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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



A reconstruction of Euler data

Authors: Bong H. Lian, Chien-Hao Liu and Shing-Tung Yau
Journal: J. Algebraic Geom. 12 (2003), 269-284
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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Additional Information

Bong H. Lian
Affiliation: National University of Singapore, Department of Mathematics, Singapore, 117543, Republic of Singapore

Chien-Hao Liu
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138

Shing-Tung Yau
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
MR Author ID: 185480
ORCID: 0000-0003-3394-2187

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