Positivity properties of toric vector bundles
Annales de l'Institut Fourier, Volume 60 (2010) no. 2, pp. 607-640.

We show that a torus-equivariant vector bundle on a complete toric variety is nef or ample if and only if its restriction to every invariant curve is nef or ample, respectively. Furthermore, we show that nef toric vector bundles have a nonvanishing global section at every point and deduce that the underlying vector bundle is trivial if and only if its restriction to every invariant curve is trivial. We apply our methods and results to study, in particular, the vector bundles L that arise as the kernel of the evaluation map H 0 (X,L)𝒪 X L, for ample line bundles L. We give examples of twists of such bundles that are ample but not globally generated.

Nous prouvons qu’un fibré vectoriel équivariant sur une variété torique complète est nef ou ample si et seulement si sa restriction à chaque courbe invariante est nef ou ample, respectivement. Nous montrons également qu’étant donne un fibré vectoriel torique nef et un point xX, il existe une section de non-nulle en x ; on déduit de cela que est trivial si et seulement si sa restriction à chaque courbe invariante est triviale. Nous appliquons ces résultats et méthodes pour étudier en particulier les fibrés vectoriels L , définis en tant que noyau des applications d’évaluation H 0 (X,L)𝒪 X L, ou L est un fibré en droites ample. Finalement, nous donnons des exemples des fibrés vectoriels toriques qui sont amples mais non engendrés par leur sections globales.

DOI: 10.5802/aif.2534
Classification: 14M25, 14F05
Keywords: Toric variety, toric vector bundle
Hering, Milena 1; Mustaţă, Mircea 2; Payne, Sam 3

1 University of Connecticut Department of Mathematics 196 Auditorium Road U-3009 Storrs CT 06269-3009 (USA)
2 University of Michigan Department of Mathematics East Hall Ann Arbor, MI 48109 (USA)
3 Stanford University Department of Mathematics Bldg 380 Stanford, CA 94305 (USA)
     author = {Hering, Milena and Musta\c{t}\u{a}, Mircea and Payne, Sam},
     title = {Positivity properties of toric vector bundles},
     journal = {Annales de l'Institut Fourier},
     pages = {607--640},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {60},
     number = {2},
     year = {2010},
     doi = {10.5802/aif.2534},
     mrnumber = {2667788},
     zbl = {1204.14024},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2534/}
AU  - Hering, Milena
AU  - Mustaţă, Mircea
AU  - Payne, Sam
TI  - Positivity properties of toric vector bundles
JO  - Annales de l'Institut Fourier
PY  - 2010
SP  - 607
EP  - 640
VL  - 60
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2534/
DO  - 10.5802/aif.2534
LA  - en
ID  - AIF_2010__60_2_607_0
ER  - 
%0 Journal Article
%A Hering, Milena
%A Mustaţă, Mircea
%A Payne, Sam
%T Positivity properties of toric vector bundles
%J Annales de l'Institut Fourier
%D 2010
%P 607-640
%V 60
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2534/
%R 10.5802/aif.2534
%G en
%F AIF_2010__60_2_607_0
Hering, Milena; Mustaţă, Mircea; Payne, Sam. Positivity properties of toric vector bundles. Annales de l'Institut Fourier, Volume 60 (2010) no. 2, pp. 607-640. doi : 10.5802/aif.2534. https://aif.centre-mersenne.org/articles/10.5802/aif.2534/

[1] Boucksom, S.; Demailly, J.-P.; Păun, M.; Peternell, T. The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension (preprint, math.AG/0405285) | Zbl

[2] Brion, M.; Kumar, S. Frobenius splitting methods in geometry and representation theory, Progress in Mathematics, 231, Birkhäuser Boston, Inc., Boston, MA, 2005 | MR | Zbl

[3] Bruns, W.; Gubeladze, J.; Trung, N. Normal polytopes, triangulations, and Koszul algebras, Journal für die Reine und Angewandte Mathematik, Volume 485 (1997), p. 123-160. | EuDML | MR | Zbl

[4] Campana, F.; Flenner, H. A characterization of ample vector bundles on a curve, Math. Ann., Volume 287 (1990), pp. 571-575 | DOI | EuDML | MR | Zbl

[5] Cox, D. The homogeneous coordinate ring of a toric variety, J. Algebraic Geom., Volume 4 (1995), pp. 17-50 | MR | Zbl

[6] Di Rocco, S. Generation of k-jets on toric varieties, Math. Z., Volume 231 (1999), pp. 169-188 | DOI | MR | Zbl

[7] Digne, F.; Michel, J. Representations of finite groups of Lie type, London Mathematical Society Student Texts, 21, Cambridge University Press, Cambridge, 1991 | MR | Zbl

[8] Ein, L.; Lazarsfeld, R. Stability and restrictions of Picard bundles, with an application to the normal bundles of elliptic curves, Complex projective geometry (Trieste, 1989/Bergen, 1989) (London Math. Soc. Lecture Note Ser.), Volume 179, Cambridge Univ. Press, Cambridge, 1992, pp. 149-156 | MR | Zbl

[9] Elizondo, J. The ring of global sections of multiples of a line bundle on a toric variety, Proc. Amer. Math. Soc., Volume 125 (1997), pp. 2527-2529 | DOI | MR | Zbl

[10] Ewald, G.; Wessels, U. On the ampleness of invertible sheaves in complete projective toric varieties, Results in Mathematics, Volume 19 (1991), pp. 275-278 | MR | Zbl

[11] Fakhruddin, N. Multiplication maps of linear systems on projective toric surfaces (preprint, arXiv:math.AG/0208178)

[12] Fujino, O. Multiplication maps and vanishing theorems for toric varieties, Math. Z., Volume 257 (2007), pp. 631-641 | DOI | MR | Zbl

[13] Fulton, W. Introduction to toric varieties, Ann. of Math. Stud., 131, Princeton Univ. Press, Princeton, NJ, 1993 (The William H. Rover Lectures in Geometry) | MR | Zbl

[14] Fulton, W.; Lazarsfeld, R. Positive polynomials for ample vector bundles, Ann. Math., Volume 118 (1983) no. 2, pp. 35-60 | DOI | MR | Zbl

[15] Green, M. L. Koszul cohomology and the geometry of projective varieties, J. Differential Geom., Volume 19 (1984), pp. 125-171 | MR | Zbl

[16] Green, M. L. Koszul cohomology and the geometry of projective varieties II, J. Differential Geom., Volume 20 (1984), pp. 279-289 | MR | Zbl

[17] Griffiths, P. Hermitian differential geometry, Chern classes, and positive vector bundles, Global Analysis (Papers in Honor of K. Kodaira) (Princeton Math. Series), Volume 29, Princeton University Press, 1969, pp. 185-251 | MR | Zbl

[18] Haase, Christian; Nill, Benjamin; Paffenholz, Andreas; Santos, Francisco Lattice points in Minkowski sums, Electron. J. Combin., Volume 15 (2008) no. 1, Note 11, pp. 5 | MR | Zbl

[19] Hacon, C. Remarks on Seshadri constants of vector bundles, Ann. Inst. Fourier, Volume 50 (2000), pp. 767-780 | DOI | Numdam | MR | Zbl

[20] Hartshorne, R. Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics, 156, Springer-Verlag, Berlin-New York, 1970 (Notes written in collaboration with C. Musili) | MR | Zbl

[21] Hu, Y.; Keel, S. Mori Dream Spaces and GIT, Michigan Math. J., Volume 48 (2000), pp. 331-348 | DOI | MR | Zbl

[22] Katz, E.; Payne, S. Piecewise polynomials, Minkowski weights, and localization on toric varieties, Algebra Number Theory, Volume 2 (2008), pp. 135-155 | DOI | MR | Zbl

[23] Klyachko, A. Equivariant vector bundles on toral varieties, Math. USSR-Izv., Volume 35 (1990), pp. 337-375 | DOI | MR | Zbl

[24] Kumar, S. Equivariant analogue of Grothendieck’s theorem for vector bundles on 1 , A tribute to C. S. Seshadri (Chennai, 2002) (Trends Math.), Birkhäuser, Basel, 2003, pp. 500-501 | MR | Zbl

[25] Lazarsfeld, R. Positivity in algebraic geometry I, II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 48 and 49, Springer-Verlag, Berlin, 2004 | MR | Zbl

[26] Liu, J.; Trotter Jr., L.; Ziegler, G. On the height of the minimal Hilbert basis, Results in Mathematics, Volume 23 (1993), pp. 374-376 | MR | Zbl

[27] Manivel, L. Théorèmes d’annulation sur certaines variétés projectives, Comment. Math. Helv., Volume 71 (1996), pp. 402-425 | DOI | MR | Zbl

[28] Mustaţă, Mircea Vanishing theorems on toric varieties, Tohoku Math. J. (2), Volume 54 (2002) no. 3, pp. 451-470 | DOI | MR | Zbl

[29] Oda, T. Problems on Minkowski sums of convex lattice polytopes (preprint, arXiv:0812.1418)

[30] Oda, T. Convex bodies and algebraic geometry, Ergeb. Math. grenzgeb., Springer Verlag, Berlin Heidelberg New York, 1988 no. 3, vol. 15 | MR | Zbl

[31] Okonek, C.; Schneider, M.; Spindler, H. Vector bundles on complex projective spaces, Progress in Mathematics, 3, Birkhäuser, Boston, Mass., 1980 | MR | Zbl

[32] Paranjape, K.; Ramanan, S. On the canonical ring of a curve, Algebraic geometry and commutative algebra, Volume II, Kinokuriya, 1988, pp. 503-516 | MR | Zbl

[33] Payne, S. Equivariant Chow cohomology of toric varieties, Math. Res. Lett., Volume 13 (2006), pp. 29-41 | MR | Zbl

[34] Payne, S. Stable base loci, movable curves, and small modifications, for toric varieties, Math. Z., Volume 253 (2006), pp. 421-431 | DOI | MR | Zbl

[35] Payne, S. Moduli of toric vector bundles, Compositio Math., Volume 144 (2008), pp. 1199-1213 | DOI | MR | Zbl

[36] Payne, S. Toric vector bundles, branched covers of fans, and the resolution property, J. Alg. Geom., Volume 18 (2009), pp. 1-36 | DOI | MR | Zbl

Cited by Sources: