Positivity properties of toric vector bundles

[Positivité pour les fibrés vectoriels toriques]

Hering, Milena ; Mustaţă, Mircea ; Payne, Sam

Annales de l'Institut Fourier, Tome 60 (2010), p. 607-640 / Harvested from Numdam

Résumé

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 $\mathcal{E}$ et un point $x\in X$, il existe une section de $\mathcal{E}$ non-nulle en $x$ ; on déduit de cela que $\mathcal{E}$ 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 $\mathcal{M}_L$, définis en tant que noyau des applications d’évaluation $H^0(X,L) \otimes \mathcal{O}_X \rightarrow L$, 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.

Abstract

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 $\mathcal{M}_L$ that arise as the kernel of the evaluation map $H^0(X,L) \otimes \mathcal{O}_X \rightarrow L$, for ample line bundles $L$. We give examples of twists of such bundles that are ample but not globally generated.

Détail

Publié le : 2010-01-01

MR 2667788 | Zbl 1204.14024

DOI : https://doi.org/10.5802/aif.2534
Classification:  14M25,  14F05
Mots clés: variété torique, fibré vectoriel torique

Bibliographie

[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 1267.32017

[2] Brion, M.; Kumar, S. Frobenius splitting methods in geometry and representation theory, Birkhäuser Boston, Inc., Boston, MA, Progress in Mathematics, Tome 231 (2005) | MR 2107324 | Zbl 1072.14066

[3] Bruns, W.; Gubeladze, J.; Trung, N. Normal polytopes, triangulations, and Koszul algebras, Journal für die Reine und Angewandte Mathematik, Tome 485 (1997), p. 123-160. | MR 1442191 | Zbl 0866.20050

[4] Campana, F.; Flenner, H. A characterization of ample vector bundles on a curve, Math. Ann., Tome 287 (1990), pp. 571-575 | Article | MR 1066815 | Zbl 0728.14033

[5] Cox, D. The homogeneous coordinate ring of a toric variety, J. Algebraic Geom., Tome 4 (1995), pp. 17-50 | MR 1299003 | Zbl 0846.14032

[6] Di Rocco, S. Generation of $k$-jets on toric varieties, Math. Z., Tome 231 (1999), pp. 169-188 | Article | MR 1696762 | Zbl 0941.14020

[7] Digne, F.; Michel, J. Representations of finite groups of Lie type, Cambridge University Press, Cambridge, London Mathematical Society Student Texts, Tome 21 (1991) | MR 1118841 | Zbl 0815.20014

[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), Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Tome 179 (1992), pp. 149-156 | MR 1201380 | Zbl 0768.14012

[9] Elizondo, J. The ring of global sections of multiples of a line bundle on a toric variety, Proc. Amer. Math. Soc., Tome 125 (1997), pp. 2527-2529 | Article | MR 1401739 | Zbl 0883.14002

[10] Ewald, G.; Wessels, U. On the ampleness of invertible sheaves in complete projective toric varieties, Results in Mathematics, Tome 19 (1991), pp. 275-278 | MR 1100674 | Zbl 0739.14031

[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., Tome 257 (2007), pp. 631-641 | Article | MR 2328817 | Zbl 1129.14029

[13] Fulton, W. Introduction to toric varieties, Princeton Univ. Press, Princeton, NJ, Ann. of Math. Stud., Tome 131 (1993) (The William H. Rover Lectures in Geometry) | MR 1234037 | Zbl 0813.14039

[14] Fulton, W.; Lazarsfeld, R. Positive polynomials for ample vector bundles, Ann. Math., Tome 118 (1983) no. 2, pp. 35-60 | Article | MR 707160 | Zbl 0537.14009

[15] Green, M. L. Koszul cohomology and the geometry of projective varieties, J. Differential Geom., Tome 19 (1984), pp. 125-171 | MR 739785 | Zbl 0559.14008

[16] Green, M. L. Koszul cohomology and the geometry of projective varieties II, J. Differential Geom., Tome 20 (1984), pp. 279-289 | MR 772134 | Zbl 0559.14009

[17] Griffiths, P. Hermitian differential geometry, Chern classes, and positive vector bundles, Global Analysis (Papers in Honor of K. Kodaira), Princeton University Press (Princeton Math. Series) Tome 29 (1969), pp. 185-251 | MR 258070 | Zbl 0201.24001

[18] Haase, Christian; Nill, Benjamin; Paffenholz, Andreas; Santos, Francisco Lattice points in Minkowski sums, Electron. J. Combin., Tome 15 (2008) no. 1, Note 11, pp. 5 | MR 2398828 | Zbl 1160.52009

[19] Hacon, C. Remarks on Seshadri constants of vector bundles, Ann. Inst. Fourier, Tome 50 (2000), pp. 767-780 | Article | Numdam | MR 1779893 | Zbl 0956.14034

[20] Hartshorne, R. Ample subvarieties of algebraic varieties, Springer-Verlag, Berlin-New York, Lecture Notes in Mathematics, Tome 156 (1970) (Notes written in collaboration with C. Musili) | MR 282977 | Zbl 0208.48901

[21] Hu, Y.; Keel, S. Mori Dream Spaces and GIT, Michigan Math. J., Tome 48 (2000), pp. 331-348 | Article | MR 1786494 | Zbl 1077.14554

[22] Katz, E.; Payne, S. Piecewise polynomials, Minkowski weights, and localization on toric varieties, Algebra Number Theory, Tome 2 (2008), pp. 135-155 | Article | MR 2377366 | Zbl 1158.14042

[23] Klyachko, A. Equivariant vector bundles on toral varieties, Math. USSR-Izv., Tome 35 (1990), pp. 337-375 | Article | MR 1024452 | Zbl 0706.14010

[24] Kumar, S. Equivariant analogue of Grothendieck’s theorem for vector bundles on $\mathbb{P}^1$, A tribute to C. S. Seshadri (Chennai, 2002), Birkhäuser, Basel (Trends Math.) (2003), p. 500-501 | MR 2017599 | Zbl 1054.14528

[25] Lazarsfeld, R. Positivity in algebraic geometry I, II, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Tome 48 and 49 (2004) | MR 2095471 | Zbl 1093.14500

[26] Liu, J.; Trotter Jr., L.; Ziegler, G. On the height of the minimal Hilbert basis, Results in Mathematics, Tome 23 (1993), pp. 374-376 | MR 1215222 | Zbl 0779.52002

[27] Manivel, L. Théorèmes d’annulation sur certaines variétés projectives, Comment. Math. Helv., Tome 71 (1996), pp. 402-425 | Article | MR 1418945 | Zbl 0883.14004

[28] Mustaţă, Mircea Vanishing theorems on toric varieties, Tohoku Math. J. (2), Tome 54 (2002) no. 3, pp. 451-470 | Article | MR 1916637 | Zbl 1092.14064

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

[30] Oda, T. Convex bodies and algebraic geometry, Springer Verlag, Berlin Heidelberg New York, Ergeb. Math. grenzgeb. (1988) no. 3, vol. 15 | MR 922894 | Zbl 0628.52002

[31] Okonek, C.; Schneider, M.; Spindler, H. Vector bundles on complex projective spaces, Birkhäuser, Boston, Mass., Progress in Mathematics, Tome 3 (1980) | MR 561910 | Zbl 0438.32016

[32] Paranjape, K.; Ramanan, S. On the canonical ring of a curve, Algebraic geometry and commutative algebra, Kinokuriya, Tome II (1988), pp. 503-516 | MR 977775 | Zbl 0699.14041

[33] Payne, S. Equivariant Chow cohomology of toric varieties, Math. Res. Lett., Tome 13 (2006), pp. 29-41 | MR 2199564 | Zbl 1094.14036

[34] Payne, S. Stable base loci, movable curves, and small modifications, for toric varieties, Math. Z., Tome 253 (2006), pp. 421-431 | Article | MR 2218709 | Zbl 1097.14007

[35] Payne, S. Moduli of toric vector bundles, Compositio Math., Tome 144 (2008), pp. 1199-1213 | Article | MR 2457524 | Zbl 1159.14021

[36] Payne, S. Toric vector bundles, branched covers of fans, and the resolution property, J. Alg. Geom., Tome 18 (2009), pp. 1-36 | Article | MR 2448277 | Zbl 1161.14039