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Top ology of Kempf­Ness sets for algebraic torus actions
Taras Panov Moscow State University Osaka City University

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Abstract. In the theory of algebraic group actions on affine varieties, the concept of a Kempf­Ness set is used to replace the geometric quotient by the quotient with resp ect to a maximal compact subgroup. By making use of the recent achievements of "toric top ology" we show that an appropriate notion of a Kempf­Ness set exists for a class of algebraic torus actions on quasiaffine varieties (co ordinate subspace arrangement complements) arising in the "geometric invariant theory" approach to toric varieties. We pro ceed by studying the cohomology of these Kempf­Ness sets.

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1. Kempf­Ness sets for affine varieties. G reductive algebraic group, S affine G-variety. S,G : S S/ G dual to C[S ]G C[S ]. S,G establishes a bijection: / closed G-orbits of S p oints of S/ G. / S/ G is called the categorical quotient. / Let : G GL(V ) b e a representation of G, K b e a maximal compact subgroup of G, , b e a K-invariant hermitian form on V with asso ciated norm .

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Given v V , consider the function Fv : G R, g g v 2.

It has a critical p oint if and only if Gv is closed, and all critical p oints of Fv are minima. Define KN V by one of the following equivalent conditions: KN = {v V : (dFv )e = 0} = { v V : Tv G v v } (e G is the unit) (1)

= {v V : v , v = 0 for all g} = {v V : v , v = 0 for all k},

where g (resp. k) is the Lie algebra of G (resp. K). Therefore, any p oint v KN is a closest p oint to the origin in its orbit Gv . Then KN is called the Kempf­Ness set of V .
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We may assume that the affine G-variety S is equivariantly emb edded as a closed subvariety in a representation V of G. Then the Kempf­ Ness set KN S of S is defined as KN S . The imp ortance of Kempf­Ness sets for the study of orbit quotients is due to the following result (Kempf­Ness, Neeman). Thm 2. a) The comp osition KN S S S/ G is prop er and induces / a homeomorphism KN S /K S/ G. / b) There is a K-equivariant deformation retraction S KN S .

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2. Algebraic torus actions. N = Zn integral lattice, NR = N Z R. A convex subset NR is a cone if there are vectors a1, . . . , ak N such that = {µ1a1 + . . . + µk ak : µi R, µi 0}.

A cone is called regular (resp. simplicial) if a1, . . . , ak is a subset of a Z-basis of N (resp. an R-basis of NR). A finite collection = {1, . . . , s} of cones in NR is called a fan if a face of every cone in b elongs to and the intersection of any two cones in is a face of each. A fan = {1, . . . , s} is called complete if NR = 1 . . . s.

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TC = N Z C = (C)n algebraic torus, T = N Z S1 = (S1)n, a maximal compact subgroup, the (compact) torus. A toric variety is a normal algebraic variety X containing TC as a Zariski op en subset so that the natural action of TC on itself extends to an action on X . A classical construction establishes a one-to-one corresp ondence: fans in NR complex n-dimensional toric varieties, regular fans non-singular varieties, complete fans compact varieties.

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Assume has m one-dimensional cones, and consider the map Zm N sending the ith generator of Zm to the integer primitive vector ai generating the ith one-dimensional cone. Get an exact sequence 1 - G - (C)m - TC - 1, where G = (C)m-n â (finite group), and 1 - K - T m - T - 1 (here and b elow we denote T m = (S1)m). (3)

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Say that {i1, . . . , ik } [m] = {1, . . . , m} is a g-subset if {ai1 , . . . , aik } is a subset of the generator set of a cone in . K = {g -subsets} an abstract simplicial complex on the set [m]. is complete simplicial K is a triangulation of S n-1. Now set A() = and U () = Cm \ A(). Both sets dep end only on the combinatorial structure of the simplicial complex K; and U () is the co ordinate subspace arrangement complement U (K).
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{i1,...,ik } is not a g -subset

{z Cm : zi1 = . . . = zik = 0}


The set A() is an affine variety, while its complement U () admits a simple affine cover, as describ ed in the following statement. Prop 4. Given a cone , denote g ( ) [m] the set of its genera= tors, z ^ j g ( ) zj , and define /
^ V () = {z Cm : z = 0 for all }

and U ( ) = {z Cm : zj = 0 if j g ( )}. / Then A() = V () and U () = Cm \ V () =


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U ( ).


U () Cm is (C)m-invariant, and G (C)m acts on U () with finite isotropy subgroups if is simplicial. The corresp onding quotient is identified with the toric variety X determined by . The more precise statement is as follows (Batyrev­Cox). Thm 5. (a) X = U ()/ G (categorical quotient). / (b) X is the geometric quotient is simplicial. However, the analysis of the previous section do es not apply here, as U () is not an affine variety in Cm (it is only quasiaffine in general). E.g., if is a complete fan, then the G-action on the whole Cm has only one closed orbit 0, and the quotient Cm/ G consists of a single / p oint. Below we show that an appropriate notion of the Kempf­Ness set still exists for this class of torus actions.
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Consider the unit p olydisc (D2)m = {z Cm : |zj | Given , define Z ( ) = {z (D2)m : |zj | = 1 if j g ( )}, / and Z () =



1 for all j }.

Z ( ).

The subset Z () (D2)m is invariant with resp ect to the T m-action. Prop 6. Assume that is a complete simplicial fan. Then Z () is a compact T m-manifold of dimension m + n.
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Note that Z ( ) U ( ), and therefore, Z () U (). Thm 7. Assume that is a simplicial fan. a) The comp osition Z () U () U ()/G is prop er and induces a homeomorphism Z ()/K U ()/G. b) There is a T m-equivariant deformation retraction of U () to Z (). Pro of. b): there are obvious equivariant deformation retractions U ( ) Z ( ) for all , which patch together to get the necessary map U () Z (). By comparing this result with Theorem 2, we see that Z () has the same prop erties with resp ect to the G-action on U () as the set KN S with resp ect to a G-action on an affine variety S . We therefore refer to Z () as the Kempf­Ness set of U ().
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3. Normal fans. The next step in our study of the Kempf­Ness set for torus actions on quasiaffine varieties like U () would b e to obtain an explicit description like the one given by (1) in the affine case. Although we do not now of such a description in general, it do es exist in the particular case when is the normal fan of a simple p olytop e.

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MR = (NR) the dual vector space. Given primitive vectors a1, . . . , am N and b1, . . . , bm Z, consider P = {x MR : ai, x + bi 0, i Assume that P is b ounded, the affine hull of there are no redundant inequalities. Then with exactly m facets Fi and normal vectors = 1 , . . . , m }. P is the whole MR, and P is a convex p olytop e ai , i = 1 , . . . , m.

If G is an l-dimensional face, then the set of all its normal vectors {ai1 , . . . , aik } spans an (n - l)-dimensional normal cone G. P = {G : G a face of P } normal fan of P (a complete fan). From now on, assume: P is simplicial P is simple. In this case, {ai1 , . . . , aik } spans a cone of P Fi1 . . . Fik = .
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Kempf­Ness sets Z (P ) admit a very transparent interpretation as complete intersections of real algebraic quadrics. We may sp ecify P by a matrix inequality AP x + bP 0, where AP is the m â n matrix of row vectors ai, and bP is the column vector of scalars bi. The linear transformation AP : MR Rm is exactly the one obtained from the map T m T from (3) by applying HomZ( · , S1) Z R. The formula iP (x) = AP x + bP defines an affine injection iP : MR - Rm, which emb eds P into the p ositive cone Rm.
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Now define the space ZP by a pullback diagram
P

ZP -Z Cm


i

P

-P Rm

i

where (z1, . . . , zm) is given by (|z1|2, . . . , |zm|2). Prop 8. a) We have ZP U (P ). b) There is a T m-equivariant homeomorphism ZP = Z (P ).

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Applying HomZ( · , S1) Z R to 1 - K - T m - T - 1 we get
AP C 0 - MR - Rm - Rm-n - 0,

where C = (cj k ) is an (m - n) â m-matrix. The map iZ emb eds ZP in Cm as the space of solutions of the m - n real quadratic equations
m k=1

cj,k |zk |2 - bk

(

)

= 0, for 1 j m - n.

(9)

This intersection of real quadrics is non-degenerate, and therefore, ZP R2m is a smo oth submanifold with trivial normal bundle.
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4. Projective toric varieties and moment maps. Let fv = (dFv )e : g R, g v , v . Think fv g.

G is reductive g = k ik. K-action is norm preserving fv vanishes on k, so we consider fv as an element of ik = k. Varying v V we get the moment map µ : V k, which sends v V , k to iv , v . The Kempf­Ness set is the set of zero es of µ: KN = µ-1(0).
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This description do es not directly apply to algebraic torus actions on U (): the set µ-1(0) = {z Cm : z , z = 0 for all k} consist only of the origin in this case. Assume now that P is a regular fan. Then XP is a smo oth projective variety. This implies that XP is K¨ ahler, and therefore, a symplectic manifold. In this case there is a symplectic version of the previous constructions.

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(W, ) symplectic manifold with symplectic K-action. Given k, denote by the corresp onding K-invariant vector field on W . The K-action is Hamiltonian if the 1-form ( · , ) is exact for every k, that is, there is a function H on W such that ( , ) = dH( ) = (H) for every vector field on W . Under this assumption, the moment map µ : W k, is defined.
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(x, ) H(x)


m Ex 10. 1. = 2 k=1 dxk dyk , zk = xk + iyk . The co ordinatewise action of K = T m is Hamiltonian with moment map

W = Cm,

µ : Cm Rm,

(z1, . . . , zm) (|z1|2, . . . , |zm|2).

2. a simplicial fan, 1 K T m T 1. We can restrict the previous example to the K-action on the invariant subvariety U () Cm. The moment map µ : Cm - Rm - k. A choice of an isomorphism k = Rm-n allows to identify the map Rm k with the linear transformation given by matrix C in
C 0 - MR - Rm - Rm-n - 0.
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AP


Unlike the affine case, the Kempf­Ness set ZP for the G-action on U (P ) do es not coincide with the level set µ-1(0) = {z Cm : z , z = 0 for all k}. The latter is given by the equations which have only zero solution. The right statement is as follows. Prop 11. Then the Kempf­Ness set Z (P ) is given by Z (P ) = µ-1 (C bP ).
P

m cj,k |zk |2 = 0, 1 j m - n, k=1

In other words, the difference with the affine situation is that we have to take C bP instead of 0 as the value of the moment map. The reason is that C bP is a regular value of µ, unlike 0.
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In the case of normal fans the following version of our Theorem 7 (a) is known in toric geometry (Audin, Batyrev, Guillemin): Thm 12. Assume X a projective simplicial toric variety and c is in its K¨ ahler cone. Then µ-1(c) U (), and the natural map µ-1(c)/K U ()/G = X is a diffeomorphism. This statement is the essence of the construction of smo oth projective toric varieties via symplectic reduction. The submanifold µ-1(c) Cm may fail to b e symplectic as the restriction of the standard symplectic form on Cm to µ-1(c) may fail to b e non-degenerate. However, the restriction of descends to the quotient µ-1(c)/K as a symplectic form.
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Ex 13. P = n MR standard simplex defined by ei, x 0, i = 1, . . . , n, and (-1, . . . , -1), x + 1 0

Cones of P are generated by prop er subsets of the set of vectors {e1, . . . , en, (-1, . . . , -1)}. G = C and K = S1 are the diagonal subgroups in (C)n+1 and T n+1 resp ectively, while U () = Cn+1 \ {0}. C is a row of units. Moment map is given by µ(z1, . . . , zn+1) = |z1|2 + . . . + |zn+1|2. Since C bP = 1, the S2n+1 Cn+1, and projective space CP Kempf­Ness set ZP = µ-1(1) is the unit sphere n+1 \ {0})/G = S2n+1 /K is the complex X = (C n.
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Problem 14. As is known, there are many complete regular fans which cannot b e realised as normal fans of convex p olytop es. The corresp onding toric varieties X are not projective (although b eing non-singular). In this case the Kempf­Ness set Z () is still defined, as well as the moment map. However, the rest of the analysis do es not apply here. Can one still describ e Z () as a complete intersection of real quadratic (or other) hyp ersurfaces? And do es the moment map µ : U () k have any regular values?

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5. Cohomology of Kempf­Ness sets. Given an abstract simplicial complex K on the set [m], the face ring (or the Stanley­Reisner ring) Z[K ] is defined as the following quotient of the p olynomial ring on m generators: Z[K ] = Z[v1, . . . , vm]/(vi1 · · · vik : {i1, . . . , ik } is not a simplex of K ). Grading deg vi = 2, i = 1, . . . , m. Thm 15. For every simplicial fan there are algebra isomorphisms
( ) [ ] ( ) Z (); Z Tor H [u , . . . , u ] Z[K ], d , H = m 1 Z[v1,...,vm] Z[K ], Z =

where the latter denotes the cohomology of a dga with deg ui = 1, deg vi = 2, dui = vi, dvi = 0 for 1 i m.
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Given I [m], denote by K (I ) the corresp onding full sub complex of K , or the restriction of K to I . H i(K (I )) the ith reduced simplicial cohomology group of K (I ) with integer co efficients. A theorem due to Ho chster expresses TorZ[v ,...,v ](Z[K ], Z) in terms m 1 of full sub complexes of K , which leads to the following description of the cohomology of Z (). Thm 16. We have
( ) k Z () H =
I [m] -i,2j

( ) k-|I |-1 K (I ) . H

There is also a similar description of the pro duct in H (Z ()).
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References

[1] Victor Buchstab er, Taras Panov and Nigel Ray. Analogous p olytop es, circle actions, and toric manifolds. Preprint, 2005, available from http://higeom.math.msu.su/people/taras/english.html#publ

[2] Taras Panov. Top ology of Kempf-Ness sets for algebraic torus actions. Preprint, arXiv:math.AG/0603556.

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