Mappings into compact complex manifolds with negative first Chern class

For compact complex manifolds with vanishing first Chern class that are compact torus principal bundles over Kähler manifolds, we prove that all holomorphic geometric structures on them, of affine type, are locally homogeneous. For a compact simply connected complex manifold in Fujiki class [Formula: see text], whose dimension is strictly larger than the algebraic dimension, we prove that it does not admit any holomorphic rigid geometric structure, and also it does not admit any holomorphic Cartan geometry of algebraic type. We prove that compact complex simply connected manifolds in Fujiki class [Formula: see text] and with vanishing first Chern class do not admit any holomorphic Cartan geometry of algebraic type.

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The existence problem of Kähler-Einstein metrics oncompact complex manifolds with positive first Chern class

Scientia Sinica Mathematica ◽

10.1360/ssm-2019-0265 ◽

2020 ◽

Vol 50 (3) ◽

pp. 339

Author(s):

Zhu Xiaohua

Keyword(s):

Chern Class ◽

Einstein Metrics ◽

Complex Manifolds ◽

Existence Problem ◽

First Chern Class

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A Bochner principle and its applications to Fujiki class 𝒞 manifolds with vanishing first Chern class

Communications in Contemporary Mathematics ◽

10.1142/s0219199719500512 ◽

2019 ◽

Vol 22 (06) ◽

pp. 1950051 ◽

Cited By ~ 1

Author(s):

Indranil Biswas ◽

Sorin Dumitrescu ◽

Henri Guenancia

Keyword(s):

Chern Class ◽

Geometric Structure ◽

Riemannian Metric ◽

Vanishing Theorem ◽

Translation Invariant ◽

Zariski Open Subset ◽

First Chern Class ◽

Locally Homogeneous ◽

Affine Type ◽

Compact Complex Manifolds

We prove a Bochner-type vanishing theorem for compact complex manifolds [Formula: see text] in Fujiki class [Formula: see text], with vanishing first Chern class, that admit a cohomology class [Formula: see text] which is numerically effective (nef) and has positive self-intersection (meaning [Formula: see text], where [Formula: see text]). Using it, we prove that all holomorphic geometric structures of affine type on such a manifold [Formula: see text] are locally homogeneous on a non-empty Zariski open subset. Consequently, if the geometric structure is rigid in the sense of Gromov, then the fundamental group of [Formula: see text] must be infinite. In the particular case where the geometric structure is a holomorphic Riemannian metric, we show that the manifold [Formula: see text] admits a finite unramified cover by a complex torus with the property that the pulled back holomorphic Riemannian metric on the torus is translation invariant.

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The Kodaira Problem for Kähler Spaces with Vanishing First Chern Class

Forum of Mathematics Sigma ◽

10.1017/fms.2021.18 ◽

2021 ◽

Vol 9 ◽

Author(s):

Patrick Graf ◽

Martin Schwald

Keyword(s):

Chern Class ◽

Cohomology Group ◽

Canonical Bundle ◽

Deformation Space ◽

Projective Varieties ◽

Quotient Singularities ◽

Finite Quotient ◽

First Chern Class ◽

Small Deformations

Abstract Let X be a normal compact Kähler space with klt singularities and torsion canonical bundle. We show that X admits arbitrarily small deformations that are projective varieties if its locally trivial deformation space is smooth. We then prove that this unobstructedness assumption holds in at least three cases: if X has toroidal singularities, if X has finite quotient singularities and if the cohomology group ${\mathrm {H}^{2} \!\left ( X, {\mathscr {T}_{X}} \right )}$ vanishes.

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Schottky groups acting on homogeneous rational manifolds

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2016-0065 ◽

2019 ◽

Vol 2019 (753) ◽

pp. 23-56 ◽

Cited By ~ 1

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.

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Higher-page Bott–Chern and Aeppli cohomologies and applications

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2021-0014 ◽

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.

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