Voir la notice de l'article provenant de la source Annals of Mathematics website
A conjecture of Hirschowitz’s predicts that a globally generated vector bundle $W$ on a compact complex manifold $A$ satisfies the formal principle; i.e., the formal neighborhood of its zero section determines the germ of neighborhoods in the underlying complex manifold of the vector bundle $W$. By applying Cartan’s equivalence method to a suitable differential system on the universal family of the Douady space of the complex manifold, we prove that this conjecture is true if $A$ is a Fano manifold, or if the global sections of $W$ separate points of $A$. Our method shows more generally that for any unobstructed compact submanifold $A$ in a complex manifold, if the normal bundle is globally generated and its sections separate points of $A$, then a sufficiently general deformation of $A$ satisfies the formal principle. In particular, a sufficiently general smooth free rational curve on a complex manifold satisfies the formal principle.
@article{10_4007_annals_2019_189_3_8,
author = {Jun-Muk Hwang},
title = {An application of {Cartan{\textquoteright}s} equivalence method to {Hirschowitz{\textquoteright}s} conjecture on the formal principle},
journal = {Annals of mathematics},
pages = {979--1000},
publisher = {mathdoc},
volume = {189},
number = {3},
year = {2019},
doi = {10.4007/annals.2019.189.3.8},
zbl = {07097494},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.189.3.8/}
}
TY - JOUR AU - Jun-Muk Hwang TI - An application of Cartan’s equivalence method to Hirschowitz’s conjecture on the formal principle JO - Annals of mathematics PY - 2019 SP - 979 EP - 1000 VL - 189 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.189.3.8/ DO - 10.4007/annals.2019.189.3.8 LA - en ID - 10_4007_annals_2019_189_3_8 ER -
%0 Journal Article %A Jun-Muk Hwang %T An application of Cartan’s equivalence method to Hirschowitz’s conjecture on the formal principle %J Annals of mathematics %D 2019 %P 979-1000 %V 189 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.189.3.8/ %R 10.4007/annals.2019.189.3.8 %G en %F 10_4007_annals_2019_189_3_8
Jun-Muk Hwang. An application of Cartan’s equivalence method to Hirschowitz’s conjecture on the formal principle. Annals of mathematics, Tome 189 (2019) no. 3, pp. 979-1000. doi: 10.4007/annals.2019.189.3.8
Cité par Sources :