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A counterexample to the semipositivity of these semi-flat metrics (on elliptic $K3$ surfaces) was constructed by Cao, Guenancia, Paun, Tosatti in this paper, see Theorem 3.1 and the Appendix.

This is not literally the answer that the OP wanted (a reference to the literature, which I am not aware of), but following the comments above let me write down the simple gluing argument. Let $\wide …

There is an "elliptic" proof in this paper by Eminenti, La Nave, Mantegazza, see Theorem 3.1. It does still use Perelman's $\mathcal{W}$ entropy (but it does not use the Ricci flow, it just uses a min …

Let me write your equations instead as $$\Theta^{(1)}_{i\bar{j}}=\lambda_1 g_{i\bar{j}},$$ $$\Theta^{(2)}_{i\bar{j}}=\lambda_2 g_{i\bar{j}},$$ where $\lambda_1,\lambda_2$ are real-valued functions. Th …

The first example of this phenomenon was discovered by Iku Nakamura in his 1975 paper Complex parallelisable manifolds and their small deformations, J. Differential Geom. 10 (1975), 85-112. The manif …

This result was proved at around the same time independently by Xiaowei Wang in Moment map, Futaki invariant and stability of projective manifolds, Comm. Anal. Geom. 12 (2004), no. 5, 1009–1037 (li …

The answer is contained in the book by W. Fulton "Intersection theory", chapter 18 "Riemann-Roch for singular varieties".

This is still an open problem. See this paper for some progress, which was prompted by this MO question.

This problem is solved affirmatively in Theorem 1.5 in this paper. The idea is that after some blowups we obtain a compact Kähler manifold whose canonical bundle is effective after twisting by a num …

As pointed out by Ruadhai Dervan in the comments, a paper by Fujino contains the answer to this question: the canonical ring of any compact Kähler manifold is finitely generated. By bimeromorphic inva …

A general result of Hulin shows that if $M$ is a compact complex manifold with a holomorphic embedding $f:M\to\mathbb{CP}^n$ such that $f^*g_{FS}$ is Einstein, then the Einstein constant is positive. …

Since your manifold $X$ has positive Ricci curvature, the line bundles $K_X^m$ are all ${ negative}$ for $m\geq 1$, i.e. they admit a smooth Hermitian metric with negative curvature. By Kodaira Vanish …

You made several mistakes. First, you should apply the result with $R=1$ to the manifold $(M,R^{-2}g)$, so that the unit ball in this scaled metric equals the $R$-size ball in the metric $g$. Secon …

You seem to have some basic misunderstandings here. For non-Kähler manifolds, like the Hopf manifold that you consider, the Dolbeault cohomology is not in general topological, and it depends a lot on …

If $\Omega$ is a rational Kähler class, so $M$ is projective algebraic, then Donaldson was the first to prove the Chen-Hwang lower bound that you stated. In fact, in this case he proved more, namely t …