## Thesis etd-01072013-231626 |

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Thesis type

Tesi di dottorato di ricerca

Author

ANGELLA, DANIELE

URN

etd-01072013-231626

Thesis title

Cohomological aspects of non-Kähler manifolds

Academic discipline

MAT/03

Course of study

SCIENZE DI BASE

Supervisors

**tutor**Prof. Tomassini, Adriano

Keywords

- $p$-convexity
- $\mathbf{D}$-complex structures
- $\partial\overline{\partial}$-Lemma
- almost-complex structures
- Bott-Chern cohomology
- cohomological decomposition
- complex manifolds
- symplectic manifolds

Graduation session start date

21/01/2013

Availability

Full

Summary

In 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, $\mathbf{D}$-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 $\partial\overline{\partial}$-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 $\mathcal{C}^\infty$-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 $\mathcal{C}^\infty$-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 $\mathcal{C}^\infty$-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 $\mathbf{D}$-complex manifolds in the sense of F. R. Harvey and H. B. Lawson; finally, we consider domains in $\mathbb{R}^n$ endowed with a smooth proper strictly p-convex exhaustion function, and, using $\mathrm{L}^2$ -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.

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 $\partial\overline{\partial}$-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 $\mathcal{C}^\infty$-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 $\mathcal{C}^\infty$-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 $\mathcal{C}^\infty$-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 $\mathbf{D}$-complex manifolds in the sense of F. R. Harvey and H. B. Lawson; finally, we consider domains in $\mathbb{R}^n$ endowed with a smooth proper strictly p-convex exhaustion function, and, using $\mathrm{L}^2$ -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.

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