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Apr
27
comment How much of modern algebraic geometry is there in modern complex(algebraic, analytic, differential) geometry?
Just to be clear, this very just special case, is already just very important (and, moreover, the only just known case!) :)
Apr
27
comment How much of modern algebraic geometry is there in modern complex(algebraic, analytic, differential) geometry?
What do you mean by "If you want to solve an algebraic question for example invariance of plurigenera, you need to know Kähler Ricci flow theory, it has been solved by accepting Minimal Model Program, but at the moment the only method that I think give a solution to conjecture of invariance of plurigenera is the method of professor Tian"? Invariance of plurigenera is known to hold in the projective case. It was proved by Y.-T. Siu by a clever use of Oshawa-Takegoshi extension theorem.
Apr
14
comment A question about invariance of plurigenera
Ok, I was't aware of that paper by Tsuji. Anyway, as you can of course see by yourself, it doesn't say nothing about the Kähler case.
Apr
14
comment A question about invariance of plurigenera
What kind of assumption is it? You assume that your central fiber has the property that its canonical bundle spreads out pseudoeffectiveness around? I still don't understand. It doesn't seem to me that pseudoeffectiveness is an open property in family (but maybe I am wrong). You might say that it could be at least for the canonical class. But it seems to me as well that even in the projective case where the invariance of plurigenera is known, this would use the non vanishing conjecture (which is, indeed, a conjecture).
Apr
14
comment A question about invariance of plurigenera
By the way, what do you mean by "such that psudoeffectiveness of $K_{X_0}$ gives the psudoeffectiveness of $K_{X_t}$"?
Apr
14
comment A question about invariance of plurigenera
The invariance of plurigenera in the Kähler setting is still nowadays an open problem. Even in this (slightly) more particular case you mention.
Apr
12
comment A necessary condition for existence of Ricci flat metric on pair (X,D)
It would be nice if some more effort was made in writing the question. $K_X+D=0$ in which sense? Linear equivalence? Ricci-flat metric on what? On $X$? On $X\setminus D$? In the latter case, complete or whatever? You suppose you manifold to be Kähler I guess? So, let's try to motivate people to answer your question by writing a real question!
Mar
31
answered Bott Chern cohomology via currents
Feb
18
comment Inequality on Kähler classes
very nice proof!
Nov
12
comment Stability of automorphism group of complex manifolds
Could you please explain a little bit better which kind of stability theorems you are looking for?
Nov
6
awarded  dg.differential-geometry
Nov
5
revised Let $X$ be a projective variety and let $Pic(X) \cong \mathbb Z$, then $(X, −K_X) $ is $K$-semistable
added "Fano"
Nov
5
revised Griffiths-positive metric
deleted 23 characters in body
Oct
7
awarded  Yearling
Aug
2
reviewed Approve Product of Positive Matrices
Aug
2
reviewed Approve What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem?
Jul
30
comment Isotrivial families with non-zero Kodaira spencer map
If all fibers are isomorphic, then by the Fischer-Grauert theorem the family is locally trivial. Then, the Kodaira-Spencer map -being "local"- should be zero, isn't it?
Jul
14
awarded  Nice Question
Jul
2
awarded  Popular Question
Jun
12
comment Moduli space of line-bundle holomorphic structures and sections over a Kahler-Hodge manifold
Chern connection with respect to which hermitian metric?