Timeline for Decomposition of hermitian form used in the definition of Griffiths/Nakano positivity
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19 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Aug 9, 2013 at 7:59 | comment | added | Michael Albanese | Do you know of a reference which proves that $E\otimes L^k$ is Nakano positive for $k$ large enough ($X$ compact, $L$ positive)? I thought I could prove it, but I keep getting stuck. | |
| Jul 15, 2013 at 6:38 | history | edited | diverietti | CC BY-SA 3.0 |
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| Jun 23, 2013 at 15:23 | comment | added | diverietti | I am sorry I didn't have much time during the last days. I'll give you the answer later today! | |
| Jun 21, 2013 at 0:50 | comment | added | Michael Albanese | I have asked another question which will hopefully clear things up for me: mathoverflow.net/questions/134232/… | |
| Jun 20, 2013 at 5:13 | vote | accept | Michael Albanese | ||
| Jun 18, 2013 at 3:56 | comment | added | Michael Albanese | If $i\Theta(E) = ic_{jk\lambda\mu} dz_j\wedge d\bar{z}_k\otimes e_{\lambda}^*\otimes e_{\mu}$ as in Demailly's book (page 338, equation 6.1) then $h(i\Theta(E)(v,\bar{v})\cdot\sigma, \sigma) = h(ic_{jk\lambda\mu} v_j\bar{v_k}\sigma_{\lambda}e_{\mu}, \sigma) = ic_{jk\lambda\mu} v_j\bar{v_k}\sigma_{\lambda}h(e_{\mu}, \sigma) = ic_{jk\lambda\mu} v_j\bar{v_k}\sigma_{\lambda}\bar{\sigma_{\mu}}$ which has a factor of $i$ that isn't present in Demailly's local expression for $\theta_E$ (page 338, equation 6.2). What am I missing? | |
| Jun 17, 2013 at 21:07 | comment | added | diverietti | My expression of $\theta_E$ of course agrees with the one in Demailly's book. It's just a coordinate-free way to write it. Try to write it down in a orthonormal frame and it will be clear! | |
| Jun 17, 2013 at 10:35 | comment | added | Michael Albanese | Either way, do you have a reference which uses this expression for $\theta_E$? | |
| Jun 16, 2013 at 3:56 | comment | added | Michael Albanese | Does the expression of $\theta_E$ you've written agree with the one stated in Chapter VII, section 6 of Demailly's book? I don't think he includes the factor of $i$, but I may be mistaken. | |
| Jun 16, 2013 at 3:49 | comment | added | Michael Albanese | I suppose I was wondering about the map $\theta_E$ and the map $h_E([i\Theta(E), \Lambda]\cdot,\cdot)$, but the latter does not take a tangent vector as an input. | |
| Jun 13, 2013 at 9:14 | comment | added | diverietti | What do you mean? The former is an hermitian form, the latter an operator acting on forms... | |
| Jun 13, 2013 at 6:45 | comment | added | Michael Albanese | Of course. Is there a slick way to see this? At the moment I'm using Cauchy-Schwarz and an operator norm, then by compactness, all the relative terms are bounded. Finally, as $\theta_{L^k} = k\theta_L$, it is just a matter of choosing $k$ large enough. It is precisely how Serre's Theorem is proved in Huybrechts' Complex Geometry, except there is no mention of $\theta_E$, but rather the endomorphism $[i\Theta(E), \Lambda] : \bigwedge^{p,q}X\otimes E \to \bigwedge^{p,q}X\otimes E$. Do you know if there is some direct relationship between $\theta_E$ and $[i\Theta(E), \Lambda]$? | |
| Jun 12, 2013 at 11:36 | comment | added | diverietti | More than this, you can use it to show that it is Nakano positive! | |
| Jun 12, 2013 at 8:13 | comment | added | Michael Albanese | If I'm not mistaken, you can use this decomposition to show that for any hermitian holomorphic vector bundle $E$ on a compact complex manifold with a positive line bundle $L$ (so it is algebraic by the Kodaira Embedding Theorem), there is some $k_0 \in \mathbb{N}$ such that $E\otimes L^k$ is Griffiths positive for all $k \geq k_0$. | |
| Jun 12, 2013 at 5:40 | history | edited | diverietti | CC BY-SA 3.0 |
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| Jun 11, 2013 at 15:29 | history | edited | diverietti | CC BY-SA 3.0 |
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| Jun 11, 2013 at 13:24 | history | edited | diverietti | CC BY-SA 3.0 |
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| Jun 11, 2013 at 13:16 | history | answered | diverietti | CC BY-SA 3.0 |