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Toric Kempf­Ness sets
Taras Panov Moscow State University

Toric Geometry meeting MFO 4-9 January 2009


1. Categorical quotient. G a reductive algebraic group, S an affine G-variety. S,G : S S/ G morphism dual to C[S ]G C[S ]. / S,G is surjective and establishes a bijection closed G-orbits of S p oints of S/ G. /

S,G is universal in the class of morphisms from S constant on G-orbits in the category of algebraic varieties. S/ G is called the categorical quotient. /

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2. Kempf­Ness sets for affine varieties. : G GL(V ) a representation, K G a maximal compact subgroup, , a K-invariant hermitian form on V with asso ciated norm . Given v V , consider the function 1 Fv : G R, g g v 2. 2 It has a critical p oint iff Gv is closed, and all critical p oints of Fv are minima. Define the subset KN V by KN = = = = {v {v {v {v V V V V : : : : (dFv )e = Tv Gv v v , v = v , v = 0} (e G is the unit) } 0 for all g} 0 for all k}.

(1)

Any v KN is a closest p oint to the origin in its orbit Gv . KN is called the Kempf­Ness set of V .
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Assume that S is G-equivariantly emb edded as a closed subvariety in a representation V of G. Then KN S := KN S , the Kempf­Ness set of S . The imp ortance of Kempf­Ness sets for the study of orbit quotients is due to the following result. Thm 2. (a) [Kempf­Ness] The comp osition KN S S S/ G / is prop er and induces a homeomorphism KN S /K - S/ G. / (b) [Neeman] There is a K-equivariant deformation retraction S KN S .
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=


3. Toric varieties. N = Zn an integral lattice, NR = N Z R. C = C \ {0} the multiplicative group of complex numb ers, S1 the subgroup of complex numb ers of absolute value one. TC = N T = N Z
ZC S1

= (C )n the algebraic torus, = (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 . fans in NR complex n-dim toric varieties X non-singular varieties compact varieties
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regular fans complete fans


4. Batyrev­Cox construction. Assume that one-dimensional cones of span NR as a vector space. m the numb er of one-dimensional cones. ai N the primitive generator of the ith one-dim cone, 1 Consider the map Zm N , ei ai. i m.

The corresp onding maps of tori fit into exact sequences 1 - G - (C)m - TC - 1, 1 - K - T m - T - 1 (3) (4)

where G is isomorphic to a 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, and similarly for K.
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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. 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() = and U () = Cm \ A().
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{i1,...,ik } is not a g -subset

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


Unlike G and K, b oth A() and U () dep end only on the combinatorial structure of the simplicial complex K; the set U () coincides with the co ordinate subspace arrangement complement U (K). The set A() is an affine variety, while its complement U () admits a simple affine cover, as describ ed in the following statement.
= Prop 5. Given a cone , set 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 ( ).


The complement U () Cm is (C)m-invariant. If is simplicial, the subgroup isotropy subgroups (or freely if identified with the toric variety G (C)m acts on U () with finite is regular). The quotient can b e X determined by :

Thm 6. [Cox] (a) The toric variety X is 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, then all the orbits of the G-action on U () are closed and we have U ()/ G = U ()/G. / However, the corresp onding Kempf­Ness set cannot constructed in the standard way, as U () is not an affine variety in Cm.
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5. The moment-angle complex. Consider the unit p olydisc (D2)m = {z Cm : |zj | Given a cone , define Z ( ) = {z (D2)m : |zj | = 1 if j g ( )}, / and the moment-angle complex Z () =



1 for all j }.

Z ( ) (D2)m.

Z () is T m-invariant. Also, Z () U (). Prop 7. Assume is complete simplicial. Then Z () is a compact T m-manifold of dimension m + n.
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6. Toric Kempf­Ness sets. Z () has the same prop erties with resp ect to the G-action on U () as KN S with resp ect to the G-action on an affine variety S : Thm 8 (Buchstab er-P.'00). Assume is simplicial. (a) If is complete, then the comp osition Z () U () U ()/G induces a homeomorphism Z ()/K U ()/G. (b) There is a T m-equivariant deformation retraction U () Z (). We therefore refer to Z () as the toric Kempf­Ness set of U ().
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Ex 9. 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 spanned by e2 and -e1 - e2, and the third spanned by -e1 - e2 and e1. The simplicial complex K is a complete graph on 3 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. Then G is the diagonal subtorus in (C)3, and K is the diagonal sub circle in T 3. Therefore, X = U ()/G = Z ()/K = CP 2.
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2. Now consider the fan consisting of three 1-dimensional cones generated by vectors e1, e2 and -e1 - e2. This fan is not complete, but its 1-dimensional cones span NR as a vector space. So Cox' Thm 6 applies, but Thm 8 (a) do es not. We have K = 3 disjoint p oints, U () = C3 \ {z1 = z2 = 0, z1 = z3 = 0, z2 = z3 = 0}, 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. G is again a diagonal subtorus in (C)3. By Thm 6, X = U ()/G = CP 2 \ {3 p oints}. This in non-compact, and cannot b e identified with Z ()/K.
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7. Polytop es and normal fans.
NR the dual vector space. Given primitive vectors a1, . . . , am N and integer numb ers b1, . . . , bm Z, consider P = {x NR : ai, x + bi

0 for 1

i

m}.

Assume: · P is b ounded;
· the affine hull of P is the whole NR;

· no redundant inequalities; · no (n + 1) hyp erlanes ai, x + bi = 0 meet at a p oint. Then P is a convex simple p olytop e with m facets Fi = {x P : ai, x + bi = 0} with normal vectors ai, for 1 i m.
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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 affine injection
iP : NR - Rm,

x AP x + bP 0}.

emb eds P into Rm = {y Rm : yi

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Now define the space ZP by a pullback diagram
µP

ZP -Z Cm
µ

i

P

- Rm

iP

where µ(z1, . . . , zm) = (|z1|2, . . . , |zm|2). Here iZ is a T m-equivariant emb edding. The normal fan P consists of the cones spanned by the sets {ai1 , . . . , aik } such that the intersection Fi1 . . . Fik of the corresp onding facets is non-empty. P is a simplicial fan. Prop 10. (a) We have ZP U (P ). (b) There is a T m-homeomorphism ZP = Z (P ).
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8. Complete intersections of real quadrics.
The linear transformation AP : NR Rm is exactly the one obtained from T m T by applying HomZ( · , S1) Z R.

Applying HomZ( · , S1) Z R to the whole exact sequence of tori, we obtain
C AP 0 - NR - Rm - Rm-n - 0,

where Rm-n = HomZ(G, S1) Z R. Assume the first n normal vectors a1, . . . , an span a cone of P , and take these vectors as a basis of NR. In this basis, we may take an+1,1 . . . -an+1,n a +2 . . . -an+2,n C = (cij ) = n. ,1 . . . . . . . . -am,1 . . . -am,n


- -

1 0 . . . 0

0 1 . . . 0

... 0 . . . 0 ... . . . . ... 1
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Then ZP emb eds in Cm as the space of common solutions of m - n real quadratic equations
m k=1

cj k |zk |2 - bk

(

)

= 0, for 1 j m - n.

This intersection is non-degenerate, so ZP Cm is a smo oth submanifold with trivial normal bundle (Buchstab er-P-Ray'07). The projective toric variety XP = XP can b e obtained from the action of K on U (P ) Cm via the pro cess of symplectic reduction. The moment map µP is given by the comp osition
µ C Cm - Rm - Lie(K) = Rm-n,

where µ(z1, . . . , zm) = (|z1|2, . . . , |zm|2) and C = (cj k ), so ZP = µ-1 (C bP ). P is its level surface. Then XP = ZP /K.
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Question 11. 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 non-singular, but not projective. In this case the Kempf­Ness set Z () is still defined. Is there a description of Z () similar to that of Z (P ) as a complete intersection of real quadrics?

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9. Cohomology of Kempf­Ness sets. Given an abstract simplicial complex K on the set [m], the face ring (or the Stanley­Reisner ring) Z[K] is the quotient Z[K] = Z[v1, . . . , vm]/(vi1· · · vik : {i1, . . . , ik } K). / Thm 12. [Buchstab er-P., Franz] For every simplicial fan there are algebra isomorphisms
( ) ( ) Z (); Z Tor H = Z[v1 ,...,vm ] Z[K ], Z [ ]

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

where deg ui = 1, deg vi = 2, dui = vi, dvi = 0, for 1

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T

z

F1
c d d d d d d

Ex 13. Let P b e the simple p olytop e obtained by cutting two non adjacent edges off a cub e in NR = R3. We may sp ecify P by 8 inequalities: x 0, y 0, 0, 0, z 0, 0, 0, 0. x

d d r r rr r

F2

F6

F8

F7
rrr r r ©

F4


F5
E

-x + 3 -x + y + 2

-y + 3

0


y

-z + 3

-y - z + 5

T



F3

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Toric variety XP is obtained by blowing up the pro duct CP 1 â CP 1 â CP 1 at two complex 1-dimensional subvarieties {} â {0} â CP 1 and CP 1 â {} â {}. The Kempf­Ness set ZP is given by 5 real quadratic equations: |z1|2 + |z4|2 - 3 = 0, |z3|2 + |z6|2 - 3 = 0, |z2|2 + |z5|2 - 3 = 0,

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

|z2|2 + |z3|2 + |z8|2 - 5 = 0. It is an 11-dimensional manifold with Betti vector (1, 0, 0, 10, 16, 5, 5, 16, 10, 0, 0, 1) and non-trivial Massey pro ducts of 3-dimensional classes (Baskakov'03).
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