Journal of Algebraic Geometry

Author(s): Bong H. Lian; Chien-Hao Liu; Shing-Tung Yau
Journal: J. Algebraic Geom. 12 (2003), 269-284.
Posted: September 18, 2002
Retrieve article in: PDF DVI PostScript

Abstract: We apply the mirror principle (see Mirror principle, I, Asian J. Math. 1 (1997), pp. 729-763) to reconstruct the Euler data $Q=\{Q_d\}_{d\in{\mathbb N}\cup\{0\}}$ associated to a vector bundle

on ${\mathbb C}{\mathrm P}^n$ and a multiplicative class

. This gives a direct way to compute the intersection number

without referring to any other Euler data linked to

. Here

is the integral of the cohomology class

of the induced bundle

on a stable map moduli space. A package ``EulerData_MP.m'' in Maple V that carries out the actual computation is provided in the electronic version math.AG/0003071 of the current paper. For

, the Chern polynomial, the computation of

for the bundle $V=T_{\ast}{\mathbb C}{\mathrm P}^2$ , and

, for the bundles ${\mathcal O}_{{\mathbb C}{\mathrm P}^4}(l)$ with $6\le l\le 10$ are done using the code and are also included.

[Au]: M. Audin, The topology of torus actions on symplectic manifolds, Prog. Math. 93, Birkhäuser, 1991.
[A-B]: M.F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology, 23 (1984), pp. 1 - 28.
[B-DC-P-P]: G. Bini, C. De Concini, M. Polito, and C. Procesi, On the work of Givental relative to mirror symmetry, math.AG/9805097.
[C-dlO-G-P]: P. Candelas, X. de la Ossa, P. Green, and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal field theory, Nucl. Phys. B359 (1991), pp. 21 - 74.
[C-K]: D.A. Cox and S. Katz, Mirror symmetry and algebraic geometry, Math. Surv. Mono. 68, Amer. Math. Soc. 1999.
[C-K-Y-Z]: T.-M. Chiang, A. Klemm, S.-T. Yau, and E. Zaslow, Local mirror symmetry: calculations and interpretations, hep-th/9903053.
[F-P]: W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, in Algebraic geometry - Stanta Cruz 1995, J. Kollár, R. Lazarsfeld, and D. Morrison eds., Proc. Symp. Pure Math. vol. 62, part 2, pp. 45 - 96, Amer. Math. Soc. 1997.
[Gi]: A. Givental, Equivariant Gromov-Witten invariants, alg-geom/9603021.
[H-K-T-Y]: S. Hosono, A. Klemm, S. Theisen, and S.-T. Yau, Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces, pp. 545 - 606 in Mirror symmetry II, B. Greene and S.T. Yau eds., Amer. Math. Soc. and International Press, 1997.
[Ko]: M. Kontsevich, Enumeration of rational curves via torus actions, in The moduli space of curves, R. Dijkgraaf, C. Faber, and G. van der Geer eds., pp. 335 - 368, Birkhäuser, 1995.
[Li]: Kefeng Liu, private communications.
[L-L-Y]: B.H. Lian, K. Liu, and S.-T. Yau, Mirror principle I, Asian J. Math. 1 (1997), pp. 729 - 763; II, math.AG/9905006; III, math.AG/9912038.
[M-P]: D.R. Morrison and M. Plesser, Summing the instantons: quantum cohomology and mirror symmetry in toric varieties, Nucl. Phys. B440 (1995), pp. 279 - 354.
[MS]: Mirror symmetry I, S.-T. Yau ed., Amer. Math. Soc. and International Press, 1998; Mirror symmetry II, B. Greene and S.T. Yau eds., Amer. Math. Soc. and International Press, 1997; Mirror symmetry III, D.H. Phong, L.V. Vinet, and S.-T. Yau eds., Amer. Math. Soc., International Press, and Centre de Recherches Math., 1999.
[Pa]: R. Pandharipande, Rational curves on hypersurfaces (after givental), math.AG/9806133.
[Wi]: E. Witten, Phases of theories in two dimensions, Nucl. Phys. B403 (1993), pp. 159 - 222.

Bong H. Lian
Affiliation: National University of Singapore, Department of Mathematics, Singapore, 117543, Republic of Singapore
Email: lian@brandeis.edu

Chien-Hao Liu
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Email: chienliu@math.harvard.edu

Shing-Tung Yau
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Email: yau@math.harvard.edu

PII: S 1056-3911(02)00311-9
Received by editor(s): October 9, 2000
Posted: September 18, 2002
Additional Notes: B. H. Lian is on leave from Brandeis University, Department of Mathematics, Waltham, Massachusetts 02154. This work is supported by DOE grant DE-FG02-88ER25065 and NSF grants DMS-9619884 and DMS-9803347

Previous issue \| Table of contents \| Next issue Recently posted articles \| Most recent issue \| All issues