Stability of the tangent bundles of complete intersections and effective restriction

Liu, Jie

doi:10.5802/aif.3435

Stability of the tangent bundles of complete intersections and effective restriction
[Stabilité de fibré tangent d’intersections complètes et ses restrictions]

Liu, Jie ¹

¹ Institute of Mathematics Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing, 100190 (China)

Annales de l'Institut Fourier, Tome 71 (2021) no. 4, pp. 1601-1634.

Résumé
Abstract

Soient $M$ un espace hermitien symétrique irréductible de type compact et de dimension $(n + r)$ avec $n \geq 3$ et $r \geq 1$ , $𝒪_{M} (1)$ le générateur ample de $pic (M)$ . Soit $Y = H_{1} \cap \dots \cap H_{r}$ une intersection complète lisse de dimension $n$ où $H_{i} \in | 𝒪_{M} (d_{i}) |$ avec $d_{i} \geq 2$ . Nous montrons un théorème d’annulation pour le faisceau tordu des germes de $p$ -formes holomorphes $Ω_{Y}^{p} (ℓ)$ . 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 $pic (Y) \to pic (X)$ 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 $ℙ^{n + 1}$ 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 $n \geq 3$ and $r \geq 1$ , 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 $Y = H_{1} \cap \dots \cap H_{r}$ be a smooth complete intersection of dimension $n$ , where $H_{i} \in | 𝒪_{M} (d_{i}) |$ with $d_{i} \geq 2$ . 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 $pic (Y) \to pic (X)$ 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 $ℙ^{n + 1}$ 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.

Reçu le : 2018-05-30
Accepté le : 2019-01-17
Accepté après révision le : 2019-10-22
Première publication : 2021-12-15
Publié le : 2022-03-15

DOI : 10.5802/aif.3435

Classification : 14M10, 14J70, 32M15, 32M25, 32Q26
Keywords: stability, tangent bundle, Lefschetz property, complete intersection, Hermitian symmetric space
Mot clés : stabilité, fibré tangent, propriété de Lefschetz, intersection complète, espace hermitien symétrique

Affiliations des auteurs :

Liu, Jie ¹

¹ Institute of Mathematics Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing, 100190 (China)

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{AIF_2021__71_4_1601_0,
     author = {Liu, Jie},
     title = {Stability of the tangent bundles of complete intersections and effective restriction},
     journal = {Annales de l'Institut Fourier},
     pages = {1601--1634},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {71},
     number = {4},
     year = {2021},
     doi = {10.5802/aif.3435},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3435/}
}

TY  - JOUR
AU  - Liu, Jie
TI  - Stability of the tangent bundles of complete intersections and effective restriction
JO  - Annales de l'Institut Fourier
PY  - 2021
SP  - 1601
EP  - 1634
VL  - 71
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3435/
DO  - 10.5802/aif.3435
LA  - en
ID  - AIF_2021__71_4_1601_0
ER  -

%0 Journal Article
%A Liu, Jie
%T Stability of the tangent bundles of complete intersections and effective restriction
%J Annales de l'Institut Fourier
%D 2021
%P 1601-1634
%V 71
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3435/
%R 10.5802/aif.3435
%G en
%F AIF_2021__71_4_1601_0

Liu, Jie. Stability of the tangent bundles of complete intersections and effective restriction. Annales de l'Institut Fourier, Tome 71 (2021) no. 4, pp. 1601-1634. doi : 10.5802/aif.3435. https://aif.centre-mersenne.org/articles/10.5802/aif.3435/

Bibliographie
Cité par

[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] 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 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

[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 | Zbl

[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) | DOI | MR

[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 | DOI | MR | Zbl

[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 | Zbl

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

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

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

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

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

[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 | DOI | MR

[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 | DOI | MR | Zbl

[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 | DOI | MR

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | Zbl

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

[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 | DOI | MR | Zbl

[29] Reid, Miles Bogomolov’s theorem $c_{1}^{2} \leq 4 c_{2}$ , Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) (1977), pp. 623-642 | Zbl

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

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

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

[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 | DOI | MR

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

[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 | DOI | MR | Zbl

Cité par Sources :

ANNALES DE L'INSTITUT FOURIER

ARTICLES À PARAÎTRE

FEUILLETER

PRESENTATION

COMITÉ DE RÉDACTION

SOUMETTRE UN ARTICLE

ABONNEMENT