Remarks on Laufer’s formula for the Milnor number, Rochlin’s signature theorem and the analytic Euler characteristic of compact complex manifolds

2017 ◽  
Vol 24 (1) ◽  
pp. 105-123 ◽  
Author(s):  
José Seade
Keyword(s):  
Euler Characteristic ◽  
Milnor Number ◽  
Complex Manifolds ◽  
Compact Complex Manifolds

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 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

Algebraicity criteria for compact complex manifolds

1986 ◽  
Vol 275 (4) ◽  
pp. 653-672 ◽  
Author(s):  
Thomas Peternell
Keyword(s):  
Complex Manifolds ◽  
Compact Complex Manifolds

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).

scholarly journals Hermitian ranks of compact complex manifolds

2018 ◽  
Vol 146 (5) ◽  
pp. 2195-2205
Author(s):  
Daniele Angella ◽  
Adriano Tomassini
Keyword(s):  
Complex Manifolds ◽  
Compact Complex Manifolds
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