12 research outputs found
De Rham and twisted cohomology of Oeljeklaus-Toma manifolds
No full textOeljeklaus–Toma (OT) manifolds are complex non-Kähler manifolds whose construction arises from specific number fields. In this note, we compute their de Rham cohomology in terms of invariants associated to the background number field. This is done by two distinct approaches, one by averaging over a certain compact group, and the other one using the Leray–Serre spectral sequence. In addition, we compute also their twisted cohomology. As an application, we show that the low degree Chern classes of any complex vector bundle on an OT mani- fold vanish in the real cohomology. Other applications concern the OT manifolds admitting locally conformally Kähler (LCK) metrics: we show that there is only one possible Lee class of an LCK metric, and we determine all the possible twisted classes of an LCK metric, which implies the nondegeneracy of certain Lefschetz maps in cohomology
Darboux–Weinstein theorem for locally conformally symplectic manifolds
No full textA locally conformally symplectic (LCS) form is an almost symplectic form ω such that a closed one-form θ exists with dω=θ∧ω. We present a version of the well-known result of Darboux and Weinstein in the LCS setting and give an application concerning Lagrangian submanifolds
Morse-Novikov cohomology of locally conformally Kähler surfaces
No full textWe review the properties of the Morse-Novikov cohomology and compute it for all known compact complex surfaces with locally conformally Kähler metrics. We present explicit computations for the Inoue surfaces S0, S+, S- and classify the locally conformally Kähler (and the tamed locally conformally symplectic) forms on S0. We prove the nonexistence of LCK metrics with potential and more generally, of dθ-exact LCK metrics on Inoue surfaces and Oeljeklaus-Toma manifolds
Variational problems in conformal geometry
No full textWe study the Euler-Lagrange equation for several natural functionals defined
on a conformal class of almost Hermitian metrics, whose expression involves the
Lee form of the metric. We show that the Gauduchon metrics are the
unique extremal metrics of the functional corresponding to the norm of the
codifferential of the Lee form. We prove that on compact complex surfaces, in
every conformal class there exists a unique metric, up to multiplication by a
constant, which is extremal for the functional given by the -norm of
, where denotes the complex structure. These extremal metrics are
not the Gauduchon metrics in general, hence we extend their definition to any
dimension and show that they give unique representatives, up to constant
multiples, of any conformal class of almost Hermitian metrics
Cohomologies of locally conformally symplectic manifolds and solvmanifolds
No full textWe study the Morse–Novikov cohomology and its almost-symplectic counterpart on manifolds admitting locally conformally symplectic structures. More precisely, we introduce lcs cohomologies and we study elliptic Hodge theory, dualities, Hard Lefschetz condition. We consider solvmanifolds and Oeljeklaus–Toma manifolds. In particular, we prove that Oeljeklaus–Toma manifolds with precisely one complex place, and under an additional arithmetic condition, satisfy the Mostow property. This holds in particular for the Inoue surface of type S0
Hodge decomposition for Cousin groups and Oeljeklaus-Toma manifolds
No full textInternational audienceWe compute the Dolbeault cohomology of certain domains contained in Cousin groups which satisfy a strong dispersiveness condition. As a consequence we obtain a description of the Dolbeault cohomology of Oeljeklaus-Toma manifolds and in particular the fact that the Hodge decomposition holds for their cohomology
On a class of Kato manifolds
No full textIn this talk we describe Kato manifolds, also known as manifolds with global spherical shell. We revisit Brunellaâ s proof of the fact that Katosurfaces admit locally conformally K\"ahler metrics, and we show that it holdsfor a large class of higher dimensional complex manifolds containing a globalspherical shell. On the other hand, we construct manifolds containing a globalspherical shell which admit no locally conformally K\"ahler metric. We then con-sider a specific class, which can be seen as a higher dimensional analogue of Inoue-Hirzebruch surfaces, and study several of their analytical properties. Inparticular, we give new examples, in any complex dimension , of com-pact non-exact locally conformally K\"ahler manifolds with algebraic dimension , algebraic reduction bimeromorphic to \C\Proj^{n-2} and admitting non-trivialholomorphic vector fields. These results are joint work with Nicolina Istrati(University of Tel Aviv) and Massimiliano Pontecorvo (Roma Tre University).Non UBCUnreviewedAuthor affiliation: Università Roma TrePostdoctora
Bott-Chern cohomology of compact Vaisman manifolds
No full textWe give an explicit description of the Bott-Chern cohomology groups of a
compact Vaisman manifold in terms of the basic cohomology. We infer that the
Bott-Chern numbers and the Dolbeault numbers of a Vaisman manifold determine
each other. On the other hand, we show that the cohomological invariants
introduced by Angella-Tomassini are unbounded for Vaisman manifolds.
Finally, we give a cohomological characterization of the Dolbeault and
Bott-Chern formality for Vaisman metrics.Comment: final version, to appear in Trans. Amer. Math. So
Toric Kato manifolds
No full textWe introduce and study a special class of Kato manifolds, which we call toric
Kato manifolds. Their construction stems from toric geometry, as their
universal covers are open subsets of toric algebraic varieties of non-finite
type. This generalizes previous constructions of Tsuchihashi and Oda, and in
complex dimension 2, retrieves the properly blown-up Inoue surfaces. We study
the topological and analytical properties of toric Kato manifolds and link
certain invariants to natural combinatorial data coming from the toric
construction. Moreover, we produce families of flat degenerations of any toric
Kato manifold, which serve as an essential tool in computing their Hodge
numbers. In the last part, we study the Hermitian geometry of Kato manifolds.
We give a characterization result for the existence of locally conformally
K\"ahler metrics on any Kato manifold. Finally, we prove that no Kato manifold
carries balanced metrics and that a large class of toric Kato manifolds of
complex dimension do not support pluriclosed metrics.Comment: final versio
