**TOPICS IN LOCALLY CONFORMALLY KÄHLER GEOMETRY
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**Program PNCDI III, Project number PN-III-P4-ID-PCE-2016-0065, Contract number 8/12.07.2017**

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Financed by Romanian Ministry of Research and Innovation, CNCS - UEFISCDI
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Project Executive Summary**

Locally conformally Kähler (LCK) geometry is concerned with complex
manifolds of complex dimension at least two admitting a Kähler
covering with deck transformations acting by holomorphic
homotheties with respect to the Kähler metric.
Forgetting the complex structure we obtain Locally Conformally
Symplectic (LCS) manifolds. The passage between LCS and LCK
settings is both directions.
Almost all known non-Kähler compact surfaces are LCK. In higher
dimensions, we have Hopf manifolds, Oeljeklaus-Toma (OT) manifolds,
and their complex submanifolds (when they do exist).

Objectives:

1. Classification of surjective holomorphic maps from (compact) LCK
manifolds. The total space, together with natural conditions on the
fibre, should put restrictions on the Hermitian structure of the base.

2. Proving the non-existence of LCK metrics on OT manifolds with t>1.

3. OT manifolds do not have curves and surfaces, and the LCK ones do
not have subvarieties at all. We want to prove that if a compact
complex submanifold exists in an OT manifold, it must be an OT
manifold itself. The problem can be translated into one concerning
holomorphic bundles.

4. Studying holomorphic vector bundles (stability, filtrability) of small
rank on OT.

5. Initiating the systematic study of complex LCS manifolds, relations
with other geometries.

6. Classification of toric LCS manifolds. We expect a statement of this
kind: toric, compact LCS manifolds should be LCK and indeed Vaisman.

7. Study of the spectral sequence associated to the canonical foliation
of a Vaisman manifold and computing the dimensions of its terms. This
will obstruct a given 2-dimensional foliation on a compact complex
manifold to come from a Vaisman structure.

8. Study of indefinite LCK manifolds and of their submanifolds, in
particular of semi-Riemannian submersions from indefinite LCK
manifolds, with focus on statistical manifolds.

9. Studying particular functionals of Yang-Mills type on LCK and
Vaisman manifolds.

**Budget
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Last update: 15-Aug-2017
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