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ISSN 0081-5438, Pro ceedings of the Steklov Institute of Mathematics, 2008, Vol. 263, pp. 1­13. c Pleiades Publishing, Ltd., 2008. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 263, pp. 159­172 .

Toric Kempf­Ness Sets
T. E. Panov
a,b Received March 2008

Abstract--In the theory of algebraic group actions on affine varieties, the concept of a Kempf­ Ness set is used to replace the categorical quotient by the quotient with respect to a maximal compact subgroup. Using recent achievements of "toric topology," 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 Batyrev­Cox "geometric invariant theory" approach to toric varieties. We proceed by studying the cohomology of these "toric" Kempf­Ness sets. In the case of pro jective nonsingular toric varieties the Kempf­Ness sets can be described as complete intersections of real quadrics in a complex space. DOI: 10.1134/S0081543808040123

1. INTRODUCTION The concept of a Kempf­Ness set plays an imp ortant role in geometric invariant theory, as explained, for example, in [3, § 6.12] or [17]. Given an affine variety S over C with an action of a reductive group G, one can find a compact subset KN S such that the categorical quotient S/ /G is homeomorphic to the quotient KN/K of KN by a maximal compact subgroup K G. Another imp ortant prop erty of the Kempf­Ness set KN is that it is a K -equivariant deformation retract of S . Our aim here is to extend the notion of a Kempf­Ness set to a class of algebraic torus actions on complex quasiaffine varieties (co ordinate subspace arrangement complements) arising in the theory of toric varieties. Although our Kempf­Ness sets cannot b e defined exactly in the same way as in the affine case, they p ossess the ab ove two characteristic prop erties. In the case of a pro jective toric variety, our Kempf­Ness set can b e identified with the level surface for the moment map corresp onding to a compact torus action on the complex space [11, § 4]. The toric Kempf­Ness sets also constitute a particular sub class of moment­angle complexes [8], which op ens new links b etween toric top ology and geometric invariant theory. In Section 2 we review the notion of Kempf­Ness sets for reductive groups acting on affine varieties. In Section 3 we outline the "geometric invariant theory" approach to toric varieties as quotients of algebraic torus actions on co ordinate subspace arrangement complements, and introduce a toric Kempf­Ness set using our construction of moment­angle complexes. In Section 4 we restrict our attention to torus actions arising from normal fans of convex p olytop es. In this case the corresp onding Kempf­Ness set admits a transparent geometric interpretation as a complete intersection of real quadratic hyp ersurfaces. The quotient toric variety is pro jective, and the Kempf­Ness set represents the level surface for an appropriate moment map, thereby extending the analogy with the affine case even further in Section 5. In the last Section 6 we give a description of the cohomology ring of the Kempf­Ness set. As is clear from an example provided, our Kempf­Ness sets may b e quite complicated top ologically; many interesting phenomena o ccur even for the torus actions corresp onding to simple 3-dimensional fans.
a Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia. b Institute for Theoretical and Exp erimental Physics, ul. Bol'shaya Cheremushkinskaya 25, Moscow, 117218 Russia.

E-mail address: tpanov@mech.math.msu.su

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2. KEMPF­NESS SETS FOR AFFINE VARIETIES We start by briefly reviewing quotients and Kempf­Ness sets of reductive group actions on affine varieties. The details can b e found in [3, § 6.12] and [17]. Let G be a reductive algebraic group acting on a complex affine variety X . As G is noncompact, taking the standard (or geometric ) quotient with the quotient top ology may result in a badly b ehaving space (e.g., it may fail to b e Hausdorff ). An alternative notion of a categorical quotient remedies this difficulty and ensures that the result always lies within the category of algebraic varieties. Let C[X ] b e the algebra of regular functions on X , so that X = Sp ec C[X ]. Denote by X/ the /G G of G-invariant p olynomial functions complex affine variety corresp onding to the subalgebra C[X ] on X , and let : X X/ b e the morphism dual to the inclusion C[X ]G C[X ]. Then is /G surjective and establishes a bijection b etween closed G-orbits of X and p oints of X/ . Moreover, /G is universal in the class of morphisms from X that are constant on G-orbits in the category of algebraic varieties (which explains the term "categorical quotient"). The categorical quotient coincides with the geometric one if and only if all G-orbits are closed. Example 2.1. Consider the standard C -action on C (here C is the multiplicative group of complex numb ers). The categorical quotient C/ C is a p oint, while C/C is a non-Hausdorff / two-p oint space. Let : G GL(W ) be a representation of G, let K b e a maximal compact subgroup of G, and let ·, · be a K -invariant Hermitian form on W with asso ciated norm · . Given v W , consider the function Fv : G R sending g to 1 gv 2 . It has a critical p oint if and only if the orbit Gv is 2 closed, and all critical p oints of Fv are minima [3, Theorem 6.18]. Define the subset KN V by one of the following equivalent conditions: KN = v W : (dFv )e = 0 = v W : Tv Gv v = v W : v , v = 0 for all g = v W : v , v = 0 for all k , (2.1) (e G is the unit)

where g (resp ectively, k) is the Lie algebra of G (resp ectively, K ) and we consider k g End(W ). Therefore, any p oint v KN is the closest p oint to the origin in its orbit Gv . Then KN is called the Kempf­Ness set of V . We may assume that the affine G-variety X is equivariantly emb edded as a closed subvariety in a representation W of G. Then the Kempf­Ness set KNX of X is defined as KN X . The imp ortance of Kempf­Ness sets for the study of orbit quotients is due to the following result, whose pro of can b e found in [17, (4.7), (5.1)]. /G Theorem 2.2. (a) The composition KNX X X/ is proper and induces a homeomor/G phism KNX /K X/ . (b) There is a K -equivariant deformation retraction of X to KNX . 3. ALGEBRAIC TORUS ACTIONS Zn b e an integral lattice of rank n and NR = N Z R the ambient real vector space. Let N = A convex subset NR is called a cone if there exist vectors a1 ,... ,ak N such that = µ1 a1 + ... + µk ak : µi R, µi 0 . If the set {a1 ,... ,ak } is minimal, then it is called the generator set of . A cone is called strongly convex if it contains no line; all the cones b elow are assumed to b e strongly convex. A cone is
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called regular (resp ectively, simplicial ) if a1 ,... ,ak can b e chosen to form a subset of a Z-basis of N (resp ectively, an R-basis of NR ). A face of a cone is the intersection H with a hyp erplane H for which the whole is contained in one of the two closed half-spaces determined by H ; a face of a cone is again a cone. Every generator of spans a one-dimensional face, and every face of is spanned by a subset of the generator set. A finite collection = {1 ,... ,s } of cones in NR is called a fan if a face of every cone in belongs to and the intersection of any two cones in is a face of each. A fan is called regular (resp ectively, simplicial ) if every cone in is regular (resp ectively, simplicial). A fan = {1 ,... ,s } is called complete if NR = 1 ... s . Let C = C \{0} b e the multiplicative group of complex numb ers, and S1 b e the subgroup of complex numb ers of absolute value one. The algebraic torus TC = N Z C (C )n is a commutative = complex algebraic group with a maximal compact subgroup T = N Z S1 (S1 )n , the (compact) = torus. A toric variety is a normal algebraic variety X containing the algebraic torus TC as a Zariski op en subset in such a way that the natural action of TC on itself extends to an action on X . There is a classical construction (see [5]) establishing a one-to-one corresp ondence b etween fans in NR and complex n-dimensional toric varieties. Regular fans corresp ond to nonsingular varieties, while complete fans give rise to compact ones. Below we review another construction of toric varieties as certain algebraic quotients; it is due to several authors (see [6, 10]). In the rest of this section we assume that the one-dimensional cones of span NR as a vector space (this holds, e.g., if is a complete fan). Assume that has m one-dimensional cones. We order them arbitrarily and consider the map Zm N sending the ith generator of Zm to the integer primitive vector ai generating the ith one-dimensional cone. The corresp onding map of the algebraic tori fits into an exact sequence 1 G (C )m TC 1, (3.1)

where G is isomorphic to the pro duct of (C )m-n and a finite group. If is a regular fan and has at least one n-dimensional cone, then G (C )m-n . We also have an exact sequence of the = corresp onding maximal compact subgroups: 1KT
m

T 1

(3.2)

(here and b elow we denote Tm = (S1 )m ). We say that a subset {i1 ,... ,ik } [m] = {1,... ,m} is a g-subset if {ai1 ,... ,aik } is a subset of the generator set of a cone in . The collection of g-subsets is closed with resp ect to the inclusion and therefore forms an (abstract) simplicial complex on the set [m], which we denote K . Note that if is a complete simplicial fan, then K is a triangulation of an (n - 1)-dimensional sphere. Given a cone , we denote by g( ) [m] the set of its generators. Now set A() =
{i1 ,...,ik } is not a g -subset

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

and U () = Cm \ A(). Both sets dep end only on the combinatorial structure of the simplicial complex K ; the set U () coincides with the complement of the coordinate subspace arrangement U (K ) considered in [8, § 8.2] and [2, § 9.2]. The set A() is an affine variety, while its complement U () admits a simple affine cover, as describ ed in the following statement.
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Prop osition 3.1. Given a cone , set z =

j/ g () zj

and define

V () = z Cm : z = 0 for al l and / U ( ) = z Cm : zj = 0 if j g( ) . Then A() = V () and U () = Cm \ V () =


U ( ).

Pro of. We have Cm \ V () =


z Cm : z = 0 =


U ( ).

On the other hand, given a p oint z Cm , denote by (z ) [m] the set of its zero co ordinates. Then z Cm \ A() if and only if (z ) is a g-subset. This is equivalent to saying that z U ( ) for some . Therefore, Cm \ A() = U ( ), thus proving the statement. The complement U () is invariant with resp ect to the (C )m -action on Cm , and it is easy to see that the subgroup G from (3.1) acts on U () with finite isotropy subgroups if is simplicial (or even freely if is a regular fan). The corresp onding quotient is identified with the toric variety X determined by . The more precise statement is as follows. Theorem 3.2 (see [10, Theorem 2.1]). Assume that the one-dimensional cones of span NR as a vector space. (a) The toric variety X is natural ly isomorphic to the categorical quotient of U () by G. (b) X is the geometric quotient of U () by G if and only if is simplicial. Therefore, if is a simplicial (in particular, regular) fan satisfying the assumption of Theo/G rem 3.2, then all the orbits of the G-action on U () are closed and the categorical quotient U ()/ can b e identified with U ()/G. 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). For example, if is a complete fan, then the G-action on the whole Cm has only one closed orbit, the origin, and the /G quotient Cm / consists of a single p oint. In the rest of the pap er we show that an appropriate notion of the Kempf­Ness set exists for this class of torus actions, and study some of its most imp ortant top ological prop erties. Consider the unit p olydisc (D2 )m = z Cm : |zj | 1 for all j . Given , define / Z ( ) = z (D2 )m : |zj | = 1 if j g( ) , and Z () =


Z ( ).

The subset Z () (D2 )m is invariant with resp ect to the Tm -action. (We have Z () = ZK , where ZK is the moment­angle complex asso ciated with a simplicial complex K in [8, § 6.2].) Note that Z ( ) U ( ), and therefore Z () U () by Prop osition 3.1. Prop osition 3.3. Assume that is a complete simplicial fan. Then Z () is a compact (m + n)-manifold with a Tm -action.
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Pro of. As K is a triangulation of an (n - 1)-dimensional sphere, the result follows from [8, Lemma 6.13] (or [16, Lemma 3.3]). Theorem 3.4. Assume that is a simplicial fan. (a) If is complete, then the composition Z () U () U ()/G induces a homeomorphism Z ()/K U ()/G. (b) There is a Tm -equivariant deformation retraction of U () to Z (). Pro of. Denote by cone K the cone over the barycentric sub division of K and by C () the top ological space |cone K | with the dual face decomposition (see [16, § 3.1] for details). (If is a complete fan, then K is a sphere triangulation, C () can b e identified with the unit ball in NR , and the face decomp osition of its b oundary is Poincar´ dual to K .) The space C () has a face e C ( ) of dimension n - g( ) for each cone . Set / T ( ) = (t1 ,... ,tm ) Tm : tj = 1 for j g( ) . This is a g( )-dimensional co ordinate subgroup in Tm . As detailed in [12] and [16, § 3.1], the set Z () can b e describ ed as the quotient space mo dulo an equivalence relation Z () = (Tm â C ())/, where (t, x) Tm â C () is identified with (s, x) Tm â C () if x C ( ) and t-1 s T ( ) for some . The homomorphism of tori Tm T with kernel K induces a map of the quotient spaces (Tm â C ())/ (T â C ())/. Now, according to [12], if is a complete simplicial fan, then the latter quotient space is homeomorphic to the toric variety X = U ()/G. This proves (a). (Note that if is a regular fan, then K Tm-n and the pro jection Z X is a principal K -bundle.) = Statement (b) is proved in [8, Theorem 8.9]. By comparing this result with Theorem 2.2, we see that Z () has the same prop erties with resp ect to the G-action on U () as the set KNS with resp ect to a reductive group action on an affine variety S . We therefore refer to Z () as the Kempf­Ness set of U (). Example 3.5. Let n = 2 and e1 ,e2 b e a basis in NR . 1. Consider a complete fan having the following three 2-dimensional cones: the first is spanned by e1 and e2 , the second is spanned by e2 and -e1 - e2 , and the third by -e1 - e2 and e1 . The simplicial complex K is a complete graph on three vertices (or the b oundary of a triangle). We have U () = C3 \{z : z1 = z2 = z3 = 0} = C3 \{0} and Z () = (D2 â D2 â S1 ) (D2 â S1 â D2 ) (S1 â D2 â D2 ) = ((D2 )3 ) S5 . = The subgroup G from the exact sequence (3.1) is the diagonal 1-dimensional subtorus in (C )3 , and K is the diagonal sub circle in T3 . Therefore, we have X = U ()/G = Z ()/K = CP2 , the complex pro jective 2-plane. 2. Now consider the fan consisting of three 1-dimensional cones generated by the vectors e1 , e2 and -e1 - e2 . This fan is not complete, but its 1-dimensional cones span NR as a vector space. So Theorem 3.2 applies, but Theorem 3.4(a) do es not. The simplicial complex K consists of three disjoint p oints. The space U () is the complement of three co ordinate lines in C3 : U () = C3 \ z : z1 = z2 = 0, z1 = z3 = 0, z2 = z3 = 0 ,
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and Z () = (D2 â S1 â S1 ) (S1 â D2 â S1 ) (S1 â S1 â D2 ). Both spaces are homotopy equivalent to S3 S3 S3 S4 S4 (see [8, Example 8.15] and [4]). As in the previous example, the subgroup G is the diagonal subtorus in (C )3 . By Theorem 3.2, X = U ()/G, a quasipro jective variety obtained by removing three p oints from CP2 . This variety is noncompact and cannot b e identified with Z ()/K . 4. NORMAL FANS The next step in our study of the Kempf­Ness set for torus actions on quasiaffine varieties U () would b e to obtain an explicit description like the one given by (2.1) in the affine case. Although we do not know 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. Let MR = (NR ) b e the dual vector space. Assume we are given primitive vectors a1 ,... ,am N and integer numb ers b1 ,... ,bm Z, and consider the set P = x MR : ai ,x + bi 0, i = 1,... ,m . (4.1)

We further assume that P is b ounded, the affine hull of P is the whole MR , and the intersection of P with every hyp erplane determined by the equation (ai ,x)+ bi = 0 spans an affine subspace of dimension n - 1 for i = 1,... ,m (or, equivalently, none of the inequalities can b e removed without enlarging P ). This means that P is a convex polytope with exactly m facets. (In general, the set P is always convex, but it may b e unb ounded, not of full dimension, or there may b e redundant inequalities.) By intro ducing a Euclidean metric in NR we may think of ai as the inward p ointing normal vector to the corresp onding facet Fi of P , i = 1,... ,m. Given a face Q P we say that ai is normal to Q if Q Fi . If Q is a q -dimensional face, then the set of all its normal vectors {ai1 ,... ,aik } spans an (n - q )-dimensional cone Q . The collection of cones {Q : Q a face of P } is a complete fan in N , which we denote P and refer to as the normal fan of P . The normal fan is simplicial if and only if the p olytop e P is simple, that is, there are exactly n facets meeting at each of its vertices. In this case the cones of P are generated by subsets {ai1 ,... ,aik } such that the intersection Fi1 ... Fik of the corresp onding facets is nonempty. The Kempf­Ness sets (or the moment­angle complexes) Z (P ) corresp onding to normal fans of simple p olytop es admit a very transparent interpretation as complete intersections of real algebraic quadrics, as describ ed in [9] (these complete intersections of quadrics were also studied in [7]). We give this construction b elow. In the rest of this section we assume that P is a simple p olytop e and, therefore, P is a simplicial fan. We may sp ecify P by a matrix inequality AP x + bP 0, where AP is the m â n matrix of the row vectors ai and bP is the column vector of the scalars bi . The linear transformation MR Rm defined by the matrix AP is exactly the one obtained from the map Tm T in (3.2) by applying HomZ (·, S1 ) Z R. Since the p oints of P are sp ecified by the constraint AP x + bP 0, the formula iP (x) = AP x + bP defines an affine injection iP : MR Rm , which emb eds P in the p ositive cone Rm = {y Rm : yi 0}. Now define the space ZP by a pullback diagram -- - ZP - Z C m P - P R -- -
i m i

(4.2)

P

(4.3)

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where (z1 ,... ,zm ) is given by (|z1 |2 ,... , |zm |2 ). The vertical maps ab ove are pro jections onto the quotients by the Tm -actions, and iZ is a Tm -equivariant emb edding. (b) There is a Tm -equivariant homeomorphism ZP Z (P ). = m and let (z ) b e the set of zero co ordinates of z . Since the Pro of. Assume z ZP C facet Fi of P is the intersection of P with the hyp erplane (ai ,x)+ bi = 0, the p oint P (z ) belongs to the intersection i(z ) Fi , which is thereby nonempty. Therefore, the vectors {ai : i (z )} span a cone of P . Thus, (z ) is a g-subset and z U (P ), which proves (a). To prove (b) we lo ok more closely at the construction of the quotient space from the pro of of Theorem 3.4 in the case when is a normal fan. Then the space C (P ) may b e identified with P , and C ( ) is the face ig() Fi of P . The Kempf­Ness set Z (P ) is therefore identified with (Tm â P )/. (4.4) Prop osition 4.1. (a) We have ZP U (P ).

Now we notice that if we replace P by the p ositive cone Rm (with the obvious face structure) in the ab ove quotient space, we obtain (Tm â Rm )/ = Cm . Since the map iP from (4.3) resp ects facial co dimension, the pullback space ZP can also b e identified with (4.4), thus proving (b). Cho osing a basis for coker AP , we obtain an (m - n) â m matrix C so that the resulting short exact sequence
P - 0 MR - R

A

mC

-R

m-n

0

(4.5)

is the one obtained from (3.2) by applying HomZ (·, S1 ) Z R. We may assume that the first n normal vectors a1 ,... ,an span a cone of P (equivalently, the corresp onding facets of P meet at a vertex), and take these vectors as a basis of MR . In this basis, the first n rows of the matrix (aij ) of AP form a unit n â n matrix, and we may take -an+1,1 ... -an+1,n 1 0 ... 0 -an+2,1 ... -an+2,n 0 1 ... 0 (4.6) C= . . . . .. . . .. . . .. . . .. . . .. -am,
1

...

-am,

n

0 0 ...
m

1

Then the diagram (4.3) implies that iZ emb eds ZP in C quadratic equations
m

as the set of solutions of the m - n real (4.7)

cjk (|zk |2 - bk ) = 0
k =1

for 1 j m - n,

where C = (cjk ) is given by (4.6). This intersection of real quadrics is nondegenerate [9, Lemma 3.2] (the normal vectors are linearly indep endent at each p oint), and therefore ZP R2m is a smo oth submanifold with trivial normal bundle. 5. PROJECTIVE TORIC VARIETIES AND MOMENT MAPS In the notation of Section 2, let fv = (dFv )e : g R. This map takes (see (2.1)). We may consider fv as an element of the dual Lie algebra g . A have g = k ik. Since K is norm-preserving, fv vanishes on k; so we consider ik k . Varying v V we get the moment map µ : V k , which sends v = The Kempf­Ness set is the set of zero es of µ: KN = µ
-1

g to Re v , v s G is reductive, we fv as an element of V , k to iv , v . (5.1)

(0).
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This description do es not apply to the case of algebraic torus actions on U () considered in the two previous sections: as is seen from simple examples b elow, the set µ-1 (0) = {z Cm : z , z = 0 for all k} consists only of the origin in this case. Nevertheless, in this section we show that a description of the toric Kempf­Ness set Z () similar to (5.1) exists in the case when is a normal fan, thereby extending the analogy with Kempf­Ness sets for affine varieties even further. As explained in [5] or [8, § 5.1], the toric variety X is pro jective exactly when arises as the normal fan of a convex p olytop e. In fact, the set of integers {b1 ,... ,bm } from (4.1) determines an ample divisor on XP , thereby providing a pro jective emb edding. Note that the vertices of P are not necessarily lattice p oints in M (as they may have rational co ordinates), but this can b e remedied by simultaneously multiplying b1 ,... ,bm by an integer numb er; this corresp onds to the passage from an ample divisor to a very ample one. Assume now that P is a regular fan; therefore, XP is a smo oth pro jective variety. This implies ahler and, therefore, a symplectic manifold. There is the following symplectic version that XP is K¨ of the construction from Section 3. Let (W, ) b e a symplectic manifold with a K -action that preserves the symplectic form . For every k we denote by the corresp onding K -invariant vector field on W . The K -action is said to b e 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 , (x, ) H (x)

is defined. Example 5.1. 1. A basic example is given by W = Cm with the symplectic form = 2 m dxk dyk , where zk = xk + iyk . The co ordinatewise action of Tm is Hamiltonian with the k =1 moment map µ : Cm Rm given by µ(z1 ,... ,zm ) = (|z1 |2 ,... , |zm |2 ) (we identify the dual Lie algebra of Tm with Rm ). 2. Now let b e a regular fan and K b e the subgroup of Tm defined by (3.2). We can restrict the previous example to the K -action on the invariant subvariety U () Cm . The corresp onding moment map is then defined by the comp osition µ : C
m

R

m

k .

(5.2)

A choice of an isomorphism k Rm-n allows one to identify the map Rm k with the linear = transformation given by the matrix (4.6) (see (4.5)). A direct comparison with (5.1) prompts us to relate the level set µ-1 (0) of the moment map (5.2) to the toric Kempf­Ness set Z (P ) for the G-action on U (P ). However, this analogy is not that straightforward: the set µ-1 (0) = {z Cm : z , z = 0 for all k} is given by the equations m 2 k =1 cjk |zk | = 0, 1 j m - n, which have only the zero solution. (Indeed, as the intersection of Rm with the affine n-plane iP (MR ) = AP (MR )+ bP is b ounded, its intersection with the plane AP (MR ) consists only of the origin.) On the other hand, by comparing (5.2) with (4.7), we obtain Prop osition 5.2. Let P be the normal fan of a simple polytope given by (4.1), and (5.2) be the corresponding moment map. Then the toric Kempf­Ness set Z (P ) for the G-action on U (P ) is given by Z (P ) µ-1 (CbP ). =

P

In other words, the difference b etween our situation and the affine one is that we have to take CbP instead of 0 as the value of the moment map. The reason is that CbP is a regular value of µ, unlike 0.
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By making a p erturbation bi bi + i of the values bi in (4.1) while keeping the vectors ai unchanged for 1 i m, we obtain another convex set P () determined by (4.1). Provided that the p erturbation is small, the set P () is still a simple convex p olytop e of the same combinatorial type as P . Then the normal fans of P and P () are the same, and the manifolds ZP and ZP () defined by (4.7) are Tm -equivariantly homeomorphic. Moreover, CbP () , considered as an element of ahler cone of the toric variety XP [11, § 4]. k = HomZ (K, S1 ) Z R H 2 (XP ; R), b elongs to the K¨ = In the case of normal fans the following version of our Theorem 3.4(a) is known in toric geometry: Theorem 5.3 (see [11, Theorem 4.1]). Let X be a projective simplicial toric variety and ¨ assume that c H 2 (X ; R) is in the Kahler cone. Then µ-1 (c) U (), and the natural map µ is a diffeomorphism. This statement is the essence of the construction of smo oth pro jective toric varieties via symplectic reduction. The submanifold µ-1 (c) Cm may fail to b e symplectic b ecause the restriction of the standard symplectic form on Cm to µ-1 (c) may b e degenerate. However, the restriction of descends to the quotient µ-1 (c)/K as a symplectic form. Example 5.4. Let P = n be the standard simplex defined by n +1 inequalities ei ,x 0, i = 1,... ,n, and -e1 - ... - en ,x +1 0 in MR (here e1 ,... ,en is a chosen basis which we use to identify NR with Rn ). The cones of the corresp onding normal fan are generated by the prop er subsets of the set of vectors {e1 ,... ,en , -e1 - ... - en }. The groups G C and K S1 are the = = diagonal subgroups in (C )n+1 and Tn+1 , resp ectively, while U () = Cn+1 \{0}. The (n +1) â n matrix AP = (aij ) has aij = ij for 1 i, j n and an+1,j = -1 for 1 j n. The matrix C (4.6) is just a row of units. The moment map (5.2) is given by µ (z1 ,... ,zn+1 ) = |z1 |2 + ... + |zn+1 |2 . Since CbP = 1, the Kempf­Ness set ZP = µ-1 (1) is the unit sphere S2n+1 Cn+1 , and X = (Cn+1 \{0})/G = S2n+1 /K is the complex pro jective space CPn . In the next section we consider a more complicated example, while here we conclude with an op en question. Problem 5.5. As is known (see, e.g., [8, Ch. 5]), 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 pro jective (although b eing nonsingular). In this case the toric Kempf­Ness set Z () is still defined (see Section 3). However, the rest of the analysis of the last two sections do es not apply here; in particular, we do not have a description of Z () as in (4.7). Can one still describ e Z () as a complete intersection of real quadratic (or higher order) hyp ersurfaces? 6. COHOMOLOGY OF TORIC KEMPF­NESS SETS Here we use the results of [8] and [16] cohomology rings of toric Kempf­Ness sets. of Z () may b e quite complicated even for s Given an abstract simplicial complex K Stanley­Reisner ring ) Z[K] is defined as the ators: on moment­angle complexes to describ e the integer As we shall see from an example b elow, the top ology imple fans. on the set [m] = {1,... ,m}, the face ring (or the following quotient of the p olynomial ring on m gener-1

(c)/K U ()/G = X

Z[K] = Z[v1 ,... ,vm ]/(vi1 ... vik : {i1 ,... ,ik } is not a simplex of K). We intro duce a grading by setting deg vi = 2, i = 1,... ,m. As Z[K] may b e thought of as a Z[v1 ,... ,vm ]-mo dule via the pro jection map, the bigraded Tor-modules Tor
-i,2j Z[v1 ,...,vm ]

(Z[K], Z)
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10 z T F2 d rr r

T.E. PANOV F1 c

F6

d d F8 d d F5 E y

F7 rrr x ©

F4 0

T F3 Figure.

are defined (see [18]). They can b e calculated, for example, using the Koszul resolution of the trivial Z[v1 ,... ,vm ]-mo dule Z. This also endows Tor [v1 ,...,vm ] (Z[K], Z) with a graded commutative algebra Z structure (the grading is by the total degree); see details in [8, Ch. 7]. Theorem 6.1 (see [8, Theorems 7.6, 7.7; 16, Theorem 4.7]). For every simplicial fan there are algebra isomorphisms H Z (); Z Tor =
Z[v1 ,...,vm ]

Z[K ], Z H [u1 ,... ,um ] Z[K ],d , =

where the latter denotes the cohomology of a differential graded algebra with deg ui = 1, deg vi = 2, dui = vi , and dvi = 0 for 1 i m. Given a subset I [m], denote by K(I ) the corresp onding ful l subcomplex of K, or the restriction of K to I . We also denote by H i (K(I )) the ith reduced simplicial cohomology group of K(I ) with - 2j integer co efficients. A theorem due to Ho chster [14] expresses the Tor-mo dules TorZ[i,1 ,...,vm ] (Z[K], Z) v in terms of full sub complexes of K, which leads to the following description of the cohomology of Z (). Theorem 6.2 (see [16, Corollary 5.2]). We have H k (Z ()) =
I [m]

H

k -|I |-1

(K (I )).

There is also a description of the pro duct in H (Z ()) in terms of full sub complexes of K (see [16, Theorem 5.1]). Example 6.3. Let P b e a simple p olytop e obtained by cutting two nonadjacent edges of a cub e in MR R3 , as shown in the figure. We may sp ecify such a p olytop e by eight inequalities = x 0, y 0, z 0, -x +3 0, -y +3 0, -z +3 0,

-x + y +2 0,

-y - z +5 0,

and it has eight facets F1 ,... ,F8 , numb ered as in the figure.
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P

The 1-dimensional cones of the corresp onding normal fan primitive vectors: a1 = e1 , a2 = e2 , a3 = e3 , a7 = -e1 + e2 , a4 = -e1 ,

are spanned by the following a6 = -e3 ,

a5 = -e2 ,

a8 = -e2 - e3 .

The toric variety XP is obtained by blowing up the pro duct CP1 â CP1 â CP1 (corresp onding to the cub e) in two complex 1-dimensional subvarieties {} â {0}â CP1 and CP1 â{} â{}. The matrix (4.6) is given by 1 0 010000 0 1 0 0 1 0 0 0 C = 0 0 1 0 0 1 0 0 . 1 -1 0 0 0 0 1 0 0 1 100001 Its transp ose determines the inclusion G (C )8 (or K T 8 ), and we have XP = U (P )/G = Z (P )/K by Theorem 3.4. The toric Kempf­Ness set Z (P ) ZP (4.7) is defined by five real = quadratic equations: |z1 |2 + |z4 |2 - 3 = 0, |z2 |2 + |z5 |2 - 3 = 0, |z3 |2 + |z6 |2 - 3 = 0,

|z1 |2 -|z2 |2 + |z7 |2 - 2 = 0,

|z2 |2 + |z3 |2 + |z8 |2 - 5 = 0.

The dual triangulation K is obtained from the b oundary of an o ctahedron by applying two stellar sub divisions at nonadjacent edges [15]. The face ring is Z[K ] = Z[v1 ,... ,v8 ]/ v1 v4 ,v1 v7 ,v2 v4 ,v2 v5 ,v2 v8 ,v3 v6 ,v3 v8 ,v5 v6 ,v5 v7 ,v7 v8 . According to Theorem 6.2, the group H 3 (ZP ) has a generator for every pair of vertices of K that are not joined by an edge (equivalently, for every pair of nonadjacent facets of P ). Therefore, H 3 (ZP ) Z10 , and the generators are represented by the following 3-co cycles in the differential = graded algebra from Theorem 6.1: u1 v4 , u1 v7 , u2 v4 , u2 v5 , u2 v8 , u3 v6 , u3 v8 , u5 v6 , u5 v7 , u7 v8 .

Using Theorem 6.2 again, we see that only the reduced 0-cohomology of three-vertex full sub complexes of K may contribute to H 4 (ZP ). There are two typ es of disconnected simplicial complexes on three vertices: "three disjoint p oints" and "an edge and a p oint." K contains no full sub complexes of the first typ e and 16 sub complexes of the second typ e. The corresp onding 4-co cycles in the differential graded algebra [u1 ,... ,um ] Z[K ] are u4 u7 v1 , u6 u7 v5 , u4 u5 v2 , u3 u5 v6 , u4 u8 v2 , u1 u5 v7 , u5 u8 v2 , u1 u8 v7 , u6 u8 v3 , u5 u8 v7 , u1 u2 v4 , u2 u3 v8 , u2 u6 v5 , u2 u7 v8 , u2 u7 v5 , u3 u7 v8 .

Therefore, H 4 (ZP ) = Z16 . The fifth cohomology group of ZP is the sum of the first cohomology of three-vertex full sub complexes of K and the reduced 0-cohomology of four-vertex full sub complexes. A three-vertex full sub complex of K may have nonzero first cohomology group only if the corresp onding three facets of P form a "b elt," that is, are pairwise adjacent but do not share a common vertex. As there are no
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such three-facet b elts in P , only the reduced 0-cohomology of four-vertex sub complexes contributes to H 5 (ZP ). The corresp onding 5-co cycles are u1 u5 u8 v7 , u2 u3 u7 v8 , u4 u5 u8 v2 , u2 u6 u7 v5 , u2 u7 u5 v8 - u2 u7 u8 v5 ) Z5 . Due = 10, 0, 0, 1) of 6 (Z ) Z5 , P=

(note that the last co cycle cannot b e represented by a monomial). Therefore, H 5 (ZP to Poincar´ duality, this completely determines the Betti vector (1, 0, 0, 10, 16, 5, 5, 16, e the 11-dimensional manifold ZP . The generators of the sixth cohomology group, H corresp ond to the four-facet b elts in P , and the corresp onding 6-co cycles are u2 u3 v4 v6 , u1 u5 v4 v6 , u1 u3 v6 v7 , u1 u3 v4 v8 , u1 u3 v4 v6 .

These are the Poincar´ duals to the 5-co cycles. The fundamental class of ZP is represented (up to e a sign) by the co cycle u4 u5 u6 u7 u8 v1 v2 v3 , or by any co cycle of the form u
(4) u(5) u(6) u(7) u(8) v(1) v(2) v(3)

,

where S8 is a p ermutation such that the facets F(1) , F(2) and F(3) share a common vertex. The multiplicative structure in H (ZP ) can b e easily retrieved from this description. For example, we have the identities [u1 v4 ] · [u1 v7 ] = 0, [u1 v7 ] · [u2 v4 ] = 0, [u1 v4 ] · [u3 v6 ] = [u1 u3 v4 v6 ], etc.

[u2 v4 ] · [u3 v6 ] · [u1 u5 u8 v7 ] = [u1 u2 u3 u5 u8 v4 v6 v7 ],

Yet another interesting feature of the manifold ZP of this example is the existence of nontrivial Massey pro ducts in H (ZP ) [1]. Consider three co cycles a = u1 v4 , b = u2 v5 , and c = u3 v6 representing cohomology classes , , H 3 (ZP ). Since = 0 and = 0, a triple Massey pro duct , , is defined. It consists of the cohomology classes in H 8 (ZP ) represented by the co cycles of the form af + ec for all choices of e and f such that ab = de and bc = df (here d denotes the differential; as there may b e many choices of e and f , the Massey pro duct is a multivalued op eration in general). The Massey pro duct is said to b e trivial if it contains zero. In our case we may take e = u1 u2 u5 v4 and f = 0, so , , contains a nonzero cohomology class [u1 u2 u5 u3 v4 v6 ] H 8 (ZP ). Moreover, , , is nontrivial (see [16, Example 5.7]). This implies that ZP is a nonformal manifold. A detailed study of Massey pro ducts in the cohomology of moment­angle complexes is undertaken in [13]. ACKNOWLEDGMENTS The author gratefully acknowledges many imp ortant remarks and comments suggested by Ivan Arzhantsev and thanks him for the consultation on algebraic group actions and geometric invariant theory. This work was supp orted by the Russian Foundation for Basic Research (pro ject nos. 08-01-00541 and 08-01-91855-KO), by a grant of the President of the Russian Federation (pro ject no. NSh1824.2008.1), and by P. Deligne's 2004 Balzan prize in mathematics. REFERENCES
1. I. V. Baskakov, " Massey Triple Products in the Cohomology of Moment-Angle Complexes," Usp. Mat. Nauk 58 (5), 199­200 (2003) [Russ. Math. Surv. 58, 1039­1041 (2003)]. 2. V. M. Buchstab er and T. E. Panov, Torus Actions in Topology and Combinatorics (MTsNMO, Moscow, 2004). 3. E. B. Vinb erg and V. L. Pop ov, " Invariant Theory," Itogi Nauki Tekh., Ser.: Sovr. Probl. Mat., Fund. Napr. 55, 137­314 (1989); Engl. transl. in Algebraic Geometry IV (Springer, Berlin, 1994), Encycl. Math. Sci. 55, pp. 123­278 . PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS Vol. 263 2008


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4. J. Grbi´ and S. Theriault, " Homotopy Typ e of the Complement of a Coordinate Subspace Arrangement of c Codimension Two," Usp. Mat. Nauk 59 (6), 203­204 (2004) [Russ. Math. Surv. 59, 1207­1209 (2004)]. 5. V. I. Danilov, " The Geometry of Toric Varieties," Usp. Mat. Nauk 33 (2), 85­134 (1978) [Russ. Math. Surv. 33 (2), 97­154 (1978)]. 6. V. V. Batyrev, " Quantum Cohomology Rings of Toric Manifolds," Ast´ erisque 218, 9­34 (1993); arXiv: alg-geom/9310004. 7. F. Bosio and L. Meersseman, " Real Quadrics in Cn , Complex Manifolds and Convex Polytop es," Acta Math. 197 (1), 53­127 (2006). 8. V. M. Buchstab er and T. E. Panov, Torus Actions and Their Applications in Topology and Combinatorics (Am. Math. Soc., Providence, RI, 2002), Univ. Lect. Ser. 24. 9. V. M. Buchstab er, T. E. Panov, and N. Ray, " Spaces of Polytop es and Cob ordism of Quasitoric Manifolds," Moscow Math. J. 7 (2), 219­242 (2007); arXiv: math.AT/0609346. 10. D. A. Cox, " The Homogeneous Coordinate Ring of a Toric Variety," J. Algebr. Geom. 4 (1), 17­50 (1995); arXiv: alg-geom/9210008. 11. D. A. Cox, " Recent Developments in Toric Geometry," in Algebraic geometry: Proc. Summer Res. Inst., Santa Cruz, 1995 (Am. Math. Soc., Providence, RI, 1997), Proc. Symp. Pure Math. 62, Part 2, pp. 389­436 ; arXiv: alg-geom/9606016. 12. M. W. Davis and T. Januszkiewicz, " Convex Polytop es, Coxeter Orbifolds and Torus Actions," Duke Math. J 62 (2), 417­451 (1991). 13. G. Denham and A. I. Suciu, " Moment-Angle Complexes, Monomial Ideals, and Massey Products," Pure Appl. Math. Q. 3 (1), 25­60 (2007); arXiv: math.AT/0512497. 14. M. Hochster, " Cohen­Macaulay Rings, Combinatorics, and Simplicial Complexes," in Ring Theory II: Proc. 2nd Oklahoma Conf., Ed. by B. R. McDonald and R. A. Morris (M. Dekker, New York, 1977), pp. 171­223. 15. M. Masuda and T. Panov, " On the Cohomology of Torus Manifolds," Osaka J. Math. 43, 711­746 (2006); arXiv: math.AT/0306100. 16. T. Panov, " Cohomology of Face Rings, and Torus Actions," in Surveys in Contemporary Mathematics (Cambridge Univ. Press, Cambridge, 2008), LMS Lect. Note Ser. 347, pp. 165­201; arXiv: math.AT/0506526. 17. G. W. Schwarz, " The Top ology of Algebraic Quotients," in Topological Methods in Algebraic Transformation Groups (Birkh¨ ser, Boston, 1989), Progr. Math. 80, pp. 135­151 . au 18. R. P. Stanley, Combinatorics and Commutative Algebra , 2nd ed. (Birkh¨ ser, Boston, 1996), Progr. Math. 41. au

Translated by the author

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