1,721,039 research outputs found
Cohomologies of certain orbifolds
Full text linkWe study the Bott–Chern cohomology of complex orbifolds obtained as a quotient of a compact complex manifold by a finite group of biholomorphisms
The Cohomologies of the Iwasawa Manifold and of Its Small Deformations
Full text linkWe prove that, for some classes of complex nilmanifolds, the Bott–Chern cohomology is completely determined by the Lie algebra associated with the nilmanifold with the induced complex structure. We use these tools to compute the Bott–Chern and Aeppli cohomologies of the Iwasawa manifold and of its small deformations, completing the computations by M. Schweitzer (arXiv:0709.3528v1 [math.AG])
Cohomological aspects of non-Kähler manifolds
No full textIn this thesis, we study cohomological properties of non-Kähler manifolds. In particular, we are concerned in investigating the cohomology of compact (almost-)complex manifolds, and of manifolds endowed with special structures, e.g., symplectic structures, -complex structures in the sense of F. R. Harvey and H. B. Lawson, exhaustion functions satisfying positivity conditions.
In Chapter 0, which contains no original material, we collect the basic notions concerning almost-complex, complex, and symplectic structures, we recall the main results on Hodge theory for Kähler manifolds, and we summarize the classical results on deformations of complex structures, on currents and de Rham homology, and on solvmanifolds.
In Chapter 1, we study cohomological properties of compact complex manifolds, and in particular the Bott-Chern cohomology. By using exact sequences introduced by J. Varouchas, we prove a Frölicher-type inequality for the Bott-Chern cohomology, which also provides a characterization of the validity of the -Lemma in terms of the dimensions of the Bott-Chern cohomology groups. We then prove a Nomizu-type result for the Bott-Chern cohomology, showing that, for certain classes of complex structures on nilmanifolds, the Bott-Chern cohomology is completely determined by the associated Lie algebra endowed with the induced linear complex structure. As an application, we explicitly study the Bott-Chern and Aeppli cohomologies of the Iwasawa manifold and of its small deformations. Finally, we study the Bott-Chern cohomology of complex orbifolds of the type X/G, where X is a compact complex manifold and G a finite group of biholomorphisms of X.
In Chapter 2, we study cohomological properties of almost-complex manifolds. Firstly, we recall the notion of -pure-and-full almost-complex structure, which has been introduced by T.-J. Li and W. Zhang in order to investigate the relations between the compatible and the tamed symplectic cones on a compact almost-complex manifold and with the aim to throw light on a question by S. K. Donaldson. In particular, we are interested in studying when certain subgroups, related to the almost-complex structure, let a splitting of the de Rham cohomology of an almost-complex manifold, and their relations with cones of metric structures. Then, we focus on -pure-and-fullness on several classes of (almost-)complex manifolds, e.g., solvmanifolds endowed with left-invariant almost-complex structures, semi-Kähler manifolds, almost-Kähler manifolds. Then, we study the behaviour of -pure-and-fullness under small deformations of the complex structure and along curves of almost-complex structures, investigating properties of stability, and of semi-continuity for the dimensions of the invariant and anti-invariant subgroups of the de Rham cohomology with respect to the almost-complex structure. Then, we consider the cone of semi-Kähler structures on a compact almost-complex manifold and, in particular, by adapting the results by D. P. Sullivan on cone structures, we compare the cones of balanced metrics and of strongly-Gauduchon metrics on a compact complex manifold.
In Chapter 3, we study the cohomological properties of (differentiable) manifolds endowed with special structures, other than (almost-)complex structures. More precisely, we investigate the cohomology of symplectic manifolds; then, we study cohomological decompositions on -complex manifolds in the sense of F. R. Harvey and H. B. Lawson; finally, we consider domains in endowed with a smooth proper strictly p-convex exhaustion function, and, using -techniques, we give another proof of a consequence of J.-P. Sha’s theorem, and of H. Wu’s theorem, on the vanishing of the higher degree de Rham
cohomology groups
On Bott-Chern cohomology and formality
No full textWe study a geometric notion related to formality for Bott-Chern cohomology on
complex manifolds
Hodge theory for twisted differentials
No full textWe study cohomologies and Hodge theory for complex manifolds with twisted differentials. In
particular, we get another cohomological obstruction for manifolds in class C of Fujiki. We give a Hodgetheoretical
proof of the characterization of solvmanifolds in class C of Fujiki, first stated by D. Arapura
Inequalities à la Frölicher and cohomological decompositions
No full textWe study Bott-Chern and Aeppli cohomologies of a vector space endowed with
two anti-commuting endomorphisms whose square is zero. In particular, we prove
an inequality `a la Fr"olicher relating the dimensions of the Bott-Chern and
Aeppli cohomologies to the dimensions of the Dolbeault cohomologies. We prove
that the equality in such an inequality `a la Fr"olicher characterizes the
validity of the so-called cohomological property of satisfying the
-Lemma. As an application, we study cohomological
properties of compact either complex, or symplectic, or, more in general,
generalized-complex manifolds
A vanishing result for strictly -convex domains
Full text linkIn view of Andreotti and Grauert (Bull Soc Math France 90:193–259, 1962) vanishing theorem for q -complete domains in Cn , we reprove a vanishing result by Sha (Invent Math 83(3):437–447, 1986), and Wu (Indiana Univ Math J 36(3):525–548, 1987), for the de Rham cohomology of strictly p -convex domains in Rn in the sense of Harvey and Lawson (The foundations of p -convexity and p -plurisubharmonicity in riemannian geometry. arXiv:1111.3895v1 [math.DG]). Our proof uses the L2 -techniques developed by Hörmander (An introduction to complex analysis in several variables, 3rd edn. North-Holland Publishing Co, Amsterdam 1990), and Andreotti and Vesentini (Inst Hautes Études Sci Publ Math 25:81–130, 1965)
On the -Lemma and Bott-Chern cohomology
No full textOn a compact complex manifold X, we prove a Frölicher-type inequality for Bott-Chern cohomology and we show that the equality holds if and only if X satisfies the ∂∂− -Lemma
On non-Kähler degrees of complex manifolds
No full textWe study cohomological properties of complex manifolds. In particular, under suitable metric conditions, we extend to higher dimensions a result by A. Teleman, which provides an upper bound for the Bott– Chern cohomology in terms of Betti numbers for compact complex surfaces according to the dichotomy b1 even or odd
Hermitian ranks of compact complex manifolds
No full textWe investigate degenerate special-Hermitian metrics on compact complex manifolds; in particular, degenerate Kaehler and locally conformally Kaehler metrics on special classes of non-Kaehler manifolds
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