Simplicity of tangent bundles of smooth horospherical varieties of Picard number one

Hong, Jaehyun

Comptes Rendus. Mathématique, Tome 360 (2022) no. G3, pp. 285-290.

Récemment, Kanemitsu a découvert un contre-exemple à la conjecture de longue date selon laquelle le faisceau tangent d’une variété de Fano de nombre de Picard un est (semi) stable. Son contre-exemple est une variété horosphérique lisse. Une conjecture plus faible affirme que le fibré tangent d’une variété de Fano de nombre de Picard un est simple.

Nous prouvons que cette conjecture plus faible est valable pour les variétés horosphériques lisses de nombre de Picard un. Notre preuve découle de l’existence d’une famille irréductible de courbes rationnelles non pliables dont les vecteurs tangents engendrent les espaces tangents de la variété horosphérique en des points généraux.

Recently, Kanemitsu has discovered a counterexample to the long-standing conjecture that the tangent bundle of a Fano manifold of Picard number one is (semi)stable. His counterexample is a smooth horospherical variety. There is a weaker conjecture that the tangent bundle of a Fano manifold of Picard number one is simple.

We prove that this weaker conjecture is valid for smooth horospherical varieties of Picard number one. Our proof follows from the existence of an irreducible family of unbendable rational curves whose tangent vectors span the tangent spaces of the horospherical variety at general points.

[1] Azad, Hassan; Biswas, Indranil A note on the tangent bundle of $G/P$, Proc. Indian Acad. Sci., Math. Sci., Volume 120 (2010) no. 1, pp. 69-71 | DOI | MR | Zbl

[2] Bourbaki, Nicolas Éleménts de mathématique. Groupes et algèbres de Lie, Springer, 2006

[3] Brion, Michel; Fu, Baohua Minimal rational curves on wonderful group compactifications, J. Éc. Polytech., Math., Volume 2 (2015), pp. 153-170 | DOI | Numdam | MR | Zbl

[4] Hwang, Jun-Muk Stability of tangent bundles of low-dimensional Fano manifolds with Picard number 1, Math. Ann., Volume 312 (1998) no. 4, pp. 599-606 | DOI | MR | Zbl

[5] Hwang, Jun-Muk Hecke curves on the moduli space of vector bundles over an algebraic curve, Proceedings of the symposium on Algebraic Geometry in East Asia (Kyoto, 2001), World Scientific, 2001, pp. 155-164 | Zbl

[6] Hwang, Jun-Muk Geometry of varieties of minimal rational tangents, Current developments in algebraic geometry (Mathematical Sciences Research Institute Publications), Volume 59, Cambridge University Press, 2012, pp. 197-226 | MR | Zbl

[7] Hwang, Jun-Muk; Mok, Ngaiming Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation, Invent. Math., Volume 131 (1998) no. 2, pp. 393-418 | DOI | Zbl

[8] Hwang, Jun-Muk; Mok, Ngaiming Deformation rigidity of the rational homogeneous space associated to a long root, Ann. Sci. Éc. Norm. Supér., Volume 35 (2002) no. 2, pp. 173-184 | DOI | Numdam | MR | Zbl

[9] Kanemitsu, Akihiro Fano manifolds and stability of tangent bundles, J. Reine Angew. Math., Volume 774 (2021), pp. 163-183 | DOI | MR | Zbl

[10] Mok, Ngaiming Geometric structures on uniruled projective manifolds defined by their varieties of minimal rational tangents, Differential geometry, mathematical physics, mathematics and society (II) (Astérisque), Volume 322, Société Mathématique de France, 2008, pp. 151-205 | Numdam | Zbl

[11] Onishchik, Arkadiĭ L.; Vinberg, Èrnest B. Lie groups and algebraic groups, Springer Series in Soviet Mathematics, Springer, 1990 | DOI

[12] Pasquier, Boris On some smooth projective two-orbit varieties with Picard number 1, Math. Ann., Volume 344 (2009) no. 4, pp. 963-987 | DOI | MR | Zbl

[13] Peternell, Thomas; Wiśniewski, Jarosław A. On stability of tangent bundles of Fano manifolds with $b_2=1$, J. Algebr. Geom., Volume 4 (1995) no. 2, pp. 362-384 | MR | Zbl