scholarly journals Note on Dolbeault cohomology and Hodge structures up to bimeromorphisms

2020 ◽  
Vol 7 (1) ◽  
pp. 194-214
Author(s):  
Daniele Angella ◽  
Tatsuo Suwa ◽  
Nicoletta Tardini ◽  
Adriano Tomassini
Keyword(s):  
Complex Manifold ◽  
Compact Complex Manifold ◽  
Complex Manifolds ◽  
Dolbeault Cohomology ◽  
Hodge Structures ◽  
Balanced Metric ◽  
The Stability ◽  
Cech Cohomology ◽  
Simply Connected ◽  
Compact Complex Manifolds

AbstractWe construct a simply-connected compact complex non-Kähler manifold satisfying the ∂ ̅∂ -Lemma, and endowed with a balanced metric. To this aim, we were initially aimed at investigating the stability of the property of satisfying the ∂ ̅∂-Lemma under modifications of compact complex manifolds and orbifolds. This question has been recently addressed and answered in [34, 39, 40, 50] with different techniques. Here, we provide a different approach using Čech cohomology theory to study the Dolbeault cohomology of the blowup ̃XZ of a compact complex manifold X along a submanifold Z admitting a holomorphically contractible neighbourhood.

scholarly journals Dolbeault and J-Invariant Cohomologies on Almost Complex Manifolds

2021 ◽  
Vol 15 (7) ◽  
Author(s):  
Lorenzo Sillari ◽  
Adriano Tomassini
Keyword(s):  
Complex Manifold ◽  
Sufficient Condition ◽  
Complex Manifolds ◽  
Necessary And Sufficient Condition ◽  
Dolbeault Cohomology ◽  
Almost Complex Manifolds ◽  
Almost Complex Manifold ◽  
Almost Complex ◽  
Necessary And Sufficient

AbstractIn this paper we relate the cohomology of J-invariant forms to the Dolbeault cohomology of an almost complex manifold. We find necessary and sufficient condition for the inclusion of the former into the latter to be true up to isomorphism. We also extend some results obtained by J. Cirici and S. O. Wilson about the computation of the left-invariant cohomology of nilmanifolds to the setting of solvmanifolds. Several examples are given.

scholarly journals RC-POSITIVITY, VANISHING THEOREMS AND RIGIDITY OF HOLOMORPHIC MAPS

2019 ◽  
pp. 1-16
Author(s):  
Xiaokui Yang
Keyword(s):  
Complex Manifold ◽  
Hermitian Manifold ◽  
Compact Complex Manifold ◽  
Holomorphic Sectional Curvature ◽  
Vanishing Theorems ◽  
Holomorphic Maps ◽  
Holomorphic Map ◽  
Holomorphic Bisectional Curvature ◽  
Tautological Line Bundle ◽  
Compact Complex Manifolds

Let $M$ and $N$ be two compact complex manifolds. We show that if the tautological line bundle ${\mathcal{O}}_{T_{M}^{\ast }}(1)$ is not pseudo-effective and ${\mathcal{O}}_{T_{N}^{\ast }}(1)$ is nef, then there is no non-constant holomorphic map from $M$ to $N$ . In particular, we prove that any holomorphic map from a compact complex manifold $M$ with RC-positive tangent bundle to a compact complex manifold $N$ with nef cotangent bundle must be a constant map. As an application, we obtain that there is no non-constant holomorphic map from a compact Hermitian manifold with positive holomorphic sectional curvature to a Hermitian manifold with non-positive holomorphic bisectional curvature.

scholarly journals Correspondences, von Neumann Algebras and Holomorphic L2 Torsion

2000 ◽  
Vol 52 (4) ◽  
pp. 695-736 ◽  
Author(s):  
A. Carey ◽  
M. Farber ◽  
V. Mathai
Keyword(s):  
Complex Manifold ◽  
Von Neumann Algebras ◽  
Compact Complex Manifold ◽  
Dolbeault Cohomology ◽  
Hermitian Metrics ◽  
Variation Formula ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Finite Von Neumann Algebras

AbstractGiven a holomorphic Hilbertian bundle on a compact complex manifold, we introduce the notion of holomorphic L2 torsion, which lies in the determinant line of the twisted L2 Dolbeault cohomology and represents a volume element there. Here we utilise the theory of determinant lines of Hilbertian modules over finite von Neumann algebras as developed in [CFM]. This specialises to the Ray-Singer-Quillen holomorphic torsion in the finite dimensional case. We compute ametric variation formula for the holomorphic L2 torsion, which shows that it is not in general independent of the choice of Hermitian metrics on the complex manifold and on the holomorphic Hilbertian bundle, which are needed to define it. We therefore initiate the theory of correspondences of determinant lines, that enables us to define a relative holomorphic L2 torsion for a pair of flat Hilbertian bundles, which we prove is independent of the choice of Hermitian metrics on the complex manifold and on the flat Hilbertian bundles.

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 Yang–Mills theory and jumping curves

2015 ◽  
Vol 26 (05) ◽  
pp. 1550029
Author(s):  
Yasha Savelyev
Keyword(s):  
Vector Bundle ◽  
Holomorphic Vector Bundle ◽  
Chain Complex ◽  
Complex Manifolds ◽  
Smooth Manifolds ◽  
Hermitian Vector Bundle ◽  
Connected Manifold ◽  
Classical Sense ◽  
Yang Mills ◽  
Simply Connected

We study a smooth analogue of jumping curves of a holomorphic vector bundle, and use Yang–Mills theory over S2 to show that any non-trivial, smooth Hermitian vector bundle E over a smooth simply connected manifold, must have such curves. This is used to give new examples complex manifolds for which a non-trivial holomorphic vector bundle must have jumping curves in the classical sense (when c1(E) is zero). We also use this to give a new proof of a theorem of Gromov on the norm of curvature of unitary connections, and make the theorem slightly sharper. Lastly we define a sequence of new non-trivial integer invariants of smooth manifolds, connected to this theory of smooth jumping curves, and make some computations of these invariants. Our methods include an application of the recently developed Morse–Bott chain complex for the Yang–Mills functional over S2.

scholarly journals K-analytic versus ccm-analytic sets in nonstandard compact complex manifolds

2008 ◽  
Vol 198 (2) ◽  
pp. 139-148
Author(s):  
Rahim Moosa ◽  
Sergei Starchenko
Keyword(s):  
Complex Manifolds ◽  
Analytic Sets ◽  
Compact Complex Manifolds
Sign in / Sign up

Export Citation Format

Share Document