Timeline for Is Kähler current class representable by semipositive forms?
Current License: CC BY-SA 4.0
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| Aug 26, 2023 at 0:38 | comment | added | Invariance | Dear @HYL, sorry for my late reply. The Hironaka's example you mentioned should be a complete non-projective (but Moishezon) 3-fold, right? Following your statement, it may be better to find a non-projective Moishezon surface (which must process singularities worse than rational singularities), since you talk about the intersectional number $[l][E]$? | |
| May 16, 2022 at 2:54 | comment | added | HYL | If $X$ is in the Fujiki class $\mathcal{C}$, so is $\tilde{X}$. Hironaka's example is Moishezon, in particular in the Fujiki class $\mathcal{C}$. | |
| May 15, 2022 at 11:39 | comment | added | Tom | For the holomorphic modification $\mu:\tilde X\to X$, we say $X$ is in Fujiki class $\mathcal C$ if $\tilde X$ is a compact Kähler manifold. Then the Kähler current $T$ should be on $X$ not $\tilde X$, right? | |
| May 15, 2022 at 9:38 | comment | added | HYL | You can replace $S$ by any manifold $X$ you like. If $\omega$ is a Kähler current on the blowup $\tilde{X}$, then $[E] + \varepsilon [\omega]$ for $\varepsilon > 0$ is represented by a Kähler current where $E \subset \tilde{X}$ is the exceptional divisor. Again for small $\varepsilon$, we can't represent $[E] + \varepsilon [\omega]$ by a smooth semipositive form because $[\ell] ([E] + \varepsilon [\omega]) < 0$ where $\ell$ is a line in $E$. You can construct non-Kähler $\tilde{X}$ by e.g. blowing up Hironaka's example at a general point. | |
| May 15, 2022 at 4:38 | comment | added | Tom | In your case, both $S$ and $\tilde S$ are Kähler？But actually what I want to know is that for a non-Kähler manifold in Fujiki class $\mathcal C$, the class of a Kähler current can always be represented by a semipositive form? Sorry for the confusion. | |
| May 15, 2022 at 3:17 | history | answered | HYL | CC BY-SA 4.0 |