Spherical varieties and arc spaces

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1 Spherical varieties and arc spaces Victor Batyrev, ESI, Vienna 19, 20 January Lecture 1 This is a joint work with Anne Moreau. Let us begin with a few notations. We consider G a reductive connected algebraic group over C, with Borel subgroup B. Let X be a spherical G variety, assumed to be projective and smooth. Recall that spherical means that X admits an open B-orbit. Then it also admits an open G-orbit. Let X 0 be the open G-orbit in X, which can be written as a quotient X 0 = G/H, for some spherical subgroup H of G. Spherical varieties can be described by some combinatorial data, which involve both the group G and the space X. More precisely, this description splits in two parts. First one needs to understand spherical subgroups, and second to understand equivariant embeddings of X 0 in X. Both steps of the description are combinatorial and it turns out that the second part is easier than the first one. Equivariant embeddings are described by the Luna-Vust theory via colored fans. The spherical subgroups can be described up to conjugacy by some more involved combinatorial data. With this picture in mind, there should be an interpretation of the betti numbers of X, b i (X) = dim C H i (X, C), in terms of combinatorics. Problem. How to compute b i (X) using combinatorics involving X and G? This problem has a solution and it is the purpose of these lectures to explain it. Let us first look at some examples. Example 1. Assume that X is homogeneous, that is X = X 0, in which case H is parabolic since X is projective. For simplicity, take P = B. We want to compute B(G/B, t) = i 0 b 2i (G/B)t i. (1.1) Recall that the homogeneous space G/B is spherical since the group W = B\G/B is finite. The Betti polynomial B(G/B, t) can be expressed in terms of the length of elements in W : B(G/B, t) = t l(g). (1.2) g W When G = SL(2), G/B = P 1 and B(G/B, t) = 1 + t. 1

2 Example 2. Assume that G = T = B and that H is the trivial group {e}. In other words, X is a smooth projective toric variety. Let M = X(T ) be the character lattice, N = X (T ) the dual lattice and F the fan of cones corresponding to X. It spans N R because X is projective. There is a one-to-one correspondance between G-orbits in X and cones σ in F. These cones have different dimensions but satisfy dim(σ) = codim(o σ ), (1.3) where O σ denotes the G-orbit in X corresponding to the cone σ F. Table 1: Fan of a toric variety These orbits are themselves tori of smaller dimensions. Therefore, B(X, t) = σ B(O σ, t) = σ (t 1) codimσ. (1.4) Note that B(X, p) = X(F p ) for prime integer p. For instance, P 1 has three orbits under C. Two of 0 dimension and one of dimension 1. The previous formula gives B(P 1, t) = (t 1) = 1 + t. Example 3. Let us now consider the case of horospherical varieties which includes both Example 1 and Example 2. Recall that the spherical homogeneous space X = G/H is called horospherical if H contains the unipotent radical U of some Borel subgroup B. Any embedding X 0 X of a horospherical homogeneous space X 0 = G/H is called horospherical. To make the things more concrete, let us consider G = SL(2) and U H B. The dimension of H can be either 1 or 2. The case H = B has already been studied. So let us assume that dim H = 1. The subgroup H can be the unipotent group U itself or some finite extension: ( ) λ U k = { 0 λ 1, λ k = 1}, U k /U = µ k, (1.5) 2

3 Table 2: Fan of P 1 where µ k is the cyclic group of k-roots of unity. If G/H is horospherical, the normalizer P = N G (H) is parabolic. The commutator [P, P ] is a subset of H, therefore T = P/H is a torus of dimension 1. The following map G/H G/P (1.6) is a fibration of the projective homogeneous space G/P with toric fibers. Any horospherical homogeneous space can be determined by the choice of a parabolic group P and of a subgroup H which seats in between U and B. Consider an embedding of G/H into a smooth projective spherical varietie X: G/H X. (1.7) A naive way to proceed would be two use the decomposition of X as a disjoint union of its G-orbits through the formula (which we used in example 2): B(X, t) = x B(O x, t). (1.8) The problem here is that in full generality it is not obvious to compute the Betti polynomials of the G-orbits. We will explain another way of computing them using arc spaces. Let us go back to our example where H = U k for some k 1. There are little choices to embed G/H in a smooth projective spherical variety. When k = 1, there are two such embeddings. It is either P 2 or F 1 (which is a blowup of a point in P 2 ). If k 2, there is only one choice for X, namely F k, the blowing-up of P 2 at k general points. As for the combinatorial data needed to describe such embeddings via the Luna-Vust theory, it depends on (C 2 \0)/µ k where µ k is the group of k-roots of unity. 3

4 Let us know write down the formula which allows to compute Betti polynomials. On the Q-span of the lattice N, there is a nice negative piecewise linear (on each cone of the fan of X) function κ: The formula for the Betti polynomial reads κ : N Q Q. (1.9) B(X, t) = B(G/H, t) n N t κ(n). (1.10) In the case k = 1, we have and B(P 2, t) = B(C 2 \0, t)(1 + (t 1 + t ) + (t 2 + t )) = (t 2 1)(1 + 1 t t 2 1 ) = 1 + t + t2. B(F 1, t) = B(C 2 \0, t)(1 + (t 1 + t ) + (t 1 + t )) = (t 2 1)(1 + 1 t t 1 ) = 1 + 2t + t2. Table 3: Colored fans of P 2 and F 1 This formula works for all horospherical embeddings and involve infinite series in t 1. For instance, we can revisit the example of a toric spherical variety T X with equation (1.10) in mind. Recall that T -orbits in X are parameterized by cones σ and that B(X, t) = σ (t 1) codimσ = (t 1) d σ 4 1 (t 1) dimσ B(T, t)q((t 1 )). (1.11)

5 Now, we move on more complicated spherical varieties where we have to consider spherical roots. Spherical roots are in some sense the analogue of simple roots of reductive algebraic groups in the theory of spherical varieties. One can associate to spherical roots some finite group, called the small Weyl group, generated by reflections. The fundamental domain for this action is called the valuation cone. The Luna-Vust theory puts restrictions on how spherical and simple roots should interact each other. These sophisticated constraints did not appear in the horospherical case precisely because a horospherical variety does not have spherical roots. We now consider a fourth example of a spherical variety which admits spherical roots. Example 4. G = SL(2), H = T. In this case, G/H = P 1 P 1 \( = P 1 ). The torus T is the stabilizer of (0, ). X 0 = G/H and X = P 1 P 1. The coordinate ring of the homogeneous space G/H is filtered C[G/H] = i 0 R i, (1.12) where R i f(x 0, x 1, y 0, y 1 ) = x i 0 y 0 x 1 y 1 f is bihomogeneous of degree i. (1.13) It is clear that dim R i = (i + 1) 2. The associated graded ring has blocks of odd dimensions: grr = i 0 R i /R i 1. (1.14) As a G-module, it splits into the direct sum of all even simple modules of G, V (2i). The graded ring C[SL(2)/T ] is isomorphic to the coordinate ring of G/U 2, as a G-module, but not as a ring. C[SL(2)/U 2 ] = grr. (1.15) In some sense, the horospherical variety G/U 2 is a degeneration of G/T. We call U 2 a horospherical satellite of H. We have the following formula for the Betti polynomial of X: B(X, t) = B(G/T, t) + B(G/U 2, t) n>0 = (t 2 + t) + (t 2 1)(t 1 + t 2 + ) = (t 2 + t) + (t 2 1 1) t 1 = t2 + 2t t n

6 2 Lecture 2 Table 4: Colored fans of X = P 1 P 1 Let X be a (spherical) projective algebraic variety over C. Set E(X; u, v) = ( 1) p+q h p,q (X)u p v q, 0 p,q dim X where h p,q (X) = dim h p (Ω q ), 0 p, q dim X = d. Since X is spherical, we have E(X; u, v) = B(X, t) = i b 2i(X)t i, with t = uv, since h p,q (X) 0 implies that p = q. One can embed G/H into some projective space P(V ) for some finite dimensional vector space V, and consider the closure G/H. But in general this closure is singular. For such embeddings, the string theory is necessary to consider analog of the formula for Betti polynomials. 2.1 Stringy E-functions One needs additional assumptions. Assume that X = G/H is normal and Q-Gorenstein, that is, lk X is Cartier for some positive integer l N. To define the stringy E-function E st, one needs a resolution of singularities. Let ρ: Y X be a resolution of singularities with Y smooth projective and such that the irreducible components D 1... D r of exc(ρ) are smooth projective simple normal crossing divisors. This choice is always possible. Then r K Y = ρ (K X ) + a i D i, a i > 1, la i N for some l. i=1 Set I = {1,..., r}. For J I, set D J = Y if J =, and D J = D j otherwise. 6 j J

7 Definition 1. E st (X; u, v) = J I E(D J ; u, v) j J ( ) uv 1 (uv) a i Theorem 1 (B., 1998). E st (X; u, v) does not depend on the resolution ρ. Corollary 2. If X is smooth then E st (X; u, v) = E(X; u, v). The independence of the resolution comes from integration over the space of arcs of Y : C((s)) C[[s]] C. The motivic integral looks µ R where R Y (C[[s]]) is some ring and µ some measure. Then µ = by the properties of Y (C[[s]]) Y reg(c[[s]]) the motivic integrals... The independence property can be viewed as the Jacobi transform for motivic integrals. What foregoes actually holds for non spherical varieties. 2.2 Luna-Vust theory (1983). Recall that X is spherical. Then X is an embedding of some spherical homogeneous space X 0 = G/H. Considering the fields C(X) (B) C(X), we define the set of weights of elements of C(X) (B). Let M = Z r be the weight lattice, that is, the set of such weights. The open B-orbit B.x X 0 is isomorphic to (C ) r C d r, with r d = dim X. The number r is the rank of G/H. Set N = Hom(M, Z), N Q = N Z Q. The valuation cone V is contained in N Q. It is cosimplicial and we have V = {x N Q s i x 0 i}, where the s 1,..., s k, k r, are the spherical roots. Let O = C[[t]] and K = C((t)). The set G(O) acts on (G/H)(K) and G(O) \ (G/H)(K) = V N. Example 5. Let X be the set of n n-size matrices, and ( G = ) GL(n) ( ) GL(n). 0 Then G acts on X by (g 1, g 2 ).A = g 1 Ag2 1. Take B = { }. We 0 have X 0 = GL(n) M(n n, C) = X and G(O) \ (G/H)(K) = GL(n)(O) \ GL(n)(K)/GL(n)(O) t a 1 t a 2 =..., a 1 a 2 a n. t an 7

8 2.3 Satellites of spherical subgroups Let v V N be a primitive point. Then consider the embedding X 0 = G/H X v = X 0 D v, with D v a G-invariant divisor, corresponding to the uncolored ray Q + v. It is an elementary embedding, that is, it consists in only two G-orbits, with one of codimension one. Example 6. The spherical embedding (P 1 P 1 \ ) is elementary. The group G acts on the normal bundle N Dv of D v in X v. Then N Dv is spherical and it is an elementary embedding of some spherical homogeneous space G/H v, with dim H v = dim H. N Dv = G/H v D v, Theorem 3. The subgroup H v only depends on the minimal face of V containing v. Hence, there are only, up to G-conjugacy, 2 k such subgroups H v which we can the satellites of H. Example 7. Return to the example X = M(n n, C) and G = GL(n) GL(n). The spherical roots s 1,..., s n 1 are in bijection with the simple roots α 1,..., α n 1. In the case, the small Weyl group is S n S n. It acts on N Q, and V is a fundamental domain for it. Let I {s 1,..., s n 1 }. Consider the two opposite parabolic subgroups P + I and P I containing the root spaces associated to the α i s, i I. and set L I = P + I P I. The faces of V are in bijection with subsets I, and H I = {(g 1, g 2 ) P + I P I L(g 1 ) = L(g 2 )}, with L: P ± I L I the canonical projection. We have dim H I = n 2 = dim H. Note that the space a 0 a H = { a C } a 0 a is horospherical. On the opposite side, H {1,...,n 1} = H. Theorem 4. E st (X; u, v) = I {1,...,k} E(G/H I ; u, v) n V 0 I (uv) κ(n). Here X is a Q-Gorenstein projective spherical variety, with colored fan F in N Q V, and κ: N Q Q is a piecewise linear function, defined by some conditions depending on the canonical divisor K X = ( D i ) + c i D i. Note that c i 1 for all i. 8 D i G-inv D i B-inv, not G-inv

9 Remark. It is not so satisfying to summate over the different V I. Somehow, we would wish a more global formula. Write the formula for E st (X; u, v) as E st (X; u, v) = E(G/H; u, v) I {1,...,k} E(G/H I ; u, v) E(G/H; u, v) n V 0 I (uv) κ(n) in order to factorise by the E-function of the open G-orbit. Then it is natural to study the ratio E(G/H I; u, v) E(G/H; u, v). We conjecture that it is a polynomial in t 1 with integer coefficients. We proved it only in a few cases. For example, assume that r = 1, then V is a half-line and we observe that E(G/H I ; u, v) E(G/H; u, v) = 1 ± t a, a N. Question. Can we interpret the sign and the integer a, for instance in term of the small Weyl group? 9

Geometry and combinatorics of spherical varieties.

Geometry and combinatorics of spherical varieties. Notes of a course taught by Guido Pezzini. Abstract This is the lecture notes from a mini course at the Winter School Geometry and Representation Theory