On the Finiteness of the Number of Orbits on Quasihomogeneous $(\mathbb C^*)^k\times SL_2(\mathbb C)$-manifolds

E. V. Sharoiko

M. V. Lomonosov Moscow State University

Abstract: We obtain an effective criterion for the finiteness of the number of orbits contained in the closure of a given $G$-orbit for the case of a rational linear action of the group $G:=(\mathbb C^*)^k\times SL_2(\mathbb C)$ on a finite-dimensional linear space as well as on the projectivization of such a space.

Keywords: the group $SL_2(\mathbb C)$, rational linear action, orbit, character lattice, Borel subgroup, analytic curve, irreducible algebraic variety.

UDC: 512.743.7

Received: 03.11.2005
Revised: 30.08.2006

DOI: 10.4213/mzm3719

 English version: Mathematical Notes, 2007, 81:5, 686–694

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