A reconstruction of Euler data

Authors:
Bong H. Lian, Chien-Hao Liu and Shing-Tung Yau

Journal:
J. Algebraic Geom. **12** (2003), 269-284

DOI:
https://doi.org/10.1090/S1056-3911-02-00311-9

Published electronically:
September 18, 2002

MathSciNet review:
1949644

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Abstract |
References |
Additional Information

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 $V$ on ${\mathbb C}{\mathrm P}^n$ and a multiplicative class $b$. This gives a direct way to compute the intersection number $K_d$ without referring to any other Euler data linked to $Q$. Here $K_d$ is the integral of the cohomology class $b(V_d)$ of the induced bundle $V_d$ 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 $b$, the Chern polynomial, the computation of $K_1$ for the bundle $V=T_{\ast }{\mathbb C}{\mathrm P}^2$, and $K_d$, $d=1,2,3$, 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.

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[F-P]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.
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[L-L-Y]L-L-Y B.H. Lian, K. Liu, and S.-T. Yau, *Mirror principle I*, *Asian J. Math.* ** 1** (1997), pp. 729 - 763; *II*, arXiv:math.AG/9905006; *III*, arXiv:math.AG/9912038.
[M-P]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.
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[Wi]Wi E. Witten, * Phases of $N=2$ theories in two dimensions*, * Nucl. Phys.* ** B403** (1993), pp. 159 - 222.

[Au]Au M. Audin, *The topology of torus actions on symplectic manifolds*, Prog. Math. 93, Birkhäuser, 1991.
[A-B]A-B M.F. Atiyah and R. Bott, *The moment map and equivariant cohomology, Topology*, **23** (1984), pp. 1 - 28.
[B-DC-P-P]B-DC-P-P G. Bini, C. De Concini, M. Polito, and C. Procesi, *On the work of Givental relative to mirror symmetry*, arXiv:math.AG/9805097.
[C-dlO-G-P]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]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]C-K-Y-Z T.-M. Chiang, A. Klemm, S.-T. Yau, and E. Zaslow, *Local mirror symmetry: calculations and interpretations*, arXiv:hep-th/9903053.
[F-P]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]Gi A. Givental, *Equivariant Gromov-Witten invariants*, arXiv:alg-geom/9603021.
[H-K-T-Y]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]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]Li Kefeng Liu, private communications.
[L-L-Y]L-L-Y B.H. Lian, K. Liu, and S.-T. Yau, *Mirror principle I*, *Asian J. Math.* ** 1** (1997), pp. 729 - 763; *II*, arXiv:math.AG/9905006; *III*, arXiv:math.AG/9912038.
[M-P]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]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]Pa R. Pandharipande, * Rational curves on hypersurfaces (after givental)*, arXiv:math.AG/9806133.
[Wi]Wi E. Witten, * Phases of $N=2$ theories in two dimensions*, * Nucl. Phys.* ** B403** (1993), pp. 159 - 222.

Additional Information

**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

MR Author ID:
185480

ORCID:
0000-0003-3394-2187

Email:
yau@math.harvard.edu

Received by editor(s):
October 9, 2000

Published electronically:
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