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Let $M$ be a complex manifold, and $Z \subset M$ a closed real analytic subvariety. Suppose that the set of smooth points in $Z$ is complex analytic in $M$. Will it follow that $Z$ is complex analytic …

This function has constant Jacobian by Liouville. Then it is map of Jacobian 1 composed with a homothety or its differential is degenerate everywhere. The constant Jacobian biholomorphisms are subject …

closed (1,1)-forms and currents on X are not necessary locally $dd^c$-exact in general What makes it different when X is singular? The obstruction to local $dd^c$-lemma is $R^1\pi_*(O_{X'})$, where …

As Jason said already, there are many examples of Moishezon manifolds which are rationally connected. Indeed, any manifold bimeromorphic to a rational connected manifold is again rationally connected, …

Klemyatin proved that this action is trivial if the corresponding ${\Bbb C}$-flow is compatible with some metric (hence can be extended to a compact torus action), https://arxiv.org/abs/1909.04075, (N …

Let $M$ be a Stein manifold with smooth, strictly pseudoconvex boundary, and $x$ a point on its boundary. Is there a holomorphic function $f$ on $M$, smooth on the boundary, with strict maximum of $|f …

Let $M$ be a real analytic variety (if someone is concerned about distinction between "real analytic spaces" and "real analytic varieties" in real analytic geometry, let's assume that $M$ is both "va …

It is not hard to see that the first cohomology is infinite-dimensional. Take a complement $M:=CP^3 \ CP^1$ and consider a projection $\pi:\; M \mapsto CP^1$. It is not hard to see that $M$ is isomor …

For Kahler manifold this is true. First, notice that limits in the sense of Barlett or Douady clearly (up to multiplicity) coincide with the limits in Hausdorff topology on the set of compact subsets. …

There are manifolds without non-constant global meromorphic functions, such as generic K3 or a torus. Among these K3 surfaces, there are ones with (-2)-curves, which give a counterexample to the quest …

In complex dimension 3 or more it is still an open conjecture (which was re-stated Yau a couple of years ago in his UCLA lectures). There is not a single known example of an almost complex manifold of …

Most likely the canonical ring in Kahler situation is also finitely generated. You can check the paper http://arxiv.org/abs/1304.4013 "Minimal models for Kaehler threefolds" (Andreas Hoering, Thomas …

There are counterexamples: a Moishezon manifold, which has trivial canonical class and is birationally equivalent to a hyperkahler manifold, is also deformationally equivalent to a hyperkaehler manif …

A generic deformation of a Hilbert scheme of K3 and a generic torus have no subvarieties, hence they are "simple" in the above sense. For a torus it's well known, for a Hilbert scheme of K3 it's in my …