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 Nov 21 revised Example of a compact Kähler manifold with non-finitely generated canonical ring? added 6 characters in body Oct 24 comment What is the obstruction to the existence of a global Kahler potential? @Hassan Jolany 海桑乔朗丽 Never trust blindly any paper, unless you have checked the details yourself! Oct 24 comment What is the obstruction to the existence of a global Kahler potential? The statement of the result of Gauduchon is wrong, it should say $b^1=2h^{0,1}$. Oct 24 comment What is the obstruction to the existence of a global Kahler potential? The "standard textbook" proof shows that if $[\omega]=0$ in $H^2(X,\mathbb{R})$ and if $H^{0,1}_{\bar\partial}(X)$ vanishes, then you can write $\omega=i\partial\bar\partial K$. You can easily reconstruct the proof by yourself. Oct 22 comment Unique Kahler-Einstein metric $g$ with $\mathrm{Ricc}(g)=-g$ when first Chern class $C_1(M)<0$: $\mathrm{Ricc}(h)=-g\,\Rightarrow\,h=cg$ for $c>0$? If the dimension of $H^{1,1}$ is strictly larger than $1$, then as Henri explained there are other Kahler metrics (not of the form $cg$ with $c$ a constant) with the same Ricci curvature as $g$. Oct 21 comment Are holomorphic quasi-positive line bundles on a Kähler manifold positive? That's right, in the case that we described (the blowup of a point), if a curve is homologous to another curve which is contained in the exceptional divisor E then its intersection number with L must be zero, while if a curve passes through a point outside E then its intersection number with L is strictly positive (and the same holds for submanifolds instead of curves). Oct 18 comment Are holomorphic quasi-positive line bundles on a Kähler manifold positive? @Vesselin Dimitrov: sorry, for some reason I did not notice that you already gave that example in your comment, and I repeated it in my answer... Oct 17 comment Are holomorphic quasi-positive line bundles on a Kähler manifold positive? @Vesselin Dimitrov: the statement that $L$ is nef if and only if it admits a semipositive metric is false, see example 1.7 here www-fourier.ujf-grenoble.fr/~demailly/manuscripts/dps1.pdf Oct 17 answered Are holomorphic quasi-positive line bundles on a Kähler manifold positive? Oct 11 comment ricci flow on surfaces then the answer to your question is obvious. For all $t$ large, $4C\exp(rt)$ is smaller than $-r/2$. Oct 11 comment ricci flow on surfaces Is $r$ a constant? Sep 2 comment Vafa's semi-Ricci flat metric Theorem 1.1 (iii) Aug 23 comment Volume form on pair (X,D) The part about wedge products of currents is exactly what I mentioned above, Demailly's results. Also, in BBEGZ's definition 3.3 that you quote there is no wedge product of singular currents. Aug 19 comment Removable base loci for non-projective varieties Short answer: Yes. This is proved in Fujita's original paper. More generally he proves this result with $X$ just a compact complex analytic space, and this includes all complete algebraic varieties over $\mathbb{C}$. Aug 19 comment Volume form on pair (X,D) The "non-pluripolar" product defined in the paper that you mention is not what I had in mind in my comment, and it does not have the usual "standard" properties of the complex Monge-Ampere operator, e.g. it is in general discontinuous along decreasing sequences, and furthermore its total integral could be less than the cohomological volume of the class. For the statements of Demailly (which gives a different Monge-Ampere operator from BEGZ, which satisfies the two properties that I just mentioned), see www-fourier.ujf-grenoble.fr/~demailly/manuscripts/trento.pdf, Theorem 2.5. Aug 19 comment Volume form on pair (X,D) Not really, you need some assumptions in order to do that (e.g. that the locus where the current is singular has sufficiently high codimension). For example, to apply his result to get the top product $\omega^n$ you would need $\omega$ to be smooth everywhere except at finitely many points. Or, you could make some other assumption on $\omega$ (e.g. that it is quasi-isometric to some reasonable model near its singularities). But you did not specify anything at all in your question. Aug 18 comment Volume form on pair (X,D) And what is the volume form of a Kahler current? In general you cannot multiply currents by other currents. Aug 17 comment How can we define constant scalar curvature Kahler or cscK on pair $(X,D)$ I don't see a reasonable way to define the scalar curvature over $D$ (say e.g. as a distribution), even in the model case of $(\mathbb{C},0)$ with the standard cone metric $|z|^{2(\beta-1)}idz\wedge d\overline{z}$. Aug 9 comment Vafa's semi-Ricci flat metric This question has been solved in the affirmative by Y-J Choi in arxiv.org/abs/1508.00323 Jul 1 awarded ag.algebraic-geometry