Searcharxiv⌕ Search

arXiv subjects

Daniele Angella

Publications and source records attributed to Daniele Angella.

At least 55 records · Page 3Linked to original sources
arXiv2015

On the Bott-Chern and Aeppli cohomology

This survey summarizes the results discussed in a talk at "Bielefeld Geometry & Topology Days" held at Bielefeld University in July 2015. We are interested in quantitative and qualitative properties of Bott-Chern cohomology. We announce new results obtained in [D. Angella, N. Tardini, Quantitative and qualitative cohomological properties for non-Kähler manifolds, arXiv:1507.07108] jointly with Nicoletta Tardini. In particular, we prove an upper bound of the dimensions of Bott-Chern cohomology in terms of Hodge numbers. We also introduce a notion, called Schweitzer qualitative property, which encodes the existence of a non-degenerate pairing in Bott-Chern cohomology, like the Poincaré duality for the de Rham cohomology. We show that this property characterizes the $\partial\overline\partial$-Lemma.

math.CV↗
arXiv2015

Locally conformal Hermitian metrics on complex non-Kähler manifolds

We study complex non-Kähler manifolds with Hermitian metrics being locally conformal to metrics with special cohomological properties. In particular, we provide examples where the existence of locally conformal holomorphic-tamed structures implies the existence of locally conformal Kähler metrics, too.

math.DG↗
arXiv2015

Cohomologies of generalized complex manifolds and nilmanifolds

We study generalized complex cohomologies of generalized complex structures constructed from certain symplectic fibre bundles over complex manifolds. We apply our results in the case of left-invariant generalized complex structures on nilmanifolds and to their space of small deformations.

math.DG↗
arXiv2015

On Chern-Yamabe problem

We initiate the study of an analogue of the Yamabe problem for complex manifolds. More precisely, fixed a conformal Hermitian structure on a compact complex manifold, we are concerned in the existence of metrics with constant Chern scalar curvature. In this note, we set the problem and we provide a positive answer when the expected constant Chern scalar curvature is non-positive. In particular, this includes the case when the Kodaira dimension of the manifold is non-negative. Finally, we give some remarks on the positive curvature case, showing existence in some special cases and the failure, in general, of uniqueness of the solution.

math.DG↗
arXiv2015

On Bott-Chern cohomology of compact complex surfaces

We study Bott-Chern cohomology on compact complex non-Kähler surfaces. In particular, we compute such a cohomology for compact complex surfaces in class $\text{VII}$ and for compact complex surfaces diffeomorphic to solvmanifolds.

math.DG↗
arXiv2015

Degree of non-Kählerianity for 6-dimensional nilmanifolds

We use Bott-Chern cohomology to measure the non-Kählerianity of 6-dimensional nilmanifolds endowed with the invariant complex structures in M. Ceballos, A. Otal, L. Ugarte, and R. Villacampa's classification, [Invariant Complex Structures on 6-Nilmanifolds: Classification, Frölicher Spectral Sequence and Special Hermitian Metrics, J. Geom. Anal. (2014)]. We investigate the existence of pluriclosed metric in connection with such a classification.

math.DG↗
arXiv2014

Hodge theory for twisted differentials

We study cohomologies and Hodge theory for complex manifolds with twisted differentials. In particular, we get another cohomological obstruction for manifolds in class $\mathcal{C}$ of Fujiki. We give a Hodge-theoretical proof of the characterization of solvmanifolds in class $\mathcal{C}$ of Fujiki, first proven by D. Arapura.

math.DG↗
arXiv2014

On cohomological decomposition of generalized-complex structures

We study properties concerning decomposition in cohomology by means of generalized-complex structures. This notion includes the $\mathcal{C}^\infty$-pure-and-fullness introduced by Li and Zhang in the complex case and the Hard Lefschetz Condition in the symplectic case. Explicit examples on the moduli space of the Iwasawa manifold are investigated.

math.DG↗
arXiv2014

Inequalities à la Frölicher and cohomological decompositions

We 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 à la Frölicher 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 à la Frölicher characterizes the validity of the so-called cohomological property of satisfying the $\partial\overline{\partial}$-Lemma. As an application, we study cohomological properties of compact either complex, or symplectic, or, more in general, generalized-complex manifolds.

math.DG↗
arXiv2013

Cohomological aspects of non-Kähler manifolds

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, D-complex structures in the sense of F. R. Harvey and H. B. Lawson, exhaustion functions satisfying positivity conditions.

math.DG↗
arXiv2012

The cohomologies of the Iwasawa manifold and of its small deformations

We prove that, for some classes of complex nilmanifolds, the Bott-Chern cohomology is completely determined by the Lie algebra associated to 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 in arXiv:0709.3528v1 [math.AG] by M. Schweitzer.

math.DG↗
arXiv2012

A vanishing result for strictly p-convex domains

In view of A. Andreotti and H. Grauert's vanishing theorem for q-complete domains in C^n, (Théorème de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193--259,) we re-prove a vanishing result by J.-P. Sha, (p-convex Riemannian manifolds, Invent. Math. 83 (1986), no. 3, 437--447,) and H. Wu, (Manifolds of partially positive curvature, Indiana Univ. Math. J. 36 (1987), no. 3, 525--548,) for the de Rham cohomology of strictly p-convex domains in R^n in the sense of F. R. Harvey and H. B. Lawson, (The foundations of p-convexity and p-plurisubharmonicity in riemannian geometry, arXiv:1111.3895v1 [math.DG]). Our proof uses the L^2-techniques developed by L. Hörmander, (An introduction to complex analysis in several variables, Third edition, North-Holland Mathematical Library, 7, North-Holland Publishing Co., Amsterdam, 1990,) and A. Andreotti and E. Vesentini, (Carleman estimates for the Laplace-Beltrami equation on complex manifolds, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 81--130).

math.DG↗
arXiv2012

On the cohomology of almost-complex manifolds

Following T.-J. Li, W. Zhang [Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom.], we continue to study the link between the cohomology of an almost-complex manifold and its almost-complex structure. In particular, we apply the same argument in [T.-J. Li, W. Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom.] and the results obtained by D. Sullivan in [Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math.] to study the cone of semi-Kähler structures on a compact semi-Kähler manifold.

math.DG↗
arXiv2012

On Cohomological Decomposition of Almost-Complex Manifolds and Deformations

While small deformations of Kähler manifolds are Kähler too, we prove that the cohomological property to be $\mathcal{C}^\infty$-pure-and-full is not a stable condition under small deformations. This property, that has been recently introduced and studied by T.-J. Li and W. Zhang in [Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom.] and T. Drǎghici, T.-J. Li, and W. Zhang in [Symplectic forms and cohomology decomposition of almost complex four-manifolds, Int. Math. Res. Not.], [On the J-anti-invariant cohomology of almost complex 4-manifolds, to appear in Q. J. Math.], is weaker than the Kähler one and characterizes the almost-complex structures that induce a decomposition in cohomology. We also study the stability of this property along curves of almost-complex structures constructed starting from the harmonic representatives in special cohomology classes.

math.DG↗