Nous montrons que deux variétés projectives lisses dont les catégories dérivées sont équivalentes, ont des variétés de Picard isogènes. En particulier, elles ont la même irrégularité et le même nombre de champs de vecteurs indépendants. On en déduit l'invariance des nombres de Hodge par l'équivalence dérivée pour les variétés de dimension trois, ainsi que quelques autres conséquences numériques.
We prove that smooth projective varieties with equivalent derived categories have isogenous Picard varieties. In particular their irregularity and number of independent vector fields are the same. This implies that all Hodge numbers are the same for arbitrary derived equivalent threefolds, as well as other consequences of derived equivalence based on numerical criteria.
@article{ASENS_2011_4_44_3_527_0,
author = {Popa, Mihnea and Schnell, Christian},
title = {Derived invariance of the number of holomorphic $1$-forms and vector fields},
journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
volume = {44},
year = {2011},
pages = {527-536},
doi = {10.24033/asens.2149},
mrnumber = {2839458},
zbl = {1221.14020},
language = {en},
url = {http://dml.mathdoc.fr/item/ASENS_2011_4_44_3_527_0}
}
Popa, Mihnea; Schnell, Christian. Derived invariance of the number of holomorphic $1$-forms and vector fields. Annales scientifiques de l'École Normale Supérieure, Tome 44 (2011) pp. 527-536. doi : 10.24033/asens.2149. http://gdmltest.u-ga.fr/item/ASENS_2011_4_44_3_527_0/
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