Abstract
The tangent bundle ℐ X of a Calabi-Yau threefoldX is the only known example of a stable bundle with non-trivial restriction to any rational curve onX. By deforming the direct sum of ℐ X and the trivial line bundle one can try to obtain new examples. We use algebro-geometric techniques to derive results in this direction. The relation to the finiteness of rational curves on Calabi-Yau threefolds is discussed.
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[B] Bloch, S.: Semi-regularity and de Rham cohomology. Inv. Math.17, 51–66 (1972)
[C] Clemens, H.: Some results about Abel-Jacobi mappings. In: Topics in Transcendental Algebraic Geometry Annals of Math. Studies, Princeton NJ: Princeton University Press (1984)
[DGKM] Distler, J., Greene, B. R., Kirklin, K., Miron, P.: Calculating Endomorphism Valued Cohomology: Singlet Spectrum in Superstring Models. Commun. Math. Phys.122, 117–124 (1989)
[EH] Eastwood, M. G., Hübsch, T.: Endomorphism Valued Cohomology and Gauge-Neutral Matter. Commun. Math. Phys.132, 383 (1990)
[Gr1] Grothendieck, A.: Géométrie formelle et géométrie algébrique. Sém. Bourbaki182 (1959)
[Gr2] Grothendieck, A.: Techniques de construction et théorem d'existence en géometrie algébrique IV: Les schémas de Hilbert. Sém. Bourbaki221 (1960/61)
[K1] Katz, S.: On the finiteness of rational curves in quintic threefolds. Comp. Math.60, 151–162 (1986)
[Ka] Kawamata, Y.: Unobstructed deformations-A remark on a paper of Z. Ran. J. Alg. Geom.1, 183–190 (1992)
[KJ] Kleiman, S. L., Johnson, T.: On Clemens' conjecture. Preprint (1993)
[K] Kobayashi, S. Differential Geometry of Complex Vector Bundles. Iwanami Shoten, 1987
[Ko] Kodaira, K.: Complex Manifolds and Deformations of Complex Structures. Grundlehren Math. Wiss.189, Berlin, Heidelberg, New York: Springer 1985
[Ma] Maruyama, M.: Moduli of stable sheaves I. J. Math. Kyoto Univ.17, 91–126 (1977)
[Mi] Miyaoka, Y.: The Chern Classes and Kodaira Dimension of a Minimal Variety. Adv. Stud. Pure Math.10, 449–476 (1987)
[N] Nijsse, P.G.J.: Clemens' conjecture for octic and nonic curves. Preprint (1993)
[No] Norten, A.: Analytic moduli of complex vector bundles. Indiana Univ. Math. J.28, 365–387 (1979)
[R] Ran, Z.: Deformations of manifolds with torsion or negative canonical bundles. J. Alg. Geom.1, 279–291 (1992)
[Ra] Ramella, L.: La stratification du schéma de Hilbert des courbes rationnelles deP n par le fibré tangent restreint. C. R. Acad. Sci. Paris311, I 181–184 (1990)
[Ti] Tian, G.: Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric. In: Yau, S.-T.. Mathematical Aspects of String theory. Singapore: World Scientific, pp. 629–646 (1987)
[Ti2] Tian, G.: On stability of Fano varieties. Int. J. Math.3(3), 401–413 (1992)
[To] Todorov, A. N.: The Weil-Petersson geometry of the moduli space ofSU(n≧3) (Calabi-Yau) manifolds. Commun. Math. Phys.126, 325–346 (1989)
[Ts] Tsuji, H.: Stability of tangent bundles of minimal algebraic varieties. Topology27, 4 429–442 (1988)
[Wi] Wilson, P. M. H.: The Kähler cone on Calabi-Yau threefolds. Invent. Math.107, 561–583 (1992)
[W] Witten, E.: New issues in manifolds ofSU(3) holonomy. Nucl. Phys.B268, 79–112 (1986)
[Y] Yau, S.-T.: Open Problems in Geometry. Proc. Symp. Pure Math.54, 1–28 (1993)
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Huybrechts, D. The tangent bundle of a Calabi-Yau manifold-deformations and restriction to rational curves. Commun.Math. Phys. 171, 139–158 (1995). https://doi.org/10.1007/BF02103773
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DOI: https://doi.org/10.1007/BF02103773