no. 5
Algebraic geometry
Deformation of the product of complex Fano manifolds
[Déformation du produit de variétés de Fano complexes]

Présenté par : Claire Voisin

Li, Qifeng1
1 Korea Institute for Advanced Study, Seoul, Republic of Korea
Comptes Rendus. Mathématique, Tome 356 (2018) no. 5, pp. 538-541.

Soit X une famille connexe des variétés de Fano complexes. On montre que, si une fibre est un produit de deux variétés de dimensions inférieures, alors il en est de même pour chaque fibre. En combinant avec un résultat de Hwang et Mok, ceci implique immédiatement que, si une fibre est un espace Hermitien symétrique de type compact, alors toutes les fibres sont isomorphes à cette variété.

Let X be a connected family of complex Fano manifolds. We show that if some fiber is the product of two manifolds of lower dimensions, then so is every fiber. Combining with previous work of Hwang and Mok, this implies immediately that if a fiber is a (possibly reducible) Hermitian symmetric space of compact type, then all fibers are isomorphic to the same variety.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.04.007
Li, Qifeng 1

1 Korea Institute for Advanced Study, Seoul, Republic of Korea 
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Li, Qifeng. Deformation of the product of complex Fano manifolds. Comptes Rendus. Mathématique, Tome 356 (2018) no. 5, pp. 538-541. doi : 10.1016/j.crma.2018.04.007. http://archive.numdam.org/articles/10.1016/j.crma.2018.04.007/

[1] Höring, A. Uniruled varieties with split tangent bundle, Math. Z., Volume 256 (2007) no. 3, pp. 465-479

[2] Hwang, J.-M.; Mok, N. Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation, Invent. Math., Volume 131 (1998) no. 2, pp. 393-418

[3] Kawamata, Y.; Matsuda Katsumi, K.; Matsuki, K. Introduction to the minimal model problem, Sendai, 1985 (Advanced Studies in Pure Mathematics), Volume vol. 10, North-Holland, Amsterdam (1987), pp. 283-360

[4] Kodaira, K. On stability of compact submanifolds of complex manifolds, Amer. J. Math., Volume 85 (1963), pp. 79-94

[5] Kollár, J.; Mori, S. Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, UK, 1998 (With the collaboration of C.H. Clemens and A. Corti. Translated from the 1998 Japanese original)

[6] Molino, P. Riemannian Foliations, Progress in Mathematics, vol. 73, Birkhäuser Boston Inc., Boston, MA, USA, 1988

[7] Wiśniewski, J.A. Rigidity of the Mori cone for Fano manifolds, Bull. Lond. Math. Soc., Volume 41 (2009) no. 5, pp. 779-781

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