logo CRAS
Comptes Rendus. Mathématique
Algebraic geometry
On Ampleness of vector bundles
Comptes Rendus. Mathématique, Volume 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.

Received:
Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/crmath.222
Classification: 14J60,  14N05,  14J40,  14C17
Snehajit Misra 1; Nabanita Ray 2

1. Tata Institute of Fundamental Research (TIFR), Homi Bhabha Road, Colaba, Mumbai 400005, India.
2. Tata Institute of Fundamental Research (TIFR) Homi Bhabha Road, Colaba, Mumbai 400005, India.
@article{CRMATH_2021__359_6_763_0,
     author = {Snehajit Misra and Nabanita Ray},
     title = {On {Ampleness} of vector bundles},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {763--772},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {6},
     year = {2021},
     doi = {10.5802/crmath.222},
     zbl = {07390659},
     language = {en},
}
TY  - JOUR
AU  - Snehajit Misra
AU  - Nabanita Ray
TI  - On Ampleness of vector bundles
JO  - Comptes Rendus. Mathématique
PY  - 2021
DA  - 2021///
SP  - 763
EP  - 772
VL  - 359
IS  - 6
PB  - Académie des sciences, Paris
UR  - https://zbmath.org/?q=an%3A07390659
UR  - https://doi.org/10.5802/crmath.222
DO  - 10.5802/crmath.222
LA  - en
ID  - CRMATH_2021__359_6_763_0
ER  - 
%0 Journal Article
%A Snehajit Misra
%A Nabanita Ray
%T On Ampleness of vector bundles
%J Comptes Rendus. Mathématique
%D 2021
%P 763-772
%V 359
%N 6
%I Académie des sciences, Paris
%U https://doi.org/10.5802/crmath.222
%R 10.5802/crmath.222
%G en
%F CRMATH_2021__359_6_763_0
Snehajit Misra; Nabanita Ray. On Ampleness of vector bundles. Comptes Rendus. Mathématique, Volume 359 (2021) no. 6, pp. 763-772. doi : 10.5802/crmath.222. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.222/

[1] Arnaud Beauville An ampleness criterion for rank 2 vector bundles on surfaces, Vietnam J. Math., Volume 48 (2020) no. 1, pp. 125-129 | Article | MR 4068374 | Zbl 1461.14058

[2] Indranil Biswas Chern classes for parabolic bundles, J. Math. Kyoto Univ., Volume 37 (1997) no. 4, pp. 597-613 | MR 1625964 | Zbl 0929.14005

[3] Indranil Biswas Parabolic ample bundles, Math. Ann., Volume 307 (1997) no. 3, pp. 511-529 | Article | MR 1437053 | Zbl 0877.14013

[4] Indranil Biswas Parabolic bundles as orbifold bundles, Duke Math. J., Volume 88 (1997) no. 2, pp. 305-325 | MR 1455522 | Zbl 0955.14010

[5] Indranil Biswas; Ugo Bruzzo On semistable principal bundles over a complex projective manifold, Int. Math. Res. Not., Volume 2008 (2008), rnn035, 28 pages | Zbl 1183.14065

[6] Indranil Biswas; Krishna Hanumanthu; Donihakkalu S. Nagaraj Positivity of vector bundles on homogeneous varieties, Int. J. Math., Volume 31 (2020) no. 12, 2050097, 11 pages | MR 4184429 | Zbl 1458.14011

[7] Indranil Biswas; Sukhendu Mehrotra; A. J. Parameswaran Nef line bundles on flag bundles on a curve over 𝔽 p ¯, Arch. Math., Volume 101 (2013) no. 2, pp. 105-110 | Article | MR 3089765 | Zbl 1279.14042

[8] Vasile Brînzǎnescu Algebraic 2-vector bundles on ruled surfaces, Ann. Univ. Ferrara, Nuova Ser., Sez. VII, Volume 37 (1991), pp. 55-64 | MR 1206993 | Zbl 0795.14010

[9] Mihai Fulger The cones of effective cycles on projective bundles over curves, Math. Z., Volume 269 (2011) no. 1-2, pp. 449-459 | Article | MR 2836078 | Zbl 1230.14047

[10] Robin Hartshorne Ample Vector bundles, Publ. Math., Inst. Hautes Étud. Sci., Volume 29 (1966), pp. 63-94 | Numdam | Zbl 0173.49003

[11] Robin Hartshorne Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer, 1977 | Zbl 0367.14001

[12] Milena Hering; Mircea Mustaţă; Sam Payne Positivity properties of toric vector bundles, Ann. Inst. Fourier, Volume 60 (2010) no. 2, pp. 607-640 | Article | Numdam | MR 2667788 | Zbl 1204.14024

[13] Daniel Huybrechts; Manfred Lehn The Geometry of Moduli Spaces of Sheaves, Cambridge University Press, 2010 | Article | Zbl 1206.14027

[14] Robert Lazarsfeld Positivity in algebraic geometry. I. Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 48, Springer, 2004 | Zbl 1093.14501

[15] Robert Lazarsfeld Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 49, Springer, 204 | Zbl 1093.14500

[16] Martin Lübke Stability of Einstein–Hermitian vector bundles, Manuscr. Math., Volume 42 (1983), pp. 245-257 | Article | MR 701206 | Zbl 0558.53037

[17] Vikram B. Mehta; Madhav V. Nori Semistable sheaves on homogeneous spaces and abelian varieties, Proc. Indian Acad. Sci., Math. Sci., Volume 93 (1984) no. 1, pp. 1-12 | Article | MR 796768 | Zbl 0592.14017

[18] Shigeru Mukai Semi-homogeneous vector bundles on an abelian variety, J. Math. Kyoto Univ., Volume 18 (1978) no. 2, pp. 239-272 | MR 498572 | Zbl 0417.14029

[19] Noboru Nakayama 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 1715587 | Zbl 0977.14021

[20] Michael Schneider; Alessandro Tancredi Positive vector bundles on complex surfaces, Manuscr. Math., Volume 50 (1985), pp. 133-144 | Article | MR 784141 | Zbl 0572.32015

[21] Fernando Serrano Strictly nef divisors and Fano threefolds, J. Reine Angew. Math., Volume 464 (1995), pp. 187-206 | MR 1340341 | Zbl 0826.14006

[22] Hiroshi Umemura Some results in the theory of vector bundles, Nagoya Math. J., Volume 52 (1973), pp. 97-128 | Article | MR 337968 | Zbl 0271.14005

Cited by Sources: