arXiv subjects
Publications and source records attributed to Miron Stanciu.
We prove that no product of positive-dimensional compact complex manifolds admits a strict locally conformally Kähler (lcK) metric. This solves a long-standing conjecture in lcK geometry, previously established only under certain conditions.
Endo-Pajitnov manifolds are compact non-Kähler manifolds which generalize the Inoue surfaces $S_M$ to higher dimensions. We compute their Dolbeault cohomology and show that they satisfy the Hodge decomposition at the level of dimensions.
We give an account of old and new results concerning many types of non-Kähler metrics, with focus on the problem of their coexistence on compact complex manifolds, and their behaviour at deformations and blow-up. We also describe a mechanism that several authors have used to construct examples of nilmanifolds admitting metrics with certain properties.
As proven in a celebrated theorem due to Vaisman, pure locally conformally Kähler metrics do not exist on compact Kähler manifolds. In a previous paper, we extended this result to the singular setting, more precisely to Kähler spaces which are locally irreducible. Without the additional assumption of local irreducibility, there are counterexamples for which Vaisman's theorem does not hold. In this article, we give a much broader sufficient condition under which Vaisman's theorem still holds for compact Kähler spaces which are locally reducible.
We investigate the metric and cohomological properties of higher dimensional analogues of Inoue surfaces, that were introduced by Endo and Pajitnov. We provide a solvmanifold structure and show that in the diagonalizable case, they are formal and have invariant de Rham cohomology. Moreover, we obtain an arithmetic and cohomological characterization of pluriclosed and astheno-Kähler metrics and show they give new examples in all complex dimensions.
In this paper, we prove a stability result for the non-Kähler geometry of locally conformally Kähler (lcK) spaces with singularities. Specifically, we find sufficient conditions under which the image of an lcK space by a holomorphic mapping also admits lcK metrics, thus extending a result by Varouchas about Kähler spaces.
In this article, we prove that the blow-up of a locally irreducible lcK space $X$ along a subspace $Z$ which verifies certain conditions is lcK if and only if $X$ is induced gcK, generalizing a theorem of Ornea-Verbitsky-Vuletescu to singular locally irreducible spaces. We also show that even if modifications of lcK spaces are not always of lcK type, they always admit quasi-lcK metrics.
Vaisman's theorem for locally conformally Kähler (lcK) compact manifolds states that any lcK metric on a compact complex manifold which admits a Kähler metric is, in fact, globally conformally Kähler (gcK). In this paper, we extend this theorem to compact complex spaces with singularities.
In a previous article, we introduced a reduction procedure for locally conformally symplectic manifolds at any regular value of the momentum mapping. We use this construction to prove an analogue of a well-known theorem in the symplectic setting about the reduction of cotangent bundles.
We study the interplay between the following types of special non-Kähler Hermitian metrics on compact complex manifolds: it locally conformally Kähler, $k$-Gauduchon, balanced and locally conformally balanced and prove that a locally conformally Kähler compact nilmanifold carrying a balanced or a $k$-Gauduchon metric is necessarily a torus. Combined with a result of Fino and Vezzoni from 2016, this leads to the fact that a compact complex 2-step nilmanifold endowed with whichever two of the following types of metrics: balanced, pluriclosed and locally conformally Kähler is a torus. Moreover, we construct a family of compact nilmanifolds in any dimension carrying both balanced and locally conformally balanced metrics and finally we show a compact complex nilmanifold does not support a left-invariant locally conformally hyperKähler structure.
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 a previous paper. We then show that a complex space which projects over an LCK space with discrete fibers must also carry an LCK structure.
We present a reduction procedure for locally conformally symplectic (LCS) manifolds with an action of a Lie group preserving the conformal structure, with respect to any regular value of the momentum mapping. Under certain conditions, this reduction is compatible with the existence of a locally conformally Kähler structure. As a special consequence, we obtain a compatible contact reduction with respect to any regular value of the contact momentum mapping.
A locally conformally symplectic (LCS) form is an almost symplectic form $ω$ such that a closed one-form $θ$ exists with $dω= θ\wedge ω$. We present a version of the well-known result of Darboux and Weinstein in the LCS setting and give an application concerning Lagrangian submanifolds.