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Let $(M,I, \omega)$ be a compact Kähler manifold with $c_1(M)=0$. Denote by $\operatorname{Ric}^{1,1}(\omega)$ the Ricci (1,1)-form, that is, the curvature of the canonical bundle. It is known ("Defor …

Let $M$ be a compact complex manifold equipped with a line bundle $L$ which has curvature which is non-negative and strictly positive outside of a measure zero set $Z$. In his paper "Holomorphic Mors …

Explicit description of a Kahler cone for all hyperkahler manifolds is here: https://arxiv.org/abs/1401.0479 (Rational curves on hyperkahler manifolds, Ekaterina Amerik, Misha Verbitsky)

I don't think this is known. For hyperkahler manifolds, conjecturally, all smooth complex deformations are class C and birational to hyperkahler. If this is true, your conjecture would follow automati …

Let $(M,g)$ be a Riemannian manifold. The Riemannian cone of $M$ is $C(M) = M \times {\Bbb R}^{>0}$ with the metric $t^2 g + dt\otimes dt$. A manifold is called Sasakian if its cone is Kähler, with t …

Sure, any closed 2-form $\eta$ with integer cohomology can serve as the curvature of a connection on a line bundle. This can be seen if you take a line bundle with the same Chern class and connection …

Sure, you should assume that there are sufficiently many functions which grow polynomially. Here is an example of such a result. THEOREM: Let $M$ be a Stein variety equipped with a Kahler metric outsi …

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, …

Let $M$ be a compact smooth manifold, and $F\subset TM$ a smooth foliation. It is called transversally Kähler if the normal bundle $TM/F$ is equipped with a Hermitian structure (that is, a complex str …

Let $T$ be a compact torus, and $X$ its blow-up in a point (or in several points). It seems that $X$ is K-stable for any Kahler form on $X$. Is there a reference to this? Also, what can we say a …

Does a nonsingular fibre $\pi^{−1}(z)$ has vanishing first Chern class? Yes. Denote a regular fiber $Z$; then $K_M|_ Z= det(N^*Z)\otimes K_Z= K_Z$ by adjunction formula. On the other hand, $K_M …

Let $M$ be a compact complex manifold. The set of Kahler metric on $M$ in a given Kahler class is parametrized by the set of positive volume forms with given integral, because (by Calabi-Yau theorem) …

The most general version of Weitzenbock identities (with coefficients in appropriate universal enveloping algebras) is due to Uwe Semmelmann and Gregor Weingart: http://arxiv.org/abs/math/0702031 "The …

There are three definitions of infinite-dimensional complex manifolds: strong (existence of holomorphic charts), weak (abundance of holomorphic functions) and formal (vanishing of Nijenhuis tensor). T …

When $Y$ is compact, the blow-up is always Kahler; see e.g. Lemma 3.4 in this paper (this is a generally known folklore theorem which we had to use, and hence written down). For $Y$ non-compact the …