Vaisman theorem for lcK spaces

Blow‐ups and modifications of lcK spaces

“…An important result in the geometry of Kähler spaces by Fujiki (see [15, 3.1], [4, Lemma 2]) states that the blow‐up of a compact Kähler space along a complex subspace is also Kähler. A rewritten proof can be found in our previous paper [12, Theorem 3.1]. Moreover, a careful examination of that proof shows that we can obtain a little more by observing that the line bundle $$\mathcal {O}(1)$$ is trivial outside any neighborhood of the exceptional divisor and choosing conveniently the sections involved in the construction of the new metric, such that this new metric coincides with the pullback of the old one away from the exceptional divisor.…”

“…Vaisman's theorem [14], a fundamental result of lcK geometry, states that on a compact complex manifold, pure lcK and Kähler metrics (with respect to the same complex structure) cannot coexist. A generalization of Vaisman's theorem to locally irreducible complex spaces is [12, Theorem 4.4], stated below. Theorem Let $$(X,\omega ,\theta )$$ be a compact, locally irreducible, lcK space.…”

“…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.…”

“…The latter condition is satisfied if $$Z$$ is a locally complete intersection. We also use Vaisman's theorem for lcK spaces [12], for which we need local irreducibility of $$Z$$, guaranteed by the normality assumption. The reverse implication is true for any compact subspace $$Z$$, and this is done by refining [12, Theorem 3.1] to obtain the slightly more general Theorem 2.10.…”

LCK Spaces. A Short Survey

“…Kähler forms on singular complex spaces were first introduced by Grauert [2], using families of locally defined strictly plurisubharmonic functions and compatibility conditions. In the spirit of Grauert's idea, we can define lcK forms on singular spaces, as in [8]: Definition 1.1. Let X be a complex space.…”