Stability of the tangent bundles of complete intersections and effective restriction
[Stabilité de fibré tangent d’intersections complètes et ses restrictions]
Annales de l'Institut Fourier, Online first, 34 p.

Soient M un espace hermitien symétrique irréductible de type compact et de dimension (n+r) avec n3 et r1, 𝒪 M (1) le générateur ample de pic(M). Soit Y=H 1 H r une intersection complète lisse de dimension nH i |𝒪 M (d i )| avec d i 2. Nous montrons un théorème d’annulation pour le faisceau tordu des germes de p-formes holomorphes Ω Y p (). Comme application, nous montrons que le fibré tangent T Y de Y est stable. De plus, si X est une hypersurface lisse de degré d dans Y telle que la restriction pic(Y)pic(X) soit surjective, nous obtenons des estimations effectives liées à la stabilité de la restriction T Y | X . En particulier, si Y est une hypersurface générale dans n+1 et X est un diviseur général, nous montrons que T Y | X est stable sauf certains exemples bien connus. Nous considérons aussi le cas où le nombre de Picard augmente par restriction.

For n3 and r1, let M be an (n+r)-dimensional irreducible Hermitian symmetric space of compact type and let 𝒪 M (1) be the ample generator of pic(M). Let Y=H 1 H r be a smooth complete intersection of dimension n, where H i |𝒪 M (d i )| with d i 2. We prove a vanishing theorem for twisted holomorphic forms on Y. As an application, we show that the tangent bundle T Y of Y is stable. Moreover, if X is a smooth hypersurface of degree d in Y such that the restriction pic(Y)pic(X) is surjective, we establish some effective results for d to guarantee the stability of the restriction T Y | X . In particular, if Y is a general hypersurface in n+1 and X is a general smooth divisor in Y, we show that T Y | X is stable except for some well-known examples. We also address the cases where the Picard group increases by restriction.

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DOI : https://doi.org/10.5802/aif.3435
Classification : 14M10,  14J70,  32M15,  32M25,  32Q26
Mots clés : stabilité, fibré tangent, propriété de Lefschetz, intersection complète, espace hermitien symétrique
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Liu, Jie. Stability of the tangent bundles of complete intersections and effective restriction. Annales de l'Institut Fourier, Online first, 34 p.

[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 | Article | MR 2654899 | Zbl 1187.53053

[2] Biswas, Indranil; Chaput, Pierre-Emmanuel; Mourougane, Christophe Stability of restrictions of the cotangent bundle of irreducible Hermitian symmetric spaces of compact type, Publ. Res. Inst. Math. Sci., Volume 55 (2019) no. 2, pp. 283-318 | Article | MR 3941485 | Zbl 1425.32018

[3] Borel, Armand; Hirzebruch, Friedrich Ernst Peter Characteristic classes and homogeneous spaces. I, Am. J. Math., Volume 80 (1958), pp. 458-538 | Article | MR 0102800 | Zbl 0097.36401

[4] Bott, Raoul Homogeneous vector bundles, Ann. Math., Volume 66 (1957), pp. 203-248 | Article | MR 0089473 | Zbl 0094.35701

[5] Chen, Xiuxiong; Donaldson, Simon; Sun, Song Kähler-Einstein metrics and stability, Int. Math. Res. Not. (2014) no. 8, pp. 2119-2125 | Article | MR 3194014 | Zbl 1331.32011

[6] Chen, Xiuxiong; Donaldson, Simon; Sun, Song Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches 2π and completion of the main proof, J. Am. Math. Soc., Volume 28 (2015) no. 1, pp. 235-278 | Article | MR 3264768 | Zbl 1311.53059

[7] Dimca, Alexandru Topics on real and complex singularities. An introduction, Advanced Lectures in Mathematics, Vieweg & Sohn, 1987, xviii+242 pages | Article | MR 1013785 | Zbl 0628.14001

[8] Fahlaoui, Rachid Stabilité du fibré tangent des surfaces de del Pezzo, Math. Ann., Volume 283 (1989) no. 1, pp. 171-176 | Article | MR 973810 | Zbl 0672.14009

[9] Flenner, Hubert Divisorenklassengruppen quasihomogener Singularitäten, J. Reine Angew. Math., Volume 328 (1981), pp. 128-160 | Article | MR 636200 | Zbl 0457.14001

[10] Flenner, Hubert Restrictions of semistable bundles on projective varieties, Comment. Math. Helv., Volume 59 (1984) no. 4, pp. 635-650 | Article | MR 780080 | Zbl 0599.14015

[11] Green, Mark Lee A new proof of the explicit Noether–Lefschetz theorem, J. Differ. Geom., Volume 27 (1988) no. 1, pp. 155-159 | MR 918461 | Zbl 0674.14005

[12] Hartshorne, Robin Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics, 156, Springer, 1970, xiv+256 pages (notes written in collaboration with C. Musili) | Article | MR 0282977

[13] 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 | Article | MR 1660263 | Zbl 0932.14025

[14] Hwang, Jun-Muk Geometry of minimal rational curves on Fano manifolds, School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000) (ICTP Lecture Notes), Volume 6, Abdus Salam International Centre for Theoretical Physics, 2001, pp. 335-393 | MR 1919462 | Zbl 1086.14506

[15] Katz, Nicholas Michael; Sarnak, Peter Random matrices, Frobenius eigenvalues, and monodromy, Colloquium Publications, 45, American Mathematical Society, 1999, xii+419 pages | MR 1659828

[16] Kim, Sung-Ock Noether–Lefschetz locus for surfaces, Trans. Am. Math. Soc., Volume 324 (1991) no. 1, pp. 369-384 | Article | MR 1043861 | Zbl 0739.14019

[17] Kobayashi, Shoshichi; Ochiai, Takushiro Characterizations of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ., Volume 13 (1973), pp. 31-47 | Article | MR 0316745 | Zbl 0261.32013

[18] Kostant, Bertram Lie algebra cohomology and the generalized Borel–Weil theorem, Ann. Math., Volume 74 (1961), pp. 329-387 | Article | MR 0142696 | Zbl 0134.03501

[19] Langer, Adrian Semistable sheaves in positive characteristic, Ann. Math., Volume 159 (2004) no. 1, pp. 251-276 | Article | MR 2051393 | Zbl 1080.14014

[20] Lazarsfeld, Robert Positivity in algebraic geometry. I. Classical Setting: Line Bundles and Linear Series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 48, Springer, 2004, xviii+387 pages | Article | MR 2095471

[21] Lefschetz, Solomon On certain numerical invariants of algebraic varieties with application to abelian varieties, Trans. Am. Math. Soc., Volume 22 (1921) no. 3, pp. 327-406 | Article | MR 1501178 | Zbl 48.0428.03

[22] Mehta, Vikram Bhagvandas; Ramanathan, Annamalai Restriction of stable sheaves and representations of the fundamental group, Invent. Math., Volume 77 (1984) no. 1, pp. 163-172 | Article | MR 751136

[23] Migliore, Juan Carlos; Miró-Roig, Rosa María Ideals of general forms and the ubiquity of the weak Lefschetz property, J. Pure Appl. Algebra, Volume 182 (2003) no. 1, pp. 79-107 | Article | MR 1978001 | Zbl 1041.13011

[24] Migliore, Juan Carlos; Nagel, Uwe Survey article: a tour of the weak and strong Lefschetz properties, J. Commut. Algebra, Volume 5 (2013) no. 3, pp. 329-358 | Article | MR 3161738 | Zbl 1285.13002

[25] Naruki, Isao Some remarks on isolated singularity and their application to algebraic manifolds, Publ. Res. Inst. Math. Sci., Volume 13 (1977) no. 1, pp. 17-46 | Article | MR 0492381 | Zbl 0384.14002

[26] 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. 363-384 | MR 1311356 | Zbl 0837.14033

[27] Ramanan, Sundararaman Holomorphic vector bundles on homogeneous spaces, Topology, Volume 5 (1966), pp. 159-177 | Article | MR 0190947 | Zbl 0138.18602

[28] Reid, Les; Roberts, Leslie G.; Roitman, Moshe On complete intersections and their Hilbert functions, Can. Math. Bull., Volume 34 (1991) no. 4, pp. 525-535 | Article | MR 1136655 | Zbl 0757.13005

[29] Reid, Miles Bogomolov’s theorem c 1 2 4c 2 , Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) (1977), pp. 623-642 | Zbl 0478.14003

[30] Snow, Dennis M. Cohomology of twisted holomorphic forms on Grassmann manifolds and quadric hypersurfaces, Math. Ann., Volume 276 (1986) no. 1, pp. 159-176 | Article | MR 863714 | Zbl 0596.32016

[31] Snow, Dennis M. Vanishing theorems on compact Hermitian symmetric spaces, Math. Z., Volume 198 (1988) no. 1, pp. 1-20 | Article | MR 938025 | Zbl 0631.32025

[32] Stanley, Richard P. Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Algebraic Discrete Methods, Volume 1 (1980) no. 2, pp. 168-184 | Article | MR 578321 | Zbl 0502.05004

[33] Tian, Gang K-stability and Kähler–Einstein metrics, Commun. Pure Appl. Math., Volume 68 (2015) no. 7, pp. 1085-1156 | Article | MR 3352459

[34] Umemura, Hiroshi On a theorem of Ramanan, Nagoya Math. J., Volume 69 (1978), pp. 131-138 | Article | MR 0473243 | Zbl 0345.14017

[35] Voisin, Claire Théorie de Hodge et géométrie algébrique complexe, Contributions in Mathematical and Computational Sciences, 10, Société Mathématique de France, 2002, viii+595 pages | Article | MR 1988456

[36] Wahl, Jonathan M. A cohomological characterization of P n , Invent. Math., Volume 72 (1983) no. 2, pp. 315-322 | Article | MR 700774

[37] Watanabe, Junzo The Dilworth number of Artinian rings and finite posets with rank function, Commutative algebra and combinatorics (Kyoto, 1985) (Advanced Studies in Pure Mathematics), Volume 11, North-Holland, 1987, pp. 303-312 | Article | MR 951211 | Zbl 0648.13010

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