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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 …

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 …

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 …

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 …

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 …

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) …

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 …

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 …

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 …

Take a K3, or a general $n$-dimensional complex torus $M$, $n>1$, without any integer (1,1)-classes, and remove a point $x$. You will obtain a non-compact Kahler manifold without any non-trivial line …

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 …

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)

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 …