adic integers, ie, by the numbers of the ring Z _p . Each swarm propagation has the following 2 stages: (1) the discover of localizations of neighbour attractants and repellents and (2) the logistical optimization of the road system connecting all the reachable attractants and avoiding all the neighbour repellents. In the meanwhile, at the discovering stage, the swarm builds some direct roads and, at the logistical stage, the transporting network of the swarm gets loops (circles) and it permanently changes. So, at the first stage, the behaviour can be expressed by some linear p ‐adic valued strings. At the second stage, it is expressed by non‐linear modifications of p ‐adic valued strings. The second stage cannot be described by conventional algebraic tools; therefore, I have introduced the so‐called non‐linear group theory for describing both stages in the swarm propagation.     

GV‐subschemes and their embeddings in principally polarized abelian varieties

We prove that any embedding of a ‐subscheme in a principally polarized abelian variety does not factor through any nontrivial isogeny. As an application, we present a new proof of a theorem of Clemens–Griffiths identifying the intermediate Jacobian of a smooth cubic threefold to the Albanese variety of its Fano surface of lines.     

Cell decomposition for P‐minimal fields

In [12], P. Scowcroft and L. van den Dries proved a cell decomposition theorem for p‐adically closed fields. We work here with the notion of P‐minimal fields defined by D. Haskell and D. Macpherson in [6]. We prove that a P‐minimal field K admits cell decomposition if and only if K has definable selection. A preprint version in French of this result appeared as a prepublication [8] (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A remark on relative geometric invariant theory for quasi‐projective varieties

Relative geometric invariant theory studies the behavior of semistable points under equivariant morphisms. More precisely, suppose G is a reductive linear algebraic group over an algebraically closed field k, X and Y are quasi‐projective varieties endowed with G‐actions, is a G‐equivariant projective morphism, the G‐action on Y is linearized in the ample line bundle M, and the G‐action on X is linearized in the φ‐ample line bundle L. For any positive integer n, there is an induced linearization of the G‐action on X in the line bundle . If Y is projective and , the set of points in X that are semistable with respect to this linearization is contained in the preimage under φ of the set of points in Y that are semistable with respect to the given linearization in M. The same statement is trivially also true, if Y is affine and . In this note, we show by means of an example that the statement does not hold for arbitrary quasi‐projective varieties Y. This shows that a claim by Hu of the contrary is not true. Relative geometric invariant theory plays a role in the construction and study of degenerations of moduli spaces.     

Cohomological relation between Jacobi forms and skew‐holomorphic Jacobi forms

Eichler and Zagier developed a theory of Jacobi forms to understand and extend Maass' work on the Saito‐Kurokawa conjecture. Later Skoruppa introduced skew‐holomorphic Jacobi forms, which play an important role in understanding liftings of modular forms and Jacobi forms. In this paper, we explain a relation between Jacobi forms and skew‐holomorphic Jacobi forms in terms of a group cohomology. More precisely, we introduce an isomorphism from the direct sum of the space of Jacobi cusp forms on and the space of skew‐holomorphic Jacobi cusp forms on with the same half‐integral weight to the Eichler cohomology group of with a coefficient module coming from polynomials.     

On Euler–Jaczewski sequence and Remmert–Van de Ven problem for toric varieties

Let X be a complete toric variety and Y a smooth projective variety with . We prove that, if is a surjective morphism then . Received: 15 May 2001; in final form: 22 October 2001/ Published online: 4 April 2002     

Initial conditions for the cylindrical Korteweg‐de Vries equation

In this paper, we find suitable initial conditions for the cylindrical Korteweg‐de Vries equation by first solving exactly the initial‐value problem for localized solutions of the underlying axisymmetric linear long‐wave equation. The far‐field limit of the solution of this linear problem then provides, through matching, an initial condition for the cylindrical Korteweg‐de Vries equation. This initial condition is associated only with the leading wave front of the far‐field limit of the linear solution. The main motivation is to resolve the discrepancy between the exact mass conservation law, and the “mass” conservation law for the cylindrical Korteweg‐de Vries equation. The outcome is that in the linear initial‐value problem all the mass is carried behind the wave front, and then the “mass” in the initial condition for the cylindrical Korteweg‐de Vries equation is zero. Hence, the evolving solution in the cylindrical Korteweg‐de Vries equation has zero “mass.” This situation arises because, unlike the well‐known unidirectional Korteweg‐de Vries equation, the solution of the initial‐value problem for the axisymmetric linear long‐wave problem contains both outgoing and ingoing waves, but in the cylindrical geometry, the latter are reflected at the origin into outgoing waves, and eventually the total outgoing solution is a combination of these and those initially generated.     

Some results about zero‐cycles on abelian and semi‐abelian varieties

In this short note we extend some results obtained in [7]. First, we prove that for an abelian variety A with good ordinary reduction over a finite extension of with p an odd prime, the Albanese kernel of A is the direct sum of its maximal divisible subgroup and a torsion group. Second, for a semi‐abelian variety G over a perfect field k, we construct a decreasing integral filtration of Suslin's singular homology group, , such that the successive quotients are isomorphic to a certain Somekawa K‐group.     

Error analysis for solving the Korteweg‐de Vries equation by a Legendre pseudo‐spectral method

A Legendre pseudo‐spectral method is proposed for the Korteweg‐de Vries equation with nonperiodic boundary conditions. Appropriate base functions are chosen to get an efficient algorithm. Error analysis is given for both semi‐discrete and fully discrete schemes. The numerical results confirm to the theoretical analysis. © (2000) John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 513–534, (2000)     

Riesz transforms of the Hodge–de Rham Laplacian on Riemannian manifolds

Let M be a complete non‐compact Riemannian manifold satisfying the volume doubling property. Let be the Hodge–de Rham Laplacian acting on 1‐differential forms. According to the Bochner formula, where and are respectively the positive and negative part of the Ricci curvature and ? is the Levi–Civita connection. We study the boundedness of the Riesz transform from to and of the Riesz transform from to . We prove that, if the heat kernel on functions satisfies a Gaussian upper bound and if the negative part of the Ricci curvature is ε‐sub‐critical for some , then is bounded from to and is bounded from to for where depends on ε and on a constant appearing in the volume doubling property. A duality argument gives the boundedness of the Riesz transform from to for where Δ is the non‐negative Laplace–Beltrami operator. We also give a condition on to be ε‐sub‐critical under both analytic and geometric assumptions.     

A Fourier‐Mukai approach to the enumerative geometry of principally polarized abelian surfaces

We study twisted ideal sheaves of small length on an irreducible principally polarized abelian surface $({\mathbb T},\ell )We study twisted ideal sheaves of small length on an irreducible principally polarized abelian surface $({\mathbb T},\ell )$. Using Fourier‐Mukai techniques we associate certain jumping schemes to such sheaves and completely classify such loci. We give examples of applications to the enumerative geometry of ${\mathbb T}$ and show that no smooth genus 5 curve on such a surface can contain a $g^1_3$. We also describe explicitly the singular divisors in the linear system |2?|.