Coverings of locally conformally Kähler complex spaces

Abstract

In this paper, we study the properties of coverings of locally conformally Kähler (LCK) spaces with singularities. We begin by proving that a space is LCK if any only if its universal cover is Kähler, thereby generalizing a result from Ioniţȃ and Preda (Manuscripta Math,

Blow‐ups and modifications of lcK spaces

“…is a disjoint union of copies of 𝑉 𝑗 , such that for any 𝜂 ∈ Γ, 𝜂(𝑉 𝛾 𝑗 ) = 𝑉 𝜂𝛾 𝑗 . The proof of [11,Theorem 3.10] shows that there exists a smooth function g ∶ 𝑌 0 ⟶ ℝ such that 𝜔 0 ∶= 𝑒 −g 𝜋 * 𝑌 𝜔 is a Kähler metric on 𝑌 0 , and such that Γ acts on 𝜔 0 by positive homotheties, that is, for every 𝛾 ∈ Γ, 𝛾 * 𝜔 0 = 𝑐 𝛾 𝜔 0 , where 𝑐 𝛾 > 0.…”

“…Consider an lcK metric on $$Y$$ such that $$I$$ is finite, the open sets $$V_j$$ are all connected and Stein, and for every $$j\in I$$, $$$$\begin{equation*} \pi _Y^{-1}(V_j)=\bigcup _{\gamma \in \Gamma } V_j^\gamma \end{equation*}$$$$is a disjoint union of copies of $$V_j$$, such that for any $$\eta \in \Gamma$$, $$\eta (V_j^\gamma )=V_j^{\eta \gamma }$$. The proof of [11, Theorem 3.10] shows that there exists a smooth function $$g:Y_0\longrightarrow \mathbb {R}$$ such that $$\omega _0:=e^{-g}\pi _Y^*\omega$$ is a Kähler metric on $$Y_0$$, and such that $$\Gamma$$ acts on $$\omega _0$$ by positive homotheties, that is, for every $$\gamma \in \Gam...$…”

“…In our first two papers [11] and [12], we showed that both the characterization theorem of lcK manifolds via the universal cover, and Vaisman's theorem on the pure lcK–Kähler dichotomy for compact complex manifolds remain true for complex spaces, the latter only for locally irreducible complex spaces (for the locally reducible case, we presented a counterexample). These two results are essential for the further study of lcK spaces.…”

“…We show that if all the choices we make in that construction are well related, then $$\operatorname{Deck}(\widetilde{X}/X)$$ acts by homotheties on $$\widetilde{\omega }$$. Finally, [11, Theorem 3.10] can be easily adapted to quasi‐lcK spaces, so it can be applied to conclude that $$X$$ admits a quasi‐Kähler metric.…”

Vaisman theorem for lcK spaces

“…In Section 2, we give all the definitions needed when working with lcK spaces. We also present, using Čech cohomology, a simplified proof of [PS,Corollary 2.10], the main ingredient of that paper, which we previously proved using elementary properties of integration along curves. We then collect some known results about blow-ups of schemes, which can be found in [Ha] and [EH].…”