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On $B$, there is $\delta>0$ such that $P(h)\le h-\delta$. Next, pick a point $x\in B$ and choose a potential $\rho$ for $\omega$ on $B$ such that $\rho(x)=0$. By shrinking $B$, you can assume that $\s …

Yes, this is true. By the boundedness assumption, $u$ extends to a (bounded) $\omega$-psh function on $X$. Bedford and Taylor defined in '82 the Monge-Ampère operator of a bounded psh function, which …

For Question 1, let's take $E=E_1\oplus E_2$, and let $\mu$ be the slope of $E$ (which agrees with the slope of $E_1$ and $E_2$). Let $F\subset E$ be a subbundle and let $p:=\mathrm{pr_1}|_F:F\to E_1$ …

Let $h_0$ be any hermitian metric on $L$, with curvature $F(h_0)$. By Hodge theory, there exists a function $f\in \mathcal C^{\infty}(X)$ such that $$\Delta_{\omega} f =\Lambda_{\omega} F(h_0) - \ma …

About the formulation, I would see such a family as a smooth map from $[0,1]$ to the space of smoooth sections of $S^2T^*M$ which lands in the open set of sections with values in positively definite f …

Edit: My answer deals with the projective case. First of all, for any projective manifold, $K_X$ is pseudoeffective if and only if $X$ is not uniruled (Boucksom-Demailly-Paun-Peternell). In the cas …

Let me copy my remark to close the topic. If $\partial \bar \partial f$ is a constant matrix on $\mathbb C^n$, then it follows that for any complex line $L\subset \mathbb C^n$, $\Delta (f|_L)$ is co …

This is a consequence of the generalized Lelong-Poincaré formula for vector bundles: denoting $D'$ the $(1,0)$ part of the Chern connection of $(E,h)$ and $\Theta_h(E)$ its Chern curvature, one has: …

so here I guess $M$ is a compact Kähler manifold. Thanks to Yau theorem, we know that there exists a unique Kähler metric $h$ in each Kähler cohomology class such that $\mathrm{Ric}(h)=-g$ (more gene …

If you have a morphism $\pi$ with finite generic fiber of cardinal $d$ between compact manifolds, then you have $\pi_* \circ \pi^* = d Id_{H^k(Y,\mathbb Z)}$, where $\pi_* $ is the Gysin morphism. In …

Well, it depends on what you call a Calabi-Yau manifold (there are several possible terminologies indeed). First of all, a compact Kähler manifold with trivial canonical class does not necessarily ha …

Take the most simple example $M=\mathbb C$ and $X$ the unit circle. Then $d_X(z)=|z-\frac{z}{|z|}|^2$ is not holomorphic (so as its Laplacian $i\partial \bar \partial( d_X)$), as you can easily see by …

I guess your diffeomorphism is a biholomorphism, or at least an holomorphic map, otherwise there is no hope to say much about it. Then the best way to look at these things is using differential forms …