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Steve has more or less already answered to your question. Just to give you some more intuition, let $M$ be an oriented smooth real manifold of real dimension $m$ and consider the space $^s\mathcal E …

Here is a standard example (which you can find also in the big book of Demailly) in the complex category with the language of sheaves which perhaps may be of some interest for you. Take $X=\mathbb C^ …

I will try to clarify a little bit your question and at the same time to give you an answer. All that I say, you can find on Demailly's book "Complex Analytic and Differential Geometry". GENERAL THEO …

I think the answers you are looking for are in this paper by V. Tosatti, see in particular Proposition 1.1, point (4) and Proposition 1.3. Warning (in view of the comment below by S.S.): the holonom …

With your notations, the hermitian form $\theta_E$ on $T_X\otimes E$ defined by $\Theta_E$ is given in a somewhat more extrinsic way by $$ \theta_E(v\otimes\sigma,v\otimes\sigma):=h(\Theta_E(v,\bar v) …

My answer will consist mainly of a collection of trivial facts but which nonetheless often generate some confusion. I begin by fixing some notation. Let $V$ be a complex vector space of complex dimen …

Here is the answer. Let $(X,\omega)$ be a Kähler $n$-dimensional manifold. Fix a point $x_0\in X$ an choose local holomorphic coordinates $(z_1,\dots,z_n)$ centered at $x_0$ and such that $(\partial/ …

As soon as the base manifold has dimension greater than one, the existence of Griffiths positive metrics on an ample vector bundle is not known (and even in the one dimensional case, it is not obvious …

With your notations, the normal bundle is spanned over $\{Z_1\ne 0\}$ by $\partial/\partial v$. Now, over $\{Z_0\ne 0\}$, take affine coordinates $x=Z_1/Z_0$ and $y=Z_2/Z_0$, so that, where defined, y …

I assume that for $\operatorname{Hom}(X,Y)$ you mean $\operatorname{Hol}(X,Y)$, that is the family of all holomorphic maps from $X$ to $Y$, endowed with its universal complex structure (which exists s …

Maybe what follows is not exactly what you are looking for, but it gives you an answer at least when you don't want to restrict yourself to the tangent bundle and work rather with general (complex) ve …

Let X be a complex hermitian manifold with hermitian form $\omega$. How can you prove that if $\omega$ has negative holomorphic sectional curvature, then its scalar curvature is negative, too?

Kobayashi hyperbolicity (or Brody hyperbolicity, the two notions coincide for compact complex spaces), is an open condition with respect to the analytic topology. Thus, for instance, once you find an …

The answer to your question is exactly the same of the answer to this older question of mine. Enjoy!

The whole point is that if the metric is Kähler, then the Chern connection coincides with the complexification of Levi-Civita connection of the riemanian metric on the underlying real manifold given b …