Birational geometry and deformations of nilpotent orbits

Transcription

1 arxiv:math/ v1 [math.ag] 6 Nov 2006 Birational geometry and deformations of nilpotent orbits Yoshinori Namikawa In order to explain what we want to do in this paper, let us begin with an explicit example. Let O be the nilpotent orbit of sl(4, C) with Jordan type [3, 1] (under the adjoint action of G := SL(4,C)). We will denote by X i,j,k the cotangent bundle T (G/P i,j,k ) of the projective manifold G/P i,j,k where P i,j,k is a parabolic subgroup of G with flag type (i, j, k). Then the closure Ō of O admits three different Springer resolutions and µ 1,1,2 : X 1,1,2 Ō, µ 1,2,1 : X 1,2,1 Ō, µ 2,1,1 : X 2,1,1 Ō. The set of µ 1,1,2 -numerical classes of line bundles on X 1,1,2 becomes a finitely generated Abelian group, which we denote by N 1. We put NR 1 := N1 Z R. Then the (closure of the) movable cone Mov(X 1,1,2 /Ō) is a half space of N1 R and it is divided into three ample cones Amp(X 1,1,2 /Ō), Amp(X1,2,1/Ō) and Amp(X 2,1,1 /Ō). X 1,1,2 X 1,2,1 X 2,1,1 1

2 2 Our interest is in describing the another half space of NR 1. For this purpose, we shall construct a flat map 1 π : Y C with π 1 (0) = Ō and consider the (crepant) simultaneous resolutions of Y. There are exactly six different such resolutions of Y (cf. Theorem 1.3), say X 1,1,2 +, X 1,1,2, X 1,2,1 +, X1,2,1, X 2,1,1 + and X 2,1,1. Here X + i,j,k and X i,j,k give the Springer resolution µ i,j,k : X i,j,k Ō over 0 C. Let us denote by µ+ 1,1,2 the resolution X 1,1,2 + Y. Let N1 (µ + 1,1,2 ) be the set of µ+ 1,1,2-numerical classes of line bundles on X 1,1,2, + and put N 1 (µ + 1,1,2) R := N 1 (µ + 1,1,2) Z R. Now, the whole space N 1 (µ + 1,1,2) R is divided into six ample cones: X + 1,1,2 X 1,1,2 X + 1,2,1 X 1,2,1 X + 2,1,1 X 2,1,1 By the commutative diagram X 1,1,2 X + 1,1,2 there is a natural identification Ō Y, N 1 (µ + 1,1,2) R = N 1 R. Under this identification, Amp(X 1,1,2/Y + ), Amp(X 1,2,1/Y + ) and Amp(X 2,1,1/Y + ) are identified respectively with Amp(X 1,1,2 /Ō), Amp(X1,2,1/Ō) and Amp(X2,1,1/Ō). The main purpose of this paper is to generalize this construction to all parabolic subalgebras p of a complex simple Lie algebra g and their Richardson orbits O. The main result is Theorem 1.3 (see also Remarks 1.6, 1.8). 1 For the precise definition of Y, see 1.

3 1 3 Theorem 1.3 is valid even if the Springer map is not birational. Example 1.9 would be of interest in this sense. Finally, the author would like to thank A. Fujiki for letting him know [Fuj], where he learned a key idea for this work. 1 Let G be a complex simple Lie group and let g be its Lie algebra. We fix a maximal torus T of G and denote by h its Lie algebra. Then h is a Cartan subalgebra of g. Let p be a parabolic subalgebra of g which contains h. Let r(p) (resp. n(p)) be the solvable radical (resp. nilpotent radical) of p. We put k(p) := r(p) h. Then r(p) = k(p) n(p). On the other hand, a Levi factor l(p) of p is defined as l(p) := g k(p). Then p = l(p) n(p). The Lie algebra g becomes a G-variety via the adjoint action. Let Z g be a closed subvariety. For m N, put Z (m) := {x Z; dim Gx = m}. Z (m) becomes a locally closed subset of Z. We put m(z) := max{m; m = dim Gx, x Z}. Then Z m(z) is an open subset of Z, which will be denoted by Z reg. Theorem 1.1. Gr(p) = Gk(p) = Gk(p) reg. Proof. See [B-K], Satz 5.6. Every element x of g can be uniquely written as x = x n + x s with x n nilpotent and with x s semi-simple such that [x n, x s ] = 0. Let W be the Weyl group of G with respect to T. The set of semi-simple orbits in G is identified with h/w. Let g h/w be the map defined as x [O xs ]. There is a direct sum decomposition r(p) = k(p) n(p), (x x 1 + x 2 ) where n(p) is the nil-radical of p (cf. [Slo], 4.3). We have a well-defined map G P r(p) k(p)

4 1 4 by sending [g, x] G P r(p) to x 1 k(p) and there is a commutative diagram G P r(p) Gr(p) by [Slo], 4.3. k(p) h/w. Lemma 1.2. The induced map is a birational map. G P r(p) µp k(p) h/w Gr(p) Proof. Let h k(p) reg and denote by h h/w its image by the map k(p) h/w. Then the fiber of the map Gr(p) h/w over h coincides with a semi-simple orbit O h of g containing h. In fact, the fiber actally contains this orbit. The fiber is closed in g because Gr(p) is closed subset of g by Theorem 1.1. Note that a semi-simple orbit of g is also closed. Hence if the fiber and O h do not coincide, then the fiber contains an orbit with larger dimension than dim O h. This contradicts the fact that Gk(p) reg = Gr(p). Take a point (h, h ) k(p) reg h/w Gr(p). Then h is a semi-simple element G-conjugate to h. Fix an element g 0 G such that h = Ad g0 (h). We have (µ p ) 1 (h, h ) = {[g, x] G P r(p); x 1 = h, Ad g (x) = h }. Since x = Ad p (x 1 ) for some p P and conversely (Ad p (x)) 1 = x 1 for any p P (cf. [Slo], Lemma 2, p.48), we have (µ p ) 1 (h, h ) = {[g, Ad p (h)] G P r(p); g G, p P, Ad gp (h) = h } = {[gp, h] G P r(p); g G, p P, Ad gp (h) = h } = {[g, h] G P r(p); Ad g (h) = h } = {[g 0 g, h] G P r(p); g Z G (h)} = g 0 (Z G (h)/z P (h)). Here Z G (h) (resp. Z P (h)) is the centralizer of h in G (resp. P). By [Ko], 3.2, Lemma 5, Z G (h) is connected. Moreover, since g h p, Lie(Z G (h)) = Lie(Z P (h)). Therefore, Z G (h)/z P (h) = {1}, and (µ p ) 1 (h, h ) consists of one point.

5 1 5 By [Na], Lemma 7.3, the map G P r(p) G r(p) is proper; hence the birational morphism in the lemma above is also proper (projective). In the remainder, we shall write Y k(p) for the normalization of k(p) h/w Gr(p), and write X p for G P r(p). Then µ p : X p Y k(p) becomes a resolution of Y k(p). Moreover, as the lemma above shows, µ p,t : X p,t Y k(p),t are isomorphisms for all t k(p) reg. Here, X p,t (resp. Y k(p),t ) are fibers of X p k(p) (resp. Y k(p) k(p)) over t. Let O g be the Richardson orbit for p. For the origin 0 k(p), X p,0 = T (G/P), and the Springer map T (G/P) Ō is factorized as T (G/P) µ p,0 Y k0,0 Ō. The Springer map is not a birational morphism in general, but is only a generically finite morphism. But, µ p,0 is a birational morphism. We fix a parabolic subgroup P 0 of G containing T, and put p 0 := Lie(P 0 ) and k 0 := k(p 0 ). We define: S(p 0 ) := {p g; p : parabolic s.t. h p, k(p) = k 0 }. Let p S(p 0 ). Let b be a Borel subalgebra such that h b p. Then p corresponds to a marked Dynkin diagram D. Take a marked vertex v of the Dynkin diagram D and consider the maximal connected single marked Dynkin subdiagram D v of D containing v. We call D v the single marked diagram associated with v. When D v is one of the following, we say that D v (or v) is of the first kind, and when D v does not coincide with any of them, we say that D v (or v) is of the second kind. A n 1 (k < n/2) k n-k D n (n : odd 5) E 6,I :

6 1 6 E 6,II : In the single marked Dynkin diagrams above, two diagrams in each type (i.e. A n 1, D n, E 6,I, E 6,II ) are called duals. Let D be the marked Dynkin diagram obtained from D by making v unmarked. Let p be the parabolic subalgebra containing p corresponding to D. Now let us define a new marked Dynkin diagram D as follows. If D v is of the first kind, we replace D v D by its dual diagram D v to get a new marked Dynkin diagram D. If D v is of the second kind, we define D := D. Let g = h α Φ g α be the root space decomposition such that b = h α Φ g + α. Let Φ be the base of Φ. Each element of corresponds to a vertex of the Dynkin diagram. Then the set of unmarked vertices of D (resp. D) defines a subset I (resp. Ī ). By definition, v Ī. The unmarked vertices of D define a Dynkin subdiagram, which is decomposed into the disjoint sum of the connected component containing v and the union of other components. Correspondingly, we have a decomposition Ī = I v I v with v I v. The parabolic subalgebra p (resp. p) coincides with tha standard parabolic subalgebra p I (resp. p Ī ). Let l Ī be the (standard) Levi factor of p Ī. Let z(l Ī ) be the center of l Ī. Then l Ī /z(l Ī ) is decomposed into the direct sum of simple factors. Now let l Iv be the simple factor corresponding to I v and let l I v be the direct sum of other simple factors. Then l Ī /z(l Ī ) = l Iv l I v. The marked Dynkin diagram D v defines a standard parabolic subalgebra p v of l Iv. Let Φ v be the root subsystem of Φ generated by I v. Then the automorphism of the root system 1 : Φ v Φ v defines an automorphism of the Lie algebra (cf. [Hu], 14.2) ϕ : l Iv l Iv.

7 1 7 Then ϕ(p v ) is a parabolic subalgebra of l Iv such that p v ϕ(p v ) = l Iv {v}, where l Iv {v} is the standard Levi factor of the parabolic subalgebra p v of l Iv. When D v is of the first kind, ϕ(p v ) is conjugate to a standard parabolic subalagebra of l Iv with the dual marked Dynkin diagram D v of D v. When D v is of the second kind, ϕ(p v ) is conjugate to p v in l Iv. Let q : l Ī l Ī /z(l Ī ) be the quotient homomorphism. Note that Here we define p = l Ī n( p), p = q 1 (p v l I v ) n( p). p = q 1 (ϕ(p v ) l I v ) n( p). Then p S(p 0 ) and p is conjugate to a standard parabolic subalgebra with the marked Dynkin diagram D. This p is said to be the parabolic subalgebra twisted by v. We next define: Res(Y k0 ) := {the isomorphic classes of crepant projective resolutions of Y k0 }. Here we say that two resolutions µ : X Y k0 and µ : X Y k0 are isomorphic if there is an isomorphism φ : X X such that µ = µ φ. Theorem 1.3. There is a one-to-one correspondence between two sets S(p 0 ) and Res(Y k0 ). Proof. The correspondence is given by p [µ p : X p Y k0 ]. First note that µ p is a crepant resolution; in fact, the fiber of X p k 0 over 0 k 0 is isomorphic to the cotangent bundle T (G/P), which has trivial canonical line bundle. This implies that K Xp is µ p -trivial. In order to check that this correspondence is injective, we have to prove that, for µ p : X p Y k0 µ p is not an isomorphism. For t (k 0 ) reg, X p,t = G P (t+n(p)), Y k0,t = G t {t}, and µ p,t is an isomorphism defined by and µ p : X p Y k0 with p p, µ 1 p [g, t + x] G P (t + n(p)) (Ad g (t + x), t) G t {t}.

8 1 8 Let Z G (t) be the centralizer of t: Z G (t) := {g G; Ad g (t) = t}. Let L P be the Levi factor corresponding to l(p). Then L = Z G (t). Hence the map G G P (t + n(p)); g [g, t] has the kernel L, and it induces an isomorphism ρ t : G/L G P (t + n(p)) Note that l(p ) = l(p). Thus, in a similar way, we have an isomorphism ρ t : G/L G P (t + n(p )) for t (k 0 ) reg. The composition of the two maps ρ t (ρ t) 1 defines an isomorphism X p,t = Xp,t for each t (k 0 ) reg and it coincides with µ 1 p,t µ p,t. Assume that µ 1 p µ p is a morphism. Take g G and p P. We put g = gp. Then and lim ρ t ([g]) = [g, 0] G/P, t 0 lim ρ t ([gp]) = [gp, 0] G/P. t 0 Note that [g, 0] = [gp, 0] in G/P. On the other hand, and lim t 0 ρ t([g]) = [g, 0] G/P, lim t 0 ρ t([gp]) = [gp, 0] G/P. Since we assume that µ 1 p µ p is a morphism, [g, 0] = [gp, 0] in G/P. But, this implies that P = P, which is a contradiction. We next prove that the correspondence is surjective. Let p S(p 0 ) and let P be the corresponding parabolic subgroup of G. An irreducible proper curve C on X p is called a µ p -vertical curve if µ p (C) is a point. Two line bundles L and L on X p are called µ p -numerically equivalent if (L.C) = (L.C) for all µ p -vertical

9 1 9 curves on X p. The set of µ p -numerical classes of line bundles on X p becomes a finitely generated Abelian group, which we denote by N 1 (µ p ). We put N 1 (µ p ) R := N 1 (µ p ) Z R. The set of µ p -numerical equivalence classes of µ p -ample line bundles on X p generates an open convex cone Amp(µ p ) in N 1 (µ p ) R. For another p S(p 0 ), X p and X p are isomorphic in codimension one; thus, N 1 (µ p ) R and N 1 (µ p ) R are naturally identified. Lemma 1.4. (1) The closure Amp(µ p ) of Amp(µ p ) is a simplicial polyhedral cone. Each codimension one face F of Amp(µ p ) corresponds to a marked vertex of the marked Dynkin diagram D attached to P. (2) For each codimension one face F, there is an element p S(p 0 ) such that Amp(µ p ) Amp(µ p ) = F. Proof. (1): Let D be the marked Dynkin diagram which is obtained from D by making a marked vertex v unmarked. We then have a parabolic subgroup P with P P corresponding to D. The birational contraction map associated with F is given by: X p (G P P r(p)) h/w k 0. (2): Let p be the parabolic subalgebra twisted by v. By [Na], Proposition 6.4, (ii), we have P n(p) = P n(p ). Since k(p) = k(p ), we see that P r(p) = P r(p ). Thus, there is a diagram of birational morphisms and X p (G P P r(p)) h/w k 0 X p, Amp(µ p ) Amp(µ p ) = F. The following corollary implies the surjectivity of the correspondence. Corollary 1.5. Any crepant projective resolution of Y k0 is given by µ p : X p Y k0 for some p S(p 0 ). Proof. Take an arbitrary crepant projective resolution µ : X Y k0. Fix a µ-ample line bundle L on X and denote by L (0) Pic(X p0 ) its proper transform. If L (0) is µ p0 -nef, then X coincides with X p0. So let us assume

10 1 10 that L (0) is not µ p0 -nef. Then there is an extremal ray R + [z] NE(µ p0 ) such that (L (0).z) < 0. Let F Amp(µ p0 ) be the corresponding codimension one face. By the previous lemma, one can find p 1 S(p 0 ) such that Amp(µ p0 ) Amp(µ p1 ) = F. We let L (1) Pic(X p1 ) be the proper transform of L (0) and repeat the same procedure. Thus, we get a sequence of flops X p0 X p1 X p2. But, since S(p 0 ) is a finite set, this sequence must terminate by the same argument as [Na], Theorem 6.1. As a consequence, X = X pk for some k. Remark 1.6. The corollary above shows that, for any L Pic(X p0 ), there is p S(p 0 ) such that the proper transform L Pic(X p ) of L becomes µ p -nef. Thus N 1 (µ p0 ) R = Amp(µ p ). p S(p 0 ) As before, G is a complex simple Lie group and T is a maximal torus with Lie(T) = h. Let Φ be the root system for (G, T). Then ( 1) : Φ Φ induces an element ϕ Aut(g). For p S(p 0 ), we put p := ϕ(p). Then p S(p 0 ). Note that (p ) = p. Proposition 1.7. Let L Pic(X p ) be a µ p -ample line bundle and let L Pic(X p ) be its proper transform. Then L is µ p -ample. In other words, Amp(µ p ) = Amp(µ p ). Proof. Let P (resp. P ) be the parabolic subgroup of G with Lie(P) = p (resp. Lie(P ) = p ). Let L(P) (resp. L(P )) be the Levi factor of P (resp. P ) corresponding to l(p) (resp. l(p )). Then L(P) = L(P ); thus, we denote by L this reductive subgroup of G. As in Theorem 1.3, for t (k 0 ) reg, we define an isomorphism ρ t : G/L G P (t + n(p)).

11 1 11 Denote by the same letter ϕ the automorphism of G induced by ϕ Aut(g). Note that ϕ(p) = P and P P = L. The ϕ induces an automorphism Then ϕ : G/L G/L. ρ t ϕ (ρ t ) 1 : G P (t + n(p)) G P (t + n(p)) induces a birational involution σ : X p X p. As in the proof of Theorem 1.3, σ is not a morphism because P P. On the other hand, there is an isomorphism τ : X p X p defined by τ([g, x]) = [ϕ(g), ϕ(x)]. Then µ 1 p µ p : X p X p coincides with the composition σ τ 1. It is checked that the proper transform of L Pic(X p ) by σ is L; hence, L is µ p -ample. Remark 1.8. (1) Let O g be the Richardson orbit for P 0. When the Springer map T (G/P 0 ) Ō is a resolution, there is a natural identification (via the restriction map T (G/P 0 ) X p0 ): N 1 (µ p0,0) R = N 1 (T (G/P 0 )/Ō) R. Let S 1 (p 0 ) be the subset of S(p 0 ) consisting of the parabolic subalgebras p which are obtained from p 0 by the sequence of twists by marked vertices v of the first kind. Note that all p S(p 0 ) are obtained from p 0 by the sequence of twists by marked vertices (cf. Theorem 1.3). Then Mov(T (G/P 0 )/Ō) = Amp(µ p ), p S 1 (p 0 ) where Mov(T (G/P 0 )/Ō) is the closure of the relative movable cone of T (G/P 0 ) over Ō. This is just Theorem 6.1 of [Na]. (2) When p 0 is a Borel subalgebra b 0 of g, S(b 0 ) is the set of all Borel subalgebras b such that h b. The order of S(b 0 ) coincides with the order of the Weyl group W. In this case, k 0 = h, X b0 = G B 0 b 0, and Y k0 = g h/w h.

12 1 12 Example 1.9. Let O g be the Richardson orbit for P 0. Assume that the Springer map T (G/P 0 ) Ō is not birational. Let T (G/P 0 ) W be the Stein factorization of the Springer map. Theorem 1.3 is useful to describe crepant resolutions of W. For example, put g = so(8,c) and fix a Cartan subalgebra h and a Borel subalgebra b containing h. Consider the standard parabolic subalgebra p 0 corresponding to the marked Dynkin diagram In this case, the Springer map T (G/P 0 ) Ō has degree 2. In the marked Dynkin diagram above, there are two marked vertices. For each marked vertex, one can twist p 0 to get a new parabolic subalgebra p 1 or p 2. The marked Dynkin diagrams of p 1 and p 2 are respectively given by: Since W is the normalization of Y k0,0, three varieties X p0,0, X p1,0 and X p2,0 become crepant resolutions of W. But theses are not enough; actually, there are three more crepant resolutions. In fact, S(p 0 ) = {p 0, p 1, p 2, (p 0 ), (p 1 ), (p 2 ) }, and so three other crepant resolutions of W are X (p0 ),0, X (p1 ),0 and X (p2 ),0. Moreover, N 1 (T (G/P 0 )/W) R = Mov(T (G/P 0 )/W) and it is divided into the union of six ample cones: X p0,0 X (p2 ),0 X p1,0 X (p1 ),0 X p2,0 X (p0 ),0

13 REFERENCES 13 Remark For Springer maps of degree > 1, there is a result by Fu in a different context [F, Cor. 5.9]. References [B-K] Borho, W., Kraft, H.: Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen, Comment Math. Helv. 54 (1979), [F] Fu, B.: Extremal contractions, stratified Mukai flops and Springer maps, math.ag/ [Fuj] Fujiki, A.: A generalization of an example of Nagata, in the proceeding of Algebraic Geometry Symposium, Jan. 2000, Kyushu University, [Hu] Humphreys, J.: Introduction to Lie algebras and representation theory, Graduate Texts in Math. 9, Springer (1972) [Ko] Kostant, B.: Lie group representations on polynomial rings, Amer. J. Math. 85 (1963) [Na] Namikawa, Y.: Birational geometry of symplectic resolutions of nilpotent orbits, to appear in Advances Studies in Pure Mathematics, see also math.ag/ , math.ag/ [Slo] Slodowy, P.: Simple singularities and simple algebraic groups. Lect. Note Math. 815, Springer-Verlag, 1980 Yoshinori Namikawa Department of Mathematics, Graduate School of Science, Osaka University, JAPAN namikawa@math.sci.osaka-u.ac.jp