Some combinatorics of binomial coefficients and the Bloch-Gieseker property for some homogeneous bundles
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Abstract:
A vector bundle has the Bloch-Gieseker property if all its Chern classes are numerically positive. In this paper we show that the non-ample bundle $\Omega ^{p}_{\mathbb {P}_{n}}(p+1)$ has the Bloch-Gieseker property, except for two cases, in which the top Chern classes are trivial and the other Chern classes are positive. Our method is to reduce the problem to showing, e.g. the positivity of the coefficient of $t^{k}$ in the rational function $\frac {(1+t)^{\binom n p} (1+3t)^{\binom {n}{p-2}} \cdots (1+(p-1)t)^{\binom n2} (1+(p+1)t)}{(1+2t)^{\binom {n}{p-1}} (1+4t)^{\binom {n}{p-3}} \cdots (1+pt)^{\binom {n}{1}}}$ (for $p$ even).References
- Wolf Barth and Klaus Hulek, Monads and moduli of vector bundles, Manuscripta Math. 25 (1978), no. 4, 323–347. MR 509589, DOI 10.1007/BF01168047
- Spencer Bloch and David Gieseker, The positivity of the Chern classes of an ample vector bundle, Invent. Math. 12 (1971), 112–117. MR 297773, DOI 10.1007/BF01404655
- Mei-Chu Chang, Some remarks on Buchsbaum bundles, J. Pure Appl. Algebra 152 (2000), no. 1-3, 49–55. Commutative algebra, homological algebra and representation theory (Catania/Genoa/Rome, 1998). MR 1783984, DOI 10.1016/S0022-4049(99)00132-2
- Lawrence Ein, Generalized null correlation bundles, Nagoya Math. J. 111 (1988), 13–24. MR 961214, DOI 10.1017/S0027763000000970
- Ein, L., Private communication.
- G. Elencwajg, A. Hirschowitz, and M. Schneider, Les fibres uniformes de rang au plus $n$ sur $\textbf {P}_{n}(\textbf {C})$ sont ceux qu’on croit, Vector bundles and differential equations (Proc. Conf., Nice, 1979), Progr. Math., vol. 7, Birkhäuser, Boston, Mass., 1980, pp. 37–63 (French). MR 589220
- William Fulton and Robert Lazarsfeld, Positive polynomials for ample vector bundles, Ann. of Math. (2) 118 (1983), no. 1, 35–60. MR 707160, DOI 10.2307/2006953
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- Balwant Singh, Relations between certain numerical characters of singularities, J. Pure Appl. Algebra 16 (1980), no. 1, 99–108. MR 549707, DOI 10.1016/0022-4049(80)90045-6
- Robin Hartshorne, Varieties of small codimension in projective space, Bull. Amer. Math. Soc. 80 (1974), 1017–1032. MR 384816, DOI 10.1090/S0002-9904-1974-13612-8
- G. Horrocks, Examples of rank three vector bundles on five-dimensional projective space, J. London Math. Soc. (2) 18 (1978), no. 1, 15–27. MR 502651, DOI 10.1112/jlms/s2-18.1.15
- G. Horrocks and D. Mumford, A rank $2$ vector bundle on $\textbf {P}^{4}$ with $15,000$ symmetries, Topology 12 (1973), 63–81. MR 382279, DOI 10.1016/0040-9383(73)90022-0
- Kumar, M., Peterson, C., Rao, P., Construction of low rank vector bundles on $\mathbb P^4$ and $\mathbb P^5$ (preprint).
- Katz, S., Stromme, S., A Maple package for intersection theory.
- R. Lazarsfeld and A. Van de Ven, Topics in the geometry of projective space, DMV Seminar, vol. 4, Birkhäuser Verlag, Basel, 1984. Recent work of F. L. Zak; With an addendum by Zak. MR 808175, DOI 10.1007/978-3-0348-9348-0
- Christian Okonek, Michael Schneider, and Heinz Spindler, Vector bundles on complex projective spaces, Progress in Mathematics, vol. 3, Birkhäuser, Boston, Mass., 1980. MR 561910
- Hiroshi Tango, An example of indecomposable vector bundle of rank $n-1$ on $P^{n}$, J. Math. Kyoto Univ. 16 (1976), no. 1, 137–141. MR 401766, DOI 10.1215/kjm/1250522965
- Vogelaar, J.A., Constructing vector bundles from codimension - two subvarieties, Thesis, Leiden 1978.
- F. L. Zak, Tangents and secants of algebraic varieties, Translations of Mathematical Monographs, vol. 127, American Mathematical Society, Providence, RI, 1993. Translated from the Russian manuscript by the author. MR 1234494, DOI 10.1090/mmono/127
Additional Information
- Mei-Chu Chang
- Affiliation: Department of Mathematics, University of California, Riverside, California 92521
- Email: mcc@math.ucr.edu
- Received by editor(s): September 10, 2000
- Published electronically: September 19, 2001
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 975-992
- MSC (2000): Primary 14F05; Secondary 14J60, 05A10
- DOI: https://doi.org/10.1090/S0002-9947-01-02837-9
- MathSciNet review: 1867368