81 research outputs found
Symplectic manifolds and cohomological decomposition
Given a closed symplectic manifold, we study when the Lefschetz decomposition
induced by the -representation yields a
decomposition of the de Rham cohomology. In particular, this holds always true
for the second de Rham cohomology group, or if the symplectic manifold
satisfies the Hard Lefschetz Condition
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 of Fujiki. We give a Hodge-theoretical proof
of the characterization of solvmanifolds in class of Fujiki,
first proven by D. Arapura
Bott-Chern cohomology of solvmanifolds
We study conditions under which sub-complexes of a double complex of vector
spaces allow to compute the Bott-Chern cohomology. We are especially aimed at
studying the Bott-Chern cohomology of special classes of solvmanifolds, namely,
complex parallelizable solvmanifolds and solvmanifolds of splitting type. More
precisely, we can construct explicit finite-dimensional double complexes that
allow to compute the Bott-Chern cohomology of compact quotients of complex Lie
groups, respectively, of some Lie groups of the type
where is nilpotent. As an application, we
compute the Bott-Chern cohomology of the complex parallelizable Nakamura
manifold and of the completely-solvable Nakamura manifold. In particular, the
latter shows that the property of satisfying the
-Lemma is not strongly-closed under deformations of
the complex structure
Cohomologies of deformations of solvmanifolds and closedness of some properties
We provide further techniques to study the Dolbeault and Bott-Chern
cohomologies of deformations of solvmanifolds by means of finite-dimensional
complexes. By these techniques, we can compute the Dolbeault and Bott-Chern
cohomologies of some complex solvmanifolds, and we also get explicit examples,
showing in particular that either the -Lemma or
the property that the Hodge and Fr\"olicher spectral sequence degenerates at
the first level are not closed under deformations
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