scholarly journals The Double Complex of a Blow-up

2019 ◽  
Author(s):  
Jonas Stelzig
Keyword(s):  
Blow Up ◽  
Differential Forms ◽  
Complex Manifolds ◽  
Double Complex ◽  
Smooth Complex ◽  
Projective Bundles ◽  
De Rham ◽  
Suitable Notion ◽  
Complex Valued ◽  
Compact Complex Manifolds

AbstractWe compute the double complex of smooth complex-valued differential forms on projective bundles over and blow-ups of compact complex manifolds up to a suitable notion of quasi-isomorphism. This simultaneously yields formulas for “all” cohomologies naturally associated with this complex (in particular, de Rham, Dolbeault, Bott–Chern, and Aeppli).

VERTICAL TANGENTIAL INVARIANTS ON SOME FOLIATED LAGRANGE SPACES

2013 ◽  
Vol 10 (04) ◽  
pp. 1320002
Author(s):  
CRISTIAN IDA
Keyword(s):  
Tangent Bundle ◽  
Differential Forms ◽  
Characteristic Classes ◽  
Exterior Derivative ◽  
Cohomology Groups ◽  
Foliated Manifold ◽  
Smooth Complex ◽  
De Rham ◽  
Poincaré Lemma ◽  
Natural Decomposition

In this paper we consider a decomposition of tangentially differential forms with respect to the lifted foliation [Formula: see text] to the tangent bundle of a Lagrange space [Formula: see text] endowed with a regular foliation [Formula: see text]. First, starting from a natural decomposition of the tangential exterior derivative along the leaves of [Formula: see text], we define some vertical tangential cohomology groups of the foliated manifold [Formula: see text], we prove a Poincaré lemma for the vertical tangential derivative and we obtain a de Rham theorem for this cohomology. Next, in a classical way, we construct vertical tangential characteristic classes of tangentially smooth complex bundles over the foliated manifold [Formula: see text].

scholarly journals Effective de Rham cohomology — The general case

2019 ◽  
Vol 21 (05) ◽  
pp. 1850067
Author(s):  
Peter Scheiblechner
Keyword(s):  
Arbitrary Dimension ◽  
Differential Forms ◽  
Affine Variety ◽  
De Rham Cohomology ◽  
Smooth Complex ◽  
Polynomial Coefficients ◽  
Dimension Formula ◽  
De Rham ◽  
Smooth Affine ◽  
Single Exponential

Grothendieck has proved that each class in the de Rham cohomology of a smooth complex affine variety can be represented by a differential form with polynomial coefficients. We prove a single exponential bound on the degrees of these polynomials for varieties of arbitrary dimension. More precisely, we show that the [Formula: see text]th de Rham cohomology of a smooth affine variety of dimension [Formula: see text] and degree [Formula: see text] can be represented by differential forms of degree [Formula: see text]. This result is relevant for the algorithmic computation of the cohomology, but is also motivated by questions in the theory of ordinary differential equations related to the infinitesimal Hilbert 16th problem.

scholarly journals Finite-dimensional complex manifolds on commutative Banach algebras and continuous families of compact complex manifolds

2019 ◽  
Vol 6 (1) ◽  
pp. 228-264
Author(s):  
Hiroki Yagisita
Keyword(s):  
Cross Sections ◽  
Hausdorff Space ◽  
Banach Algebras ◽  
Continuous Functions ◽  
Continuous Family ◽  
Complex Manifolds ◽  
Finite Dimensional ◽  
Commutative Banach Algebras ◽  
Complex Valued ◽  
Compact Complex Manifolds

AbstractLet Γ(M) be the set of all global continuous cross sections of a continuous family M of compact complex manifolds on a compact Hausdorff space X. In this paper, we introduce a C(X)-manifold structure on Γ(M). Especially, if X is contractible, then Γ(M) is a finite-dimensional C(X)-manifold. Here, C(X) denotes the Banach algebra of all complex-valued continuous functions on X.

scholarly journals From Sheaf Cohomology to the Algebraic de Rham Theorem

2017 ◽  
Author(s):  
Fouad El Zein ◽  
Loring W. Tu
Keyword(s):  
Projective Variety ◽  
Differential Forms ◽  
Sheaf Cohomology ◽  
Coherent Sheaves ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Singular Cohomology ◽  
De Rham ◽  
Cech Cohomology ◽  
Complex Projective

This chapter proves Grothendieck's algebraic de Rham theorem. It first proves Grothendieck's algebraic de Rham theorem more or less from scratch for a smooth complex projective variety X, namely, that there is an isomorphism H*(Xₐₙ,ℂ) ≃ H*X,Ω‎subscript alg superscript bullet) between the complex singular cohomology of Xan and the hypercohomology of the complex Ω‎subscript alg superscript bullet of sheaves of algebraic differential forms on X. The proof necessitates a discussion of sheaf cohomology, coherent sheaves, and hypercohomology. The chapter then develops more machinery, mainly the Čech cohomology of a sheaf and the Čech cohomology of a complex of sheaves, as tools for computing hypercohomology. The chapter thus proves that the general case of Grothendieck's theorem is equivalent to the affine case.

scholarly journals Schottky groups acting on homogeneous rational manifolds

2019 ◽  
Vol 2019 (753) ◽  
pp. 23-56 ◽  
Author(s):  
Christian Miebach ◽  
Karl Oeljeklaus
Keyword(s):  
Deformation Theory ◽  
Group Actions ◽  
Picard Group ◽  
Schottky Group ◽  
Fundamental Groups ◽  
Complex Manifolds ◽  
Schottky Groups ◽  
Algebraic Dimension ◽  
Geometric Invariants ◽  
Compact Complex Manifolds

AbstractWe systematically study Schottky group actions on homogeneous rational manifolds and find two new families besides those given by Nori’s well-known construction. This yields new examples of non-Kähler compact complex manifolds having free fundamental groups. We then investigate their analytic and geometric invariants such as the Kodaira and algebraic dimension, the Picard group and the deformation theory, thus extending results due to Lárusson and to Seade and Verjovsky. As a byproduct, we see that the Schottky construction allows to recover examples of equivariant compactifications of {{\rm{SL}}(2,\mathbb{C})/\Gamma} for Γ a discrete free loxodromic subgroup of {{\rm{SL}}(2,\mathbb{C})}, previously obtained by A. Guillot.

scholarly journals Higher-page Bott–Chern and Aeppli cohomologies and applications

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Dan Popovici ◽  
Jonas Stelzig ◽  
Luis Ugarte
Keyword(s):  
Positive Integer ◽  
Spectral Sequence ◽  
Complex Manifolds ◽  
Serre Duality ◽  
Frölicher Spectral Sequence ◽  
Compact Complex Manifolds

Abstract For every positive integer r, we introduce two new cohomologies, that we call E r {E_{r}} -Bott–Chern and E r {E_{r}} -Aeppli, on compact complex manifolds. When r = 1 {r\kern-1.0pt=\kern-1.0pt1} , they coincide with the usual Bott–Chern and Aeppli cohomologies, but they are coarser, respectively finer, than these when r ≥ 2 {r\geq 2} . They provide analogues in the Bott–Chern–Aeppli context of the E r {E_{r}} -cohomologies featuring in the Frölicher spectral sequence of the manifold. We apply these new cohomologies in several ways to characterise the notion of page- ( r - 1 ) {(r-1)} - ∂ ⁡ ∂ ¯ {\partial\bar{\partial}} -manifolds that we introduced very recently. We also prove analogues of the Serre duality for these higher-page Bott–Chern and Aeppli cohomologies and for the spaces featuring in the Frölicher spectral sequence. We obtain a further group of applications of our cohomologies to the study of Hermitian-symplectic and strongly Gauduchon metrics for which we show that they provide the natural cohomological framework.

scholarly journals The de Rham Cohomology through Hilbert Space Methods

2019 ◽  
pp. 455-465
Author(s):  
Ihsane Malass ◽  
Nikolai Tarkhanov
Keyword(s):  
Hilbert Space ◽  
Elliptic Boundary ◽  
Differential Forms ◽  
Adjoint Operator ◽  
Lagrange Equations ◽  
De Rham Cohomology ◽  
Elliptic Boundary Value ◽  
De Rham ◽  
Canonical Representations ◽  
The Neumann Problem

We discuss canonical representations of the de Rham cohomology on a compact manifold with boundary. They are obtained by minimising the energy integral in a Hilbert space of differential forms that belong along with the exterior derivative to the domain of the adjoint operator. The corresponding Euler- Lagrange equations reduce to an elliptic boundary value problem on the manifold, which is usually referred to as the Neumann problem after Spencer

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