On Ampleness of vector bundles

Misra, Snehajit; Ray, Nabanita

Comptes Rendus. Mathématique, Tome 359 (2021) no. 6, pp. 763-772.

In this article, we give a necessary and sufficient condition for ampleness of semistable vector bundles with vanishing discriminant on a smooth projective variety $X$. As an application, we show ampleness of some special vector bundles on certain ruled surfaces. We prove similar results for parabolic ampleness.

Classification : 14J60, 14N05, 14J40, 14C17

[1] Beauville, Arnaud An ampleness criterion for rank 2 vector bundles on surfaces, Vietnam J. Math., Volume 48 (2020) no. 1, pp. 125-129 | DOI | MR | Zbl

[2] Biswas, Indranil Chern classes for parabolic bundles, J. Math. Kyoto Univ., Volume 37 (1997) no. 4, pp. 597-613 | MR | Zbl

[3] Biswas, Indranil Parabolic ample bundles, Math. Ann., Volume 307 (1997) no. 3, pp. 511-529 | DOI | MR | Zbl

[4] Biswas, Indranil Parabolic bundles as orbifold bundles, Duke Math. J., Volume 88 (1997) no. 2, pp. 305-325 | MR | Zbl

[5] Biswas, Indranil; Bruzzo, Ugo On semistable principal bundles over a complex projective manifold, Int. Math. Res. Not., Volume 2008 (2008), rnn035, 28 pages | Zbl

[6] Biswas, Indranil; Hanumanthu, Krishna; Nagaraj, Donihakkalu S. Positivity of vector bundles on homogeneous varieties, Int. J. Math., Volume 31 (2020) no. 12, 2050097, 11 pages | MR | Zbl

[7] Biswas, Indranil; Mehrotra, Sukhendu; Parameswaran, A. J. Nef line bundles on flag bundles on a curve over $\overline{{𝔽}_p}$, Arch. Math., Volume 101 (2013) no. 2, pp. 105-110 | DOI | MR | Zbl

[8] Brînzǎnescu, Vasile Algebraic 2-vector bundles on ruled surfaces, Ann. Univ. Ferrara, Nuova Ser., Sez. VII, Volume 37 (1991), pp. 55-64 | MR | Zbl

[9] Fulger, Mihai The cones of effective cycles on projective bundles over curves, Math. Z., Volume 269 (2011) no. 1-2, pp. 449-459 | DOI | MR | Zbl

[10] Hartshorne, Robin Ample Vector bundles, Publ. Math., Inst. Hautes Étud. Sci., Volume 29 (1966), pp. 63-94 | Numdam | Zbl

[11] Hartshorne, Robin Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer, 1977 | Zbl

[12] Hering, Milena; Mustaţă, Mircea; Payne, Sam Positivity properties of toric vector bundles, Ann. Inst. Fourier, Volume 60 (2010) no. 2, pp. 607-640 | DOI | Numdam | MR | Zbl

[13] Huybrechts, Daniel; Lehn, Manfred The Geometry of Moduli Spaces of Sheaves, Cambridge University Press, 2010 | DOI | Zbl

[14] 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 | Zbl

[15] Lazarsfeld, Robert Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 49, Springer, 204 | Zbl

[16] Lübke, Martin Stability of Einstein–Hermitian vector bundles, Manuscr. Math., Volume 42 (1983), pp. 245-257 | DOI | MR | Zbl

[17] Mehta, Vikram B.; Nori, Madhav V. Semistable sheaves on homogeneous spaces and abelian varieties, Proc. Indian Acad. Sci., Math. Sci., Volume 93 (1984) no. 1, pp. 1-12 | DOI | MR | Zbl

[18] Mukai, Shigeru Semi-homogeneous vector bundles on an abelian variety, J. Math. Kyoto Univ., Volume 18 (1978) no. 2, pp. 239-272 | MR | Zbl

[19] Nakayama, Noboru Normalized Tautological divisors of semi-stable vector bundles, Free resolutions of coordinate rings of projective varieties and related topics (Sūrikaisekikenkyūsho Kōkyūroku), Volume 1075, RIMS, Kyoto University, 1999, pp. 167-173 | MR | Zbl

[20] Schneider, Michael; Tancredi, Alessandro Positive vector bundles on complex surfaces, Manuscr. Math., Volume 50 (1985), pp. 133-144 | DOI | MR | Zbl

[21] Serrano, Fernando Strictly nef divisors and Fano threefolds, J. Reine Angew. Math., Volume 464 (1995), pp. 187-206 | MR | Zbl

[22] Umemura, Hiroshi Some results in the theory of vector bundles, Nagoya Math. J., Volume 52 (1973), pp. 97-128 | DOI | MR | Zbl