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## Basic definitions and properties

We start by introducing the equivalent definitions of a locally conformally Kähler structure. Let (M, J) be a complex n-dimensional manifold and let Ω ∈ E1,1 M (M) be a non-degenerate (1, 1)-form on M which is positive, meaning that g(·, ·) := Ω(·, J·) is a J-invariant Riemannian metric.

Definition 2.2.1: The form Ω is called locally conformally Kähler, abbreviated LCK, if there exists a covering of M with open sets {Uα}α∈I such that Ω restricted to each of them is conformal to a Kähler form: Ω|U = eϕΩα with dΩα = 0 (2.2.1) where ϕα ∈ C∞(Uα,R) and Ωα ∈ E1,1 M (Uα), for any α ∈ I.

Let Ω be an LCK form. By differentiating the relation (2.2.1), we obtain: dΩ = dϕα ∧ Ω on Uα ∀α ∈ I. Suppose for the moment that n > 1. In this case, the morphism E1M (M) → E3M (M), η 7→ η ∧Ω is injective as Ω is non-degenerate, hence the above implies that on the intersections Uα ∩ Uβ we have dϕα = dϕβ. Thus the collection {dϕα}α∈I glues up to give a real closed 1-form θ globally defined on M which verifies dΩ = θ ∧ Ω.

For n = 1, the above argument does not work, and in fact any two (1, 1)-forms, locally or globally defined on M, are conformal. Nonetheless, for any real 1-form θ on M (and there always exist some), the relation dΩ = θ ∧Ω is trivially satisfied, since both terms are 0 because of their degree. Conversely, suppose that θ is a closed 1-form on M satisfying dΩ = θ ∧ Ω. By the Poincaré lemma, M is covered by open sets {Uα}α∈I on which θ becomes exact: θ = dϕα, ϕα ∈ C∞(Uα,R). Setting then Ωα := e−ϕΩ on Uα, one verifies easily that this is a closed form on Uα conformal to Ω, and so Ω is LCK. Thus we also have: Definition 2.2.2: The form Ω is called LCK if there exists a real closed 1-form θ on M, called the Lee form, satisfying: dΩ = θ ∧ Ω.

### Special LCK metrics

Not much is known about general LCK manifolds, or about constraints on the existence of such metrics. Moreover, this class is not necessarily well behaved under natural operations, such us deforming the complex structure. However, there are some special classes of LCK metrics which are quite well understood, which we present in this section. First of all, in any conformal class, there exists a special representative which is sometimes very useful to work with. Let us fix a Hermitian manifold (M, J, g, Ω) of complex dimension n > 1, where g = Ω(·, J·). The metric g induces a L2 inner product on E•(M). We denote by d∗ the adjoint of d with respect to this inner product, and by Δ := dd∗+d∗d the corresponding Laplacian. We recall that on a complex manifold we have the formula: d∗ = − ⋆ d⋆, where ⋆ is the Hodge star operator with respect to g. On the other hand, Ω induces a Lefschetz map Lef := Ω ∧ · acting on E•(M), so that Lef : E1(M) → E3(M) is injective, as n > 1, and we have an isomorphism Ln−1 : E1M (M) → E2n−1 M (M). Define θ := 1 n − 1 (Lefn−1 )−1d(Ωn−1).

#### Infinitesimal automorphisms of LCK manifolds

In this section we will take a closer look at the Lie algebra of infinitesimal automorphisms of LCK manifolds, and will distinguish a special subalgebra that will play a particular role. From now on, even if not specified, we only work with LCK structures on complex manifolds of complex dimension greater that 1.

For an LCK manifold (M, J, [Ω]), the automorphism group Aut(M, J, [Ω]) is formed by all the conformal biholomorphisms Φ : M → M, Φ∗Ω ∈ [Ω]. Denote by aut(M, J, [Ω]) the corresponding Lie algebra of infinitesimal automorphisms. We have the first well-known property of this Lie group: Proposition 2.5.1: Let (M, J, [Ω]) be a compact LCK manifold, and let Ω0 ∈ [Ω] be a Gauduchon metric. Then Aut(M, J, [Ω]) = Aut(M, J,Ω0). In particular, the automorphism group of a compact LCK manifold is compact.

Proof. Let Φ ∈ Aut(M, J, [Ω]). As Ω0 is Gauduchon, we have ddcΩn−1 0 = 0, an as Φ is a biholomorphism, Φ∗ commutes with d and with dc. Hence we also have ddcΦ∗Ωn−1 0 = 0, i.e. Φ∗Ω0 is also a Gauduchon metric. But we have Φ∗Ω0 ∈ [Ω0], so by the uniqueness up to scalars of such metrics in a given conformal class, we must have Φ∗Ω0 = λΩ0 with λ ∈ R>0. Finally, as M is compact and Φ is bijective, we have R M Ωn = R M Φ∗Ωn = λn R M Ωn > 0, implying that λ = 1. Thus, the group Aut(M, J, [Ω0]) is a closed subgroup of the compact Lie group of isometries Aut(M,Ω0(·, J·)), hence also compact Lie group. Similarly, given an LCS manifold (M, [Ω]), the group of automorphisms Aut(M, [Ω]) is formed by all the conformal diffeomorphisms Φ : M → M, Φ∗Ω ∈ [Ω]. Next, we want to investigate the algebraic structure of the corresponding Lie algebra aut(M, [Ω]). First of all, note that X ∈ aut(M, [Ω]) means LXΩ = fXΩ. This implies (fX −θ(X))Ω = dθ(ιXΩ). Hence dθ((fX − θ(X))Ω) = 0, or also (dfX −d(θ(X))) ∧Ω = 0 and since we are working under the supposition that dimCM > 1, it follows that θ(X) − fX = cX ∈ R. By straightforward computations it can be seen that the constants cX are conformally invariant. Hence we have a linear map: l : aut(M, [Ω]) → R X 7→ cX = θ(X) − fX.

**Table of contents :**

Introduction

Notation and conventions

**1 Twisted Holomorphic Symplectic Forms **

1.1 Introduction

1.2 Holomorphic symplectic manifolds

1.3 Twisted holomorphic symplectic manifolds

1.4 A characterization

1.5 Examples

**2 Locally Conformally Kähler Geometry **

2.1 Introduction

2.2 Basic definitions and properties

2.3 Connections

2.4 Special LCK metrics

2.5 Infinitesimal automorphisms of LCK manifolds

2.6 Examples

2.6.1 Diagonal Hopf manifolds

2.6.2 Non-diagonal Hopf surfaces

2.6.3 LCK manifolds obtained from ample vector bundles

2.6.4 LCK metrics on blow-ups

2.6.5 Complex surfaces

**3 Existence Criteria for LCK Metrics **

3.1 Introduction

3.2 The Lee vector field

3.3 Existence of LCK metrics with potential

3.4 Existence of Vaisman metrics

3.5 Torus principal bundles

3.6 Analytic irreducibility of complex manifolds of LCK type

3.7 Weyl reducible manifolds

**4 Toric LCK Manifolds **

4.1 Introduction

4.2 Twisted Hamiltonian Vector Fields

4.3 Torus actions on LCS manifolds

4.4 Proof of the Main Theorem

4.5 Examples

4.6 Final remarks and questions

**5 Cohomological properties of OT manifolds **

5.1 Introduction

5.2 Oeljeklaus-Toma manifolds

5.2.1 The construction

5.2.2 Metric properties

5.3 Technical Preliminaries

5.3.1 Leray-Serre spectral sequence of a locally trivial fibration

5.3.2 Twisted cohomology

5.4 The de Rham cohomology

5.5 The Leray-Serre spectral sequence of OT manifolds

5.6 Twisted cohomology of OT manifolds

5.7 Applications and Examples

**Bibliography**