Geodesic
Chern classes of ample bundles
Sbornik. Mathematics, Tome 14 (1971) no. 1, pp. 85-98 
V. M. Barenbaum. Chern classes of ample bundles. Sbornik. Mathematics, Tome 14 (1971) no. 1, pp. 85-98. http://geodesic.mathdoc.fr/item/SM_1971_14_1_a4/
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Voir la notice de l'article provenant de la source Math-Net.Ru

In the article “Ample vector bundles”, Publ. Math., No 29, R. Hartshorne has extended the notion of ample vector bundle to vector bundles of arbitrary rank and has raised the following question. Let $\mathscr E$ be an ample vector bundle over a nonsingular algebraic variety $X$ and assume that the rank of $\mathscr E$ is equal to $n$. Is it true that the $i$th Chern class $c_i(\mathscr E)$ is numerically positive for $i\leqslant n$? In this paper it is proved that in the case $\operatorname{dim}X=2$ the degree of the point-cycle $c_2(\mathscr E)$ is positive. Bibliography: 8 titles.

[1] Anneaux de Chow et applications, Seminaire C. Chevalley, 'Ecole Norm, Supér., Paris, 1958

[2] Ph. A. Griffiths, “Hermitian Differential Geometry and the theory of Positive and Ample Harmonic Vector Bundles”, J. Math. and Mech., 14:1 (1965), 117–140 | MR | Zbl

[3] A. Grothendieck, “Elements de geometrie algebrique”, Publ. Math. IHES, 1960, no. 4; 1961, no. 8

[4] A. Grothendieck, “La theorie des classes de Chern”, Bull. Soc. Math. France, 86:2 (1958), 137–154 | MR | Zbl

[5] R. Hartshorne, “Ample vector bundles”, Publ. Math. IHES, 1966, no. 29, 63–94 | MR | Zbl

[6] Zh. P. Serr, “Lokalnaya algebra i kratnosti peresechenii”, Matematika, 7:5 (1963), 3–95

[7] O. Zarisskii, “Teoriya algebraicheskikh punktov”, Matematika, 4:2 (1960), 3–24 | MR

[8] O. Zariski, “Pensils on an algebraic variety and a new proof of a theorem of Bertini”, Trans. Amer. Math. Soc., 50 (1941), 48–70 | DOI | MR | Zbl