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DOI: 10.1007/s00208-005-0705-8 Math. Ann. 334, 557607 (2006) MathematischeAnnalen

Summary: DOI: 10.1007/s00208-005-0705-8
Math. Ann. 334, 557­607 (2006) MathematischeAnnalen
Polyhedral divisors and algebraic torus actions
Klaus Altmann · J¨urgen Hausen
Received: 29 October 2003 / Revised version: 15 June 2005
Published online: 26 December 2005 ­ © Springer-Verlag 2005
Abstract. We provide a complete description of normal affine varieties with effective algebraic
torus action in terms of what we call proper polyhedral divisors on semiprojective varieties.
Our approach extends classical cone constructions of Dolgachev, Demazure and Pinkham to the
multigraded case, and it comprises the theory of affine toric varieties.
Mathematics Subject Classification (2000): 14L24,14L30,14M25,13A50
We present a complete description of n-dimensional, normal, affine varieties with
an effective action of a k-dimensional algebraic torus in terms of "proper polyhe-
dral divisors" living on semiprojective varieties of dimension n-k. Our approach
comprises two well known theories: on the one hand, for varieties with an almost
transitive torus action (k = n), our description specializes to the theory of affine
toric varieties [9], and on the other, for C
-actions (k = 1), we recover classical
constructions of generalized affine cones of Dolgachev [6], Demazure [5] and


Source: Altmann, Klaus - Fachbereich Mathematik und Informatik & Institut für Mathematik, Freie Universität Berlin


Collections: Mathematics