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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 …

The function $\psi:=\log(fh^{-1})$ satisfies $\partial\bar\partial \psi=0$, because $ \partial\bar\partial\log f=\partial\bar\partial\log h =\omega$. Such functions are called pluriharmonic. Locally a …

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) …

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 …

It seems that the answer is very simple (and should have been known to Uhlenbeck-Yau). Let $E$ be a stable bundle, equipped with an Hermitian-Einstein metric and connection, and $End(E)$ its authomorp …

The fundamental 4-form on a quaternionic-Kahler manifold is closed and gives the same kind of decomposition as the Kahler form on a Kahler manifold. Reference: Bonan, Edmond, Sur l'algèbre extérieure …

The main obstruction to existence of Kahler metric (in addition to Lefschetz SL(2)-action and Riemann-Hodge relations in cohomology) is homotopy formality: the cohomology ring of a Kahler manifold is …

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 …

Let $\phi$ be a stricly plurisubharmonic function on a domain in ${\Bbb C}^n$, and $S=\phi^{-1}(c)$ its level set. Consider $S$ as a Riemannian manifold equipped with a metric induced by $dd^c\phi$. I …

It is impossible, because the birational (movable) nef cone is mapped to birational nef cone, where birational nef cone is a cone of all classes which are non-negative on all curves which move in fami …

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 …

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 …

They can be used to prove the vanishing theorems (and more). Kodaira-Nakano vanishing theorem and Kodaira embedding theorem follow from these identities. One proves that the difference of the $\partia …

Lempert, László The Dolbeault complex in infinite dimensions. III. Sheaf cohomology in Banach spaces. Invent. Math. 142 (2000), no. 3, 579-603. Lempert, László The Dolbeault complex in infinite dimen …

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 …