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Dmitri A. Timashev

Torus actions of complexity one
Let T = (Cв )r be a complex algebraic torus acting on an algebraic variety X . The complexity c(T , X ) is the codimension of generic T -orbits in X . Toric varieties are exactly those of complexity 0. We give a combinatorial description of torus actions of complexity 1 in the language of convex geometry in the same spirit as for toric varieties. We restrict our consideration to normal T -varieties. This restriction, common in toric geometry, is not very essential since every T -variety admits a T -equivariant normalization. Without loss of generality we may assume that the action T X is faithful, i.e., generic T -orbits have trivial stabilizers. By Sumihiro's theorem X = Xi is covered by finitely many affine open T -stable subvarieties. Hence a description of X amounts to 2 problems: (1) Describe the affine T -varieties Xi ; (2) Indicate how to patch them together. To solve the 1-st problem, we may assume that X itself is affine. It is determined by its coordinate algebra C[X ]. The latter is a finitely generated integrally closed T -algebra, whence C[X ] = v=vD Ov over all T -stable prime divisors D X (T -divisors in short), where vD denotes the valuation of the field of rational functions C(X ) corresponding to D. Now we describe T -invariant discrete valuations of C(X ) taking values in Q (T -valuations in short). It is easy to see that they are completely determined by the restriction to the multiplicative group of T -eigenfunctions C(X )(T ) , and C(X )(T ) (C(X )T )в в , where C(X )T is the field of T -invariant functions and is the weight lattice of T . Since c(T , X ) = 1, we have C(X )T C(C ) for some smooth pro jective curve C . Restricting a valuation to C(X )T and , in turn, we deduce: Prop osition. The T -valuations are in a 11 correspondence with the triples (z , h, ), z C , h Q+ , Z := Hom(, Q), modulo the equivalence relation (z1 , 0, ) (z2 , 0, ), z1 , z2 C. Hence the set of T -valuations is V = zC Vz , where the half-spaces Vz = Q+ в Z are patched together along Z . Definition. A hypercone in V is a union C = Cz of rational polyhedral cones Cz Vz such that: (1) Cz Z =: K does not depend on z C ; (2) Cz = Q+ в K for all but finitely many z ; (3) Let Pz be the pro jections of Cz ({1} в Z ) to Z ; then P = z C Pz := { z Pz , z = 0 for all but finitely many z } K \ {0}. (P may be empty!) (4) For any face K0 K, K0 P = 0, and , , K0 = 0, , K 0, put then a multiple of z z · z is a principal divisor on C .
z

z |

= min , Pz ;
z

Note: Condition (4) holds automatically if C = P1 , i.e., if X is rational, because

=0

Theorem 1. The normal affine T -varieties of complexity 1 are in a 11 correspondence with the hypercones. The T -divisors on X correspond to the edges of the Cz 's not intersecting P . Next we address the 2-nd problem. By a hyperface C C we mean a hypercone C such that Cz is a face of Cz , z C . Theorem 2. Affine T -varieties Xi can be patched together giving a (possibly non-affine) T -variety X of complexity 1 iff the respective hypercones Ci intersect exactly in their common hyperfaces. Conclusion: Normal T -varieties of complexity 1 are in a 11 correspondence with finite collections of hypercones intersecting in their common hyperfaces, called hyperfans. In terms of a hyperfan, there are a description of all orbits, a criterion for smoothness, etc.

References
[1] D. A. Timashev, Classification of G-varieties of complexity 1, Izv. Math. 61 (1997), no. 2, 363397. [2] D. A. Timashev, Homogeneous spaces and equivariant embeddings , arXiv:math.AG/0602228, to appear in EncyclopФdia of Math. Sciences.