Timeline for Does a Kähler manifold always admit a complete Kähler metric?
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| Nov 30, 2017 at 19:36 | comment | added | user21574 | Sai-Kee extended the Griffiths result in Kahler setting : He showed if $M$ is a compact Kähler manifold of complex dimension $n$, and if $ D$ is a smooth hypersurface such that $[D]$ is positive and $K+[D]$ is negative, then the affine algebraic manifold $X=M-D$ admits a complete Kähler metric of positive Ricci curvature gdz.sub.uni-goettingen.de/dms/load/img/… | |
| Nov 30, 2017 at 16:19 | comment | added | user21574 | Let's give a Shiffman's characterization of open subsets of Stein manifolds: Let $D$ be an open subset of a Stein manifold $V$. Then the following conditions are equivalent. i) $D$ is Stein, ii) $D$ has a complete Kahler metric with non.positive holomorphic sectional curvatures, iii) The universal covering manifold of $D$ has a complete hermitian metric with non-positive holomorphic sectional curvatures. link.springer.com/article/10.1007%2FBF01350128?LI=true | |
| Nov 22, 2017 at 15:46 | comment | added | user21574 | Let $X$ be a compact Kahler Stein manifold and $Z$ be a analytic space in $X$, then $X∖Z$ pocesses a complete Kahler metric, see Theorem 0.2 of numdam.org/article/ASENS_1982_4_15_3_457_0.pdf See Theorem 1.5 also and Proposition of 1.6 which is as same as the result of Ohsawa | |
| Nov 21, 2017 at 17:41 | comment | added | user21574 | About my previous comment for proof of "Every weakly pseudoconvex Kähler manifold $X$ carries a complete metric"see Proposition 14 math.u-psud.fr/~merker/Enseignement/Geometrie-complexe/Proietti/…. | |
| Nov 21, 2017 at 17:21 | comment | added | user21574 | A Kahler manifold with a $C^\infty$ exhaustive pluri-subharmonic function is a complete Kahler manifold. So, Stein manifolds are complete Kahler manifolds. Let $D$ be a bounded domain with a smooth pseudoconvex boundary in a Kahler manifold. Then, $D$ is a complete jstage.jst.go.jp/article/kyotoms1969/20/1/20_1_21/_pdf . Ohsawa asked an interesting conjecture after Example 3 Kahler manifold. | |
| Nov 15, 2017 at 9:36 | comment | added | user21574 | Let $V$ be a smooth projective variety and $D$ an ample divisor with simple normal crossings . Then the complement $A = V-D$ is a special affine variety. Griffiths showed that There exists a complete Kahler metric $\varphi$ on $A$ whose associated Ricci form satisfies $$Ric( \varphi)=O(\omega_{Poincare})$$ see (3.5) Proposition. Analytic Cycles and Vector Bundles on Non-Compact Algebraic Varieties. Maurizio Cornalba; Phillip Griffiths Inventiones mathematicae (1975) Volume: 28, page 1-106 ISSN: 0020-9910; 1432-1297/e eudml.org/doc/142315 | |
| Jul 24, 2017 at 7:14 | comment | added | user21574 | Let $M$ be a compact complex manifold and $ S\subset M$ a proper analytic subset Let $\omega$ be a $d$-closed $(1, 1)$-current satisfying the conditions 1) $ω$ is smooth on $M-S$ , where $S$ is some proper analytic subset in $M$ 2)$ ω > εσ$ in the sense of currents, where $ε > 0$ is some real number and $σ$ is a fixed positive definite $(1, 1)$-form (not necessarily d-closed) on $M$ , Then $M - S$ admits a complete Kahler metric. See lemma 4. 1 , msp.org/pjm/1993/158-2/pjm-v158-n2-p09-p.pdf | |
| Jun 2, 2017 at 14:31 | history | edited | user21574 | CC BY-SA 3.0 |
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| Jun 2, 2017 at 14:23 | history | edited | user21574 | CC BY-SA 3.0 |
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| Jun 2, 2017 at 14:14 | history | answered | user21574 | CC BY-SA 3.0 |