## A mathematical proof of a formula of Aspinwall and Morrison.(English)Zbl 0951.14025

Summary: We give a rigorous proof of the Aspinwall-Morrison formula [cf. P. S. Aspinwall and D. R. Morrison, Commun. Math. Phys. 151, No. 2, 245-262 (1993; Zbl 0776.53043)] which expresses the cubic derivatives of the Gromov-Witten as a series depending only on the number of rational curves in each homology class, for a Calabi-Yau threefold with only rigid immersed rational curves.

### MSC:

 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 14N10 Enumerative problems (combinatorial problems) in algebraic geometry 14J30 $$3$$-folds

Zbl 0776.53043
Full Text:

### References:

 [1] Aspinwall, P.S. and Morrison, D.R. : Topological field theory and rational curves , Comm. in Math. Phys., vol. 151 (1993), 245-262. · Zbl 0776.53043 [2] Audin, M. and Lafontaine, J. (eds): Holomorphic curves in symplectic geometry, Progress in Math. 117, Birkhaüser, 1994. · Zbl 0802.53001 [3] Candelas, P. , De La Ossa, X.C. , Green, P.S. and Parkes, L. : A pair of Calabi-Yau manifolds as an exactly soluble superconformal field theory , Nucl. Phys. B359 (1991), 21-74. · Zbl 1098.32506 [4] Gromov, M. : Pseudoholomorphic curves in symplectic manifolds , Invent. Math. 82 (1985), 307-347. · Zbl 0592.53025 [5] Kontsevich, M. : Enumeration of rational curves via torus action , In: Proceedings of the conference ’The moduli space of curves’, Eds Dijgraaf, Faber, Van der Geer, Birkhaüser, 1995. · Zbl 0885.14028 [6] Kontsevich, M. : Homological algebra of mirror symmetry , In: Proceedings of the International Congress of Mathematicians, Zurich, 1994, Birkhaüser, 1995. · Zbl 0846.53021 [7] Kontsevich, M. and Manin, Yu. : Gromov-Witten classes, quantum cohomology and enumerative geometry , Communications in Math. Physics, vol. 164 (1994), 525-562. · Zbl 0853.14020 [8] Laufer, H.B. : On CP1 as an exceptional set , in Recent progress in several complex variables , 261-275, Princeton University Press, 1981. · Zbl 0523.32007 [9] Mcduff, D. and Salamon, D. : J-holomorphic curves and quantum cohomology , University Lecture Series, vol. 6, AMS, 1994. · Zbl 0809.53002 [10] Manin, Yu. : Generating functions in Algebraic Geometry and Sums over Trees , MPI preprint 94-66, Proceedings of the conference ’The moduli space of curves’, Eds Dijgraaf, Faber, Van der Geer, Birkhaüser, 1995. · Zbl 0871.14022 [11] Morrison, D. : Mirror symmetry and rational curves on quintic threefolds , Journal of the AMS, vol. 6 (1), 223-241. · Zbl 0843.14005 [12] Ruan, Y. and Tian, G. : A mathematical theory of quantum cohomology , preprint 1994. · Zbl 0860.58006 [13] Vafa, C. : Topological mirrors and quantum rings , in [16]. · Zbl 0827.58073 [14] Voisin, C. : Symétrie miroir, Panoramas et Synthèses , n^\circ 2, 1996, Societé Mathematique de France. · Zbl 0849.14001 [15] Witten, E. : Mirror manifolds and topological field theory , in [16], 120-158. · Zbl 0834.58013 [16] Yau, S. T. (ed.): Essays on mirror manifolds , International Press, Hong Kong, 1992. · Zbl 0816.00010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.