Elizondo, Javier The Euler series of restricted Chow varieties. (English) Zbl 0845.14005 Compos. Math. 94, No. 3, 297-310 (1994). 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. Reviewer: Vasile Brinzănescu (MR 96g:14003) Cited in 2 ReviewsCited in 8 Documents MSC: 14C25 Algebraic cycles 14C05 Parametrization (Chow and Hilbert schemes) 14F45 Topological properties in algebraic geometry Keywords:restricted Chow variety; Chow monoid; Euler characteristic; Euler series; toric varieties Citations:Zbl 0639.32015 PDFBibTeX XMLCite \textit{J. Elizondo}, Compos. Math. 94, No. 3, 297--310 (1994; Zbl 0845.14005) Full Text: arXiv Numdam EuDML References: [1] E. Bifet , C. De Concini , and C. Procesi , Cohomology of regular embeddings , Adv. in Math., 82 (1990), 1-34. · Zbl 0743.14018 · doi:10.1016/0001-8708(90)90082-X [2] E. Bifet , Personal letters , August 1 and December 5, 1992. [3] W.L. Chow and B.L. Van Der Waerden , Uber zugeordnete formen und algebraische system von algebraischen mannigfaltigkeiten , Math. Annalen, 113 (1937), 692-704. · Zbl 0016.04004 · doi:10.1007/BF01571660 [4] V.I. Danilov , The theory of toric varieties , Russian Math. Surveys, 33(2) (1978), 97-154. · Zbl 0425.14013 · doi:10.1070/RM1978v033n02ABEH002305 [5] J. Elizondo , The Euler-Chow Series for Toric Varieties, PhD thesis , SUNY at Stony Brook, August 1992. [6] W. Fulton , Introduction to Toric Varieties, Number 131 in Annals of Mathematics Studies . Princeton University Press, Princeton, NJ, 1st. edition, 1993. · Zbl 0813.14039 · doi:10.1515/9781400882526 [7] H.B. Lawson, Jr. and S.S.T. Yau , Holomorphic symmetries , Ann. scient. Éc. Norm. Sup., t.20 (1987), 557-577. · Zbl 0639.32015 · doi:10.24033/asens.1544 [8] I G. MacDonald, The Poincaré polynomial of a symmetric product , Proc. Cam. Phil. Soc., 58 (1962), 563-568. · Zbl 0121.39601 [9] P. Samuel , Méthodes d’Algèbre Abstrait en Géométrie Algébrique , Springer-Verlag, Heidelberg, 1955. · Zbl 0067.38904 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.