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Almost homogeneous toric varieties
Ivan Arzhantsev Moscow State University

based on a joint work with J¨ urgen Hausen
"On emb eddings of homogeneous spaces with small b oundary"

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1. Automorphisms of toric varieties M.Demazure (1970) ­ smo oth complete toric varieties (ro ots of the fan); D.Cox (1995) ­ simplicial complete toric varieties (graded automorphisms of the homogeneous co ordinate ring); D.B¨ er (1996) ­ complete toric varieties (a generalization of Cox's uhl results); W.Bruns - J.Gub eladze (1999) ­ projective toric varieties in any characteristic (graded automorphisms of semigroup rings, p olytopal linear groups).
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Let G b e a connected simply connected semisimple algebraic group over C, e.g., G = S L(n), S p(2n), S pin(n), . . . . Problem 1. Describ e all toric varieties X such that G Aut(X ) and G : X with an op en orbit Gx (1-condition). Problem 2. codimX (X \ Gx) Solve Problem 2 (2-condition). 1 under the assumption

Examples. S L(n + 1) : Pn,

S L(n + 1) : Pn â Pn (n > 1).

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2. Cox's construction
For a given toric variety X with C l(X ) = Zr there exist an op en U V = CN and T S = (Câ )N : V such that a go o d quotient : U U //T = X and (S/ T ) : X
[A go o d quotient of a T -invariant op en subset U V is an affine T -invariant morphism : U Z such that the pullback map : OZ (OU )T to the sheaf of invariants is an isomorphism]

Moreover, an op en W U with codimV (V \ W ) 2 and -1 ( (w)) is a T -orbit for any w W (in particular, one may assume that -1 (X reg ) W ) W = U (i.e., is a geometric quotient) iff X is simplicial (V , T , U ) is the Cox realization of X Lemma. If V is a T -mo dule, U V admits : U U //T , and W U : codimV (V \ W ) 2, -1 ( (w)) is a T -orbit for any w W, then (V , T , U ) is the Cox realization of X := U //T .

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Claim 1) G : X (G â T ) : V ; 2) 1-condition (G â T ) : V with an op en orbit (and linearly, [H.KraftV.Pop ov'85]); in this case V is a prehomogeneous vector space; 3) 2-condition G : V with an op en orbit. Example. S L(n + 1) : Pn â Pn (S L(n + 1) â (Câ)2) : Cn+1 â Cn+1 with an op en orbit; S L(n + 1) : Cn+1 â Cn+1 with an op en orbit n > 1. Idea: Given a prehomogeneous G-mo dule V = V1 · · · Vr , let T = (Câ)r : V , (t1, . . . , tr ) (v1, . . . , vr ) = (t1v1, . . . , tr vr ), and T T b e a subtorus. We shall obtain all X as U //T for a (G â T )-invariant op en U V , and give a combinatorial description of such U .
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3. Quotients of torus actions T is a torus, (T ) - the character lattice, (T )Q = (T ) Z Q T : V = r=1 Vi , where 1, . . . , r (T ), and i V = {v V | t v = (t)v } op en T -invariant U V is a go o d T -set if it admits a go o d quotient : U U //T W U is saturated if Y U //T such that W = -1(Y ) U V is maximal if it is a go o d T -set maximal with resp ect to op en saturated inclusions
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(V ) = {C one(i1 , . . . , ip ) (T )Q | {i1, . . . , ip} {1, . . . , r } }
0 0 a collection of cones (V ) is connected if 1, 2 : 1 2 =

is maximal if it is connected and is not a prop er sub collection of a connected collection v = vi + · · · + vip V , vj = 0 (v ) = C one(i1 , . . . , ip ) 1 U () = {v V | 0 : 0 ( v )} V

U V (U ) = { (v ) | v U, T v is closed in U } Theorem { maximal U V } { maximal (V )} ´e [A. Bialynicki-Birula - J. Swi¸cicka (1998)]
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Which maximal defines quasiprojective U ()//T ? (T ) = ( ) =
(v )

(v ) ( ) = { (V ) | 0 0}

the cones ( ) form a fan (GIT-fan) with supp ort C one(1, . . . , r ) Claim. U ()//T is quasiprojective iff = ( ) for some = ( ) U (( ))//T is projective C one(1, . . . , r ) is strictly convex and all i are non-zero is interior if C one(1, . . . , r ) generic fib ers of are T -orbits U (( ))//T is affine C one(1, . . . , r ) = (T )Q and = 0
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a toric variety X is 2-complete if X X with codimX (X \ X ) implies X = X . Examples: complete, affine

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X is 2-complete iff the fan of X cannot b e enlarged without adding new rays for an interior maximal collection the variety U ()//T is 2-complete and for a 2-complete variety X the subset U in the Cox realization is maximal in V

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4. A description of 2-complete toric G-varieties
Ingradients: · prehomogeneous (G â T)-mo dule V = V1 · · · Vr , dim Vi · a subtorus T T a primitive sublattice ST Zr = (T); · the standard basis e1 , . . . , er of Zr and the projection : Zr Zr /ST = (T ); · = {C one((ei1 ), . . . , (eip )) | {i1 , . . . , ip } {1, . . . , r }}; · an interior maximal collection 2;

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V is G-prehomogeneous X = U ()//T satisfies 2-condition V is (G â T )-prehomogeneous X = U ()//T satisfies 1-condition Interior maximal collections Bunches of cones in the divisor class group (F. Berchtold - J. Hausen'2004) Fans (via a linear Gale transformation)

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5. Examples 1) V = V1 a) T = T = Câ, (e1) = e1, = {Q+}, U () = V \ {0}, X = P(V ); b) T = {e}, (e1) = 0, = {0}, U () = X = V 2) V = V1 V2 a) T = T = (Câ)2, X = P(V1) â P(V2); b) T = {e}, X = V ;

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c) dim T = 1, here we consider only some particular cases: (1) ST = (1, -1) , t(v1, v2) = (tv1, tv2) (e1) = (e2) = 1, = {Q+}, U () = V \ {0}, X = P(V ); (2) ST = (0, 1) t(v1, v2) = (tv1, v2) (e1) = 1, (e2) = 0, = {Q+}, U () = {(v1, v2) | v1 = 0}, X = P(V1) â V2; (3) ST = (1, 1) t(v1, v2) = (tv1, t-1v2) (e1) = 1, (e2) = -1 (3.1) = {0, Q}, U () = V , X = V //T and may b e realized as the cone C = {v1 v2} of decomp osible tensors in V1 V2; (3.2) = {Q+, Q}, U () = {(v1, v2) | v1 = 0}, X is a "small blow-up" of C at the vertex with the exceptional fib er P(V1)
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6. Prehomogeneous vector spaces
1) G is simple and V is G-prehomogeneous­ E.B. Vinb erg (1960)
· G = S L(m), · G = S L(2m + 1), · G = S L(2m + 1), · G = S L(2m + 1), · G = S p(2m), · G = S pin(10), V = (Cm )r , r < m, V= V= V= V =C V =C
2

V ;

C
2

2m+1

,

V ;
2

C
2

2m+1

C

2m+1

,

V ; V ;

C
2m

2m+1

(C

2m+1 )

,

;

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2) V is irreducible and (G â T)-prehomogeneous ­ M. Sato, T. Kimura (1977) 3) G contains 3 simple factors and V is (G â T)-prehomogeneous ­ T. Kimura, K. Ueda, T. Yoshigaki (1983,...)
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