Invariance for multiples of the twisted canonical bundle

Claudon, Benoît. "Invariance for multiples of the twisted canonical bundle." Annales de l’institut Fourier 57.1 (2007): 289-300. <http://eudml.org/doc/10222>.

@article{Claudon2007,
abstract = {Let $\mathcal\{X\}\rightarrow \Delta $ a smooth projective family and $(L,h)$ a pseudo-effective line bundle on $\mathcal\{X\}$ (i.e. with a non-negative curvature current $\Theta \{h\}\{L\}$). In its works on invariance of plurigenera, Y.-T. Siu was interested in extending sections of $mK_\{\mathcal\{X\}_0\}+L$ (defined over the central fiber of the family $\mathcal\{X\}_0$) to sections of $mK_\mathcal\{X\}+L$. In this article we consider the following problem: to extend sections of $m(K_\mathcal\{X\}+L)$. More precisely, we show the following result: assuming the triviality of the multiplier ideal sheaf $\mathcal\{I\}(\mathcal\{X\}_0,h_\{\vert \mathcal\{X\}_0\})$, any section of $m(K_\{\mathcal\{X\}_0\}+L)$ extends to $\mathcal\{X\}$ ; in other words, the restriction map:\[H^0(\mathcal\{X\},m(K\_\{\mathcal\{X\}\}+L))\rightarrow H^0(\mathcal\{X\}\_0,m(K\_\{\mathcal\{X\}\_0\}+L))\]is surjective.At the end of this paper, we compare this result to the case of projective manifolds: in this situation an analogous statement (due to S. Takayama) is given to extend (twisted) pluricanonical sections. This lead us to discuss the different positivity assumptions required in extension results.},
affiliation = {Université Nancy 1 Institut Elie Cartan BP 239 54506 Vandœuvre–lès–Nancy Cedex (France)},
author = {Claudon, Benoît},
journal = {Annales de l’institut Fourier},
keywords = {Extensions of pluricanonical sections; invariance of plurigenera; pseudoeffective line bundle; singular metric; multiplier ideal sheaf; extensions of pluricanonical sections; pseudo-effective line bundle},
language = {eng},
number = {1},
pages = {289-300},
publisher = {Association des Annales de l’institut Fourier},
title = {Invariance for multiples of the twisted canonical bundle},
url = {http://eudml.org/doc/10222},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Claudon, Benoît
TI - Invariance for multiples of the twisted canonical bundle
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 1
SP - 289
EP - 300
AB - Let $\mathcal{X}\rightarrow \Delta $ a smooth projective family and $(L,h)$ a pseudo-effective line bundle on $\mathcal{X}$ (i.e. with a non-negative curvature current $\Theta {h}{L}$). In its works on invariance of plurigenera, Y.-T. Siu was interested in extending sections of $mK_{\mathcal{X}_0}+L$ (defined over the central fiber of the family $\mathcal{X}_0$) to sections of $mK_\mathcal{X}+L$. In this article we consider the following problem: to extend sections of $m(K_\mathcal{X}+L)$. More precisely, we show the following result: assuming the triviality of the multiplier ideal sheaf $\mathcal{I}(\mathcal{X}_0,h_{\vert \mathcal{X}_0})$, any section of $m(K_{\mathcal{X}_0}+L)$ extends to $\mathcal{X}$ ; in other words, the restriction map:\[H^0(\mathcal{X},m(K_{\mathcal{X}}+L))\rightarrow H^0(\mathcal{X}_0,m(K_{\mathcal{X}_0}+L))\]is surjective.At the end of this paper, we compare this result to the case of projective manifolds: in this situation an analogous statement (due to S. Takayama) is given to extend (twisted) pluricanonical sections. This lead us to discuss the different positivity assumptions required in extension results.
LA - eng
KW - Extensions of pluricanonical sections; invariance of plurigenera; pseudoeffective line bundle; singular metric; multiplier ideal sheaf; extensions of pluricanonical sections; pseudo-effective line bundle
UR - http://eudml.org/doc/10222
ER -