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The Euler series of restricted Chow varieties. (English) Zbl 0845.14005

H. B. Lawson jun. and S. S.-T. Yau [Ann. Sci. Éc. Norm. Supér., IV. Sér. 20, 557-577 (1987; Zbl 0639.32015)] introduced a series associated to the Chow monoid of a projective variety that becomes a formal power series when a basis for homology is fixed and they proved it is a rational function for the cases \(\mathbb{P}^n\) and \(\mathbb{P}^n \times \mathbb{P}^m\). In this paper the author considers the “restricted” Chow variety \(C_\lambda (X)\) of a projective algebraic variety \(X\), i.e. the space consisting of all effective cycles with a given homology class \(\lambda\), and he calculates the Euler characteristic of \(C_\lambda (X)\) (by introducing a more general Euler series) when \(X\) admits a linear action of an algebraic torus, for which there are only finitely many irreducible invariant subvarieties. Then smooth toric varieties are considered and some examples are given.

MSC:

14C25 Algebraic cycles
14C05 Parametrization (Chow and Hilbert schemes)
14F45 Topological properties in algebraic geometry

Citations:

Zbl 0639.32015
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References:

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