Intersection de Rham complexes in positive characteristic

In this article, we construct two kinds of de Rham–like complexes which compute the cohomology of complete crystals on the higher-level q -crystalline site, which was introduced in a previous article by the author. One complex is the q -analog of the higher de Rham complex constructed by Miyatani, and another complex is the q -analog of the jet complex constructed by Le Stum–Quirós. The complexes we construct can also be regarded as the higher-level analogs of the q –de Rham complex.

In this chapter, we study the topology of C ∞-manifolds. We define the de Rham cohomology of a manifold, which is the vector space of closed differential forms modulo exact forms. After sheafifying the construction, we see that the de Rham complex forms a so-called acyclic resolution of the constant sheaf ℝ. We prove a general result that sheaf cohomology can be computed using such resolutions, and deduce a version of de Rham’s theorem that de Rham cohomology is sheaf cohomology with coefficients in ℝ. It follows that de Rham cohomology depends only on the underlying topology. Using a different acyclic resolution that is dual to the de Rham complex, we prove Poincaré duality. This duality makes cohomology, which is normally contravariant, into a covariant theory. We devote a section to explaining these somewhat mysterious covariant maps, called Gysin maps. We end this chapter with the remarkable Lefschetz trace formula, which in principle, calculates the number of fixed points for a map of a manifold to itself.

We reinterpret algebraic de Rham cohomology for a possibly singular complex variety X as sheaf cohomology in the site of smooth schemes over X with Voevodsky’s h-topology. Our results extend to the algebraic de Rham complex as well. Our main technique is to extend Cech cohomology of hypercovers to arbitrary local acyclic fibrations of simplicial presheaves.

A classical result in differential geometry states that for a free and proper Lie group action, the quotient map to the orbit space induces an isomorphism between the de Rham complex of differential forms on the orbit space and the basic differential forms on the original manifold. In this paper, this result is generalized to the case of a proper Lie groupoid, in which the orbit space is equipped with the quotient diffeological structure. As an application of this, we obtain a de Rham theorem for the de Rham complex on the orbit space.KeywordsDiffeologyLie groupoidde Rham complexBasic formsLinearizationMathematics Subject Classification (2010)58H0522A22

We use group cohomology and the de Rham complex on simplicial manifolds to give explicit differential forms representing generators of the cohomology rings of moduli spaces of representations of fundamental groups of 2-manifolds. These generators are constructed using the de Rham representatives for the cohomology of classifying spaces BK where K is a compact Lie group; such representatives (universal characteristic classes) were found by Bott and Shulman. Thus our representatives for the generators of the cohomology of moduli spaces are given explicitly in terms of the Maurer-Cartan form. This work solves a problem posed by Weinstein, who gave a corresponding construction (following Karshon and Goldman) of the symplectic forms on these moduli spaces. We also give a corresponding construction of equivariant differential forms on the extended moduli space X, which is a finite dimensional symplectic space equipped with a Hamiltonian action of K for which the symplectic reduced space is the moduli space of representations of the 2-manifold fundamental group in K.

For any holomorphic function f\colon X\to\mathbb{C} on a complex manifold X , we define and study moderate growth and rapid decay objects associated to an enhanced ind-sheaf on X . These will be sheaves on the real oriented blow-up space of X along f . We show that, in the context of the irregular Riemann–Hilbert correspondence of D’Agnolo–Kashiwara, these objects recover the classical de Rham complexes with moderate growth and rapid decay associated to a holonomic \mathcal{D}_{X} -module. In order to prove the latter, we resolve a recent conjectural duality of Sabbah between these de Rham complexes of holonomic \mathcal{D}_{X} -modules with growth conditions along a normal crossing divisor by making the connection with a classic duality result of Kashiwara–Schapira between certain topological vector spaces. Via a standard dévissage argument, we then prove Sabbah’s conjecture for arbitrary divisors. As a corollary, we recover the well-known perfect pairing between the algebraic de Rham cohomology and rapid decay homology associated to integrable connections on smooth varieties due to Bloch–Esnault and Hien.

Let [Formula: see text] be an abelian surface, and [Formula: see text] be the sum of [Formula: see text] distinct theta divisors having normal crossings. We set [Formula: see text]. We study the structure of the nonvanishing twisted cohomology group [Formula: see text], where [Formula: see text] denotes a locally constant sheaf over [Formula: see text] defined by a multiplicative meromorphic function on [Formula: see text] infinitely ramified just along the divisor [Formula: see text] (as will be seen below, we will take as such a function a product of complex powers of theta functions). The de Rham complex on [Formula: see text] with logarithmic poles along [Formula: see text], associated to the twisted cohomology groups [Formula: see text], is [Formula: see text]-valued, where [Formula: see text] denotes a topologically trivial (i.e. Chern class zero) line bundle over [Formula: see text] determined by the locally constant sheaf [Formula: see text]. Therefore the main results of this paper, which give us information on the order of poles of meromorphic 2-forms on [Formula: see text] generating the cohomology group [Formula: see text], are divided into Theorems 4.5 and 4.6, according as the de Rham complex on [Formula: see text] with logarithmic poles along [Formula: see text] takes the values in a holomorphically nontrivial line bundle [Formula: see text] or a holomorphically trivial one [Formula: see text] ([Formula: see text] denoting the holomorphically trivial line bundle [Formula: see text]). Such a phenomenon does not occur in the case of the twisted cohomology of complex projective space with hyperplane arrangement.

In this note we give a description of the Steenrod squares starting with the analogue of the cup-i product in the deRhamcomplex of Cartan-Miller. The integration of forms over simplices, which induces the isomorphism on cohomology in the deRham theorem, is not an algebra map. It is proved that it induces a morphism of coalgebras . Therefore the cohomology of a pullback fibration can be computed in terms of the deRham complex in some cases.

Given a log smooth log scheme $X$ over $\operatorname{Spec} \mathbb{C}$, in this article we analyze and compare different filtrations defined on the log de Rham complex $\omega^{\bullet}_X$ associated to $X$. We mainly refer to the articles of Ogus ([23]), Danilov ([1]), Ishida ([16]). In this context, we analyze two filtrations on $\omega^{\bullet}_X$: the decreasing Ogus filtration $\tilde{L}^{\bullet}$, which is a sort of extension of the Deligne weight filtration $W_{\bullet}$ to log smooth log schemes over $\Spec \mathbb{C}$, and an increasing filtration, which we call the Ishida filtration and denote by $I_{\bullet}$, defined by using the Ishida complex $\tilde{\Omega}^{\bullet}_X$ of $X$. Moreover, we have the Danilov de Rham complex $\Omega^{\bullet}_X(\log D)$ with logarithmic poles along $D= X - X_{\mathrm{triv}}$ ($X_{\mathrm{triv}}$ being the trivial locus for the log structure on $X$), endowed with an increasing weight filtration (the Danilov weight filtration $\mathcal{W}_{\bullet}$). Then we prove that the Danilov de Rham complex $\Omega^{\bullet}_X(\log D)$ coincides with the log de Rham complex $\omega^{\bullet}_X$ and the Ishida filtration $I_{\bullet}$ (which is a globalization of the Danilov weight filtration $\mathcal W_{\bullet}$) coincides with the opposite Ogus filtration $\tilde{L}^{-\bullet}$.

We show that, for a pseudo-proper smooth noetherian formal scheme $\mathfrak{X}$ over a positive characteristic $p$ field, its truncated De Rham complex up to the characteristic $p$ is decomposable. Moreover, if the dimension of $\mathfrak{X}$ is exactly $p$, then the full De Rham complex is decomposable. Along the way we establish the Cartier isomorphism associated to a smooth morphism of positive characteristic noetherian formal schemes.

For a smooth foliated manifold $(M,\mathcal F)$, the basic and the foliated cohomologies are defined by using the de Rham complex of $M$. These cohomologies are related with the cohomology of the manifold by the de Rham spectral sequence of $\mathcal F$. A foliated manifold is an example of a space with two topologies, one coarser than the other. For these spaces one can define a continuous cohomology that, for a foliated manifold, corresponds to the continuous foliated (or leafwise) cohomology. In this paper we introduce a construction for spaces with two topologies based upon the Alexander-Spanier continuous cochains. It allows us to define a spectral sequence, similar to the de Rham spectral sequence for a foliation. In particular, continuous basic and foliated cohomologies are defined and related with the cohomology of the space. For a smooth foliated manifold, we also consider Alexander-Spanier differentiable cochains. We compare the continuous and differentiable cohomologies, and the latter with the de Rham cohomology. We prove that all three spectral sequences are isomorphic from $E_2$ onwards if $\mathcal F\/$ is a Riemannian foliation. As a consequence, we conclude that this spectral sequence is a topological invariant of the Riemannian foliation. We also compute some examples. In particular, we give an isomorphism between the $E_2$ term for a $G$-Lie foliation and the reduced cohomology of $G$ (in the sense of S.-T. Hu) with coefficients in the reduced foliated cohomology of $\mathcal F$.

Let M be a symplectic manifold over $ℝ. In [CFS] the authors construct an invariant ϕ in the cyclic cohomology of M for any closed star-product. They compute this invariant in the de Rham complex of M when M=T*V. We generalize this result by computing the image of ϕ in the de Rham complex for any symplectic manifold and any star-product and we show how this invariant is related to the general classification of Kontsevich. The proof uses the Riemann–Roch theorem for periodic cyclic chains of Nest–Tsygan.

Finite element methods for electromagnetic problems modeled by Maxwell-type equations are highly sensitive to the conformity of approximation spaces, and non-conforming methods may cause loss of convergence. This fact leads to an essential obstacle for almost all the interface-unfitted mesh methods in the literature regarding the application to electromagnetic interface problems, as they are based on non-conforming spaces. In this work, a novel immersed virtual element method for solving a three-dimensional (3D) H(curl) interface problem is developed, and the motivation is to combine the conformity of virtual element spaces and robust approximation capabilities of immersed finite element spaces. The proposed method is able to achieve optimal convergence. To develop a systematic framework, the [Formula: see text], H(curl) and H(div) interface problems and their corresponding problem-orientated immersed virtual element spaces are considered all together. In addition, the de Rham complex will be established based on which the Hiptmair–Xu (HX) preconditioner can be used to develop a fast solver for the H(curl) interface problem.

BRST and anti-BRST operators for quantum linear algebra U q( gl( N)).

When quantum field theory is formulated in terms of transition amplitudes between initial and final configurations, it is important to impose mathematically consistent boundary conditions on the fields. The choice of boundary conditions is also important when dealing with elliptic complexes and index theorems on bounded manifolds. Although Dirichlet and Neumann boundary conditions are adequate for many purposes, they are not always appropriate. However, a number of the remaining cases can be handled by imposing Dirichlet conditions on certain field components and Neumann conditions on the others. Quantum gravity and supergravity, spinor field theory, and various elliptic complexes all lend themselves to mixed boundary conditions of this sort. The mathematical significance of mixed boundary conditions is investigated, and a simple rule is obtained for determining which field components should obey which boundary conditions. It will be shown that gauge transformations on a bounded manifold give rise to intrinsic gauge transformations within the boundary. This result is used to define gauge-invariant and BRS-invariant quantum amplitudes for boundary configurations. The most general form for mixed boundary conditions on the Dirac operator and the de Rham complex will also be derived.