On stability of tangent bundle of toric varieties. (English) Zbl 1480.14033
Let \(X\) be a non-singular complex projective toric variety. In this article, the authors obtain various results on the stability and semi-stability of tangent bundle \(TX\) of such varieties. Their main results are as follows:
- 1.
- Let \(X\) be the Hirzebruch surface obtained by projectivization of the bundle \(\mathcal{O}_{\mathbb{P}^n} \oplus \mathcal{O}_{\mathbb{P}^n}(-m).\) The tangent bundle of \(X\) is unstable with respect to every polarization if \(m \geq 2\) (Theorem 6.2).
- 2.
- In Theorem 7.1, they provide an alternate proof of the result that \(T\mathbb{P}^n\) is stable with respect to the anti-canonical polarization.
- 3.
- In Theorem 8.1, authors prove that the tangent bundle of the Fano toric variety \(\mathbb{P}(\mathcal{O}_{\mathbb{P}^n} \oplus \mathcal{O}_{\mathbb{P}^n}(-m))\), where \(0 < m \leq n-1\) and \(n \geq 3\) is unstable with respect to the anti-canonical polarization.
- ●
- In Theorem 9.3, the question of semi-stability of the tangent bundle is completely solved for Fano toric 4-folds with Picard number \(\leq 2.\)
Reviewer: Amit Tripathi (Bangalore)
MSC:
| 14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |
| 32L05 | Holomorphic bundles and generalizations |
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
References:
| [1] | Atiyah, MF, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc., 85, 181-207 (1957) · Zbl 0078.16002 · doi:10.1090/S0002-9947-1957-0086359-5 |
| [2] | Aubin, T., Équations du type Monge-Ampère sur les variétés kählériennes compactes, Bull. Sci. Math., 102, 63-95 (1978) · Zbl 0374.53022 |
| [3] | Birkar, C.; Cascini, P.; Hacon, CD; McKernan, J., Existence of minimal models for varieties of log general type, J. Amer. Math. Soc., 23, 405-468 (2010) · Zbl 1210.14019 · doi:10.1090/S0894-0347-09-00649-3 |
| [4] | Batyrev, V. V., Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom., 3, 493-535 (1994) · Zbl 0829.14023 |
| [5] | Batyrev, VV, On the classification of toric Fano 4-folds, J. Math. Sci., 94, 1021-1050 (1999) · Zbl 0929.14024 · doi:10.1007/BF02367245 |
| [6] | Biswas, I.; Parameswaran, AJ, On the equivariant reduction of structure group of a principal bundle to a Levi subgroup, J. Math. Pures Appl., 85, 54-70 (2006) · Zbl 1159.14307 · doi:10.1016/j.matpur.2005.10.007 |
| [7] | Biswas, I.; Dumitrescu, S.; Lehn, M., On the stability of flat complex vector bundles over parallelizable manifolds, Com. Ren. Math. Acad. Sci. Paris, 358, 151-158 (2020) · Zbl 1444.32026 |
| [8] | Fahlaoui, R., Stabilité du fibré tangent des surfaces de del Pezzo, Math. Ann., 283, 171-176 (1989) · Zbl 0672.14009 · doi:10.1007/BF01457509 |
| [9] | Fulton W, Introduction to Toric Varieties, Annals of Mathematics Studies, 131, (1993) (Princeton: Princeton University Press) · Zbl 0813.14039 |
| [10] | Huybrechts D and Lehn M, The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, (1997) (Braunschweig: Friedr. Vieweg & Sohn) xiv+269 pp., ISBN: 3-528-06907-4 · Zbl 0872.14002 |
| [11] | Kleinschmidt, P., A classification of toric varieties with few generators, Aequationes Math., 35, 254-266 (1988) · Zbl 0664.14018 · doi:10.1007/BF01830946 |
| [12] | Klyachko A A, Equivariant bundles over toric varieties, Izv. Akad. Nauk SSSR Ser. Mat.53 (1989) 1001-1039, 1135; translation in Math. USSR-Izv. 35(2) (1990) 337-375 · Zbl 0706.14010 |
| [13] | Kobayashi S, Differential Geometry of Complex Vector Bundles, Publications of the Mathematical Society of Japan, vol. 15 (1987) (Princeton: Princeton University Press) · Zbl 0708.53002 |
| [14] | Kool M, Moduli spaces of sheaves on toric varieties, Ph.D. thesis (2010) (University of Oxford) · Zbl 1227.14018 |
| [15] | Kool, M., Fixed point loci of moduli spaces of sheaves on toric varieties, Adv. Math., 227, 1700-1755 (2011) · Zbl 1227.14018 · doi:10.1016/j.aim.2011.04.002 |
| [16] | Lübke, M., Stability of Einstein-Hermitian vector bundles, Manuscripta Math., 42, 245-257 (1983) · Zbl 0558.53037 · doi:10.1007/BF01169586 |
| [17] | Nakagawa, Y., Einstein-Kähler toric Fano fourfolds, Tohoku Math. J., 45, 297-310 (1993) · Zbl 0778.53040 · doi:10.2748/tmj/1178225923 |
| [18] | Nakagawa, Y., Classification of Einstein-Kähler toric Fano fourfolds, Tohoku Math. J., 46, 125-133 (1994) · Zbl 0838.32008 · doi:10.2748/tmj/1178225805 |
| [19] | Oda T, Convex Bodies and Algebraic Geometry: An Introduction to the Theory of Toric Varieties, translated from the Japanese, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 15 (1988) (Berlin: Springer-Verlag) · Zbl 0628.52002 |
| [20] | Perling, M., Graded rings and equivariant sheaves on toric varieties, Math. Nachr., 263, 264, 181-197 (2004) · Zbl 1043.14013 · doi:10.1002/mana.200310130 |
| [21] | Steffens, A., On the stability of the tangent bundle of Fano manifolds, Math. Ann., 304, 635-643 (1996) · Zbl 0865.14023 · doi:10.1007/BF01446311 |
| [22] | Yau, S-T, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math., 31, 339-411 (1978) · Zbl 0369.53059 · doi:10.1002/cpa.3160310304 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
