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Springer fiber components in the two columns case for types A and D are normal
Nicolas Perrin and Evgeny Smirnov September 12, 2010
Abstract We study the singularities of the irreducible components of the Springer fiber over a nilpotent element N with N 2 = 0 in a Lie algebra of type A or D (the so-called two columns case). We use Frobenius splitting techniques to prove that these irreducible components are normal, Cohen­Macaulay, and have rational singularities.

1

Intro duction

Let K be an algebraically closed field of arbitrary characteristic not equal to 2. Let N be a nilpotent element in a Lie algebra g = gln (type A) or g = so2n (type D). We consider the Springer fiber FN over N . It is the fiber of the famous Springer resolution of the nilpotent cone N g over N . This resolution can be constructed as follows. Let F be the variety of complete flags in Kn (resp. OF the variety of complete isotropic flags, see Section 3 for the description of the Springer fiber OFN in this case). A flag f = (Vi )i[0,n] , where Vi is a vector subspace of Kn of dimension i, is stabilized by N N if N (Vi ) Vi-1 for all i > 0. We shall denote this by N (f ) f . Define the variety N = {(f , N ) F â N | N (f ) f }. The pro jection N F is a smooth morphism thus N is smooth. The natural pro jection N N is birational and proper. It is a resolution of singularities for N called the Springer resolution. The Springer fibers, i.e., the fibers of the Springer resolution, are of great interest. They are connected (this can be seen directly or follows from the normality of the nilpotent cone N), equidimensional, but not irreducible. There is a natural combinatorial framework to describe them: Young diagrams and standard tableaux. The irreducible components of the Springer fibers are not well understood. For example, it is known that in general the components are singular but there is no general description of the singular components. There are only partial answers in type A. First, it is known in the so-called hook and two lines cases that all the components are smooth (see [Fun03]). The first case where singular components appear is the two columns case. A description of the singular components in the two columns case has been given by L. Fresse in [Fre09a] and [Fre09b]. In their recent work [FM10] L. Fresse and A. Melnikov describe the Young diagrams for which all irreducible components are smooth.
Keywords: Springer fiber, Frobenius splitting, normality, rational resolution, rational singularities. Mathematics Sub ject Classification: 14B05; 14N20

1


In this paper, we focus on the the two columns case, that is to say, the case of nilpotent elements N of order 2. The corresponding Young diagram = (N ) has two columns. We want to understand the type of singularities appearing in a component of the Springer fiber. Let X be an irreducible component of the Springer fiber FN , resp. OFN , in type A, resp. D, with N nilpotent such that N 2 = 0. In the two columns case, we describe a resolution : X X of the irreducible component X . We use this resolution to prove, for Char(K) > 0, that X is Frobenius split, and deduce the following result for arbitrary characteristic: Theorem 1.1. The irreducible component X is normal. We are able to prove more on the resolution . Recall that a proper birational morphism f : X Y is called a rational resolution if X is smooth and if the equalities f OX = OY and Ri f OX = Ri f X = 0 for i > 0 are satisfied. We prove the following Theorem 1.2. The morphism is a rational resolution. Corollary 1.3. The irreducible component X is Cohen­Macaulay with dualizing sheaf X . Rational singularities are well defined in characteristic zero. In this case we obtain the following Corollary 1.4. If Char(K) = 0, then X has rational singularities. Acknowledgements. We express our gratitude to Michel Brion for useful discussions, in particular, concerning the existence of Frobenius splittings using the pair (SL2n , Sp2n ). We thank X. He and J.F. Thomsen for giving us a preliminary version of their work [HT10] which simplifies the proof of Theorem 4.5. We also thank Catharina Stroppel for discussions on her paper [SW10] which were the starting point of this pro ject. Finally, we are grateful to the referee for his careful reading of our paper and for correcting some mistakes in the initial version of this paper, especially Proposition 3.10. E.S. was partially supported by the RFBR grant 10-01-00540 and by the RFBR­CNRS grant 10-01-93111.

Contents
1 Intro duction 2 Irreducible comp onents of Springer fib ers 2.1 General case . . . . . . . . . . . . . . . . . 2.2 Two columns case . . . . . . . . . . . . . 2.3 A birational transformation of the Springer 2.4 A Schubert variety containing X . . . . . in typ .... .... fiber .... e . . . . A .. .. .. .. D ers .. .. .. 1 3 3 4 4 5 5 5 7 9 10

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3 Irreducible comp onents of Springer fib ers in typ e 3.1 Preliminaries on orthogonal groups and Springer fib 3.2 Description of components in the two columns case 3.3 Birational transformation of the Springer fiber . . 3.4 A Schubert-like variety containing X . . . . . . . . 2

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4 Frob enius splitting 4.1 Bott­Samelson resolutions 4.2 Resolutions of X and Y . 4.2.1 Two groups . . . . 4.2.2 Resolutions . . . . 4.3 Existence of a splitting . . 4.4 D-splitting . . . . . . . .

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11 11 12 12 13 14 14 16 16 17 18

5 Normality 5.1 Some preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Proof of Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2
2.1

Irreducible comp onents of Springer fib ers in typ e A
General case

Let N gln be a nilpotent element, and let (mi )i[1,r] be the sizes of Jordan blocks of N . To N we assign a Young diagram = (N ) of size n with rows of lengths (mi )i[1,r] . We refer to [Ful97] for more details on Young diagrams. Definition 2.1. A standard Young tableau of shape is a bijection : [1, n] such that the numbers assigned to the boxes in each row are decreasing from left to right, and the numbers in each column are decreasing from top to bottom. Remark 2.2. Usually, one requires that the integers in the boxes of a standard tableau increase, not decrease from left to right and from top to bottom. However, using decreasing tableaux in our case simplifies the notation, so we decided to follow this (rather unusual) definition. Remark that the datum of a standard tableau is equivalent to the datum of a chain of decreasing Young diagrams = (0) (1) (2) · · · (n) = , where (i) is the set of the n - i boxes with the largest numbers, that is, -1 ({i + 1, . . . , n}). Let f = (Vi ) F be an N -stable flag. We assign to it a standard tableau of shape = (N ) in the following way. Consider the quotient spaces V (i) = V /Vi . The endomorphism N induces an endomorphism of each of these quotients N (i) : V (i) V (i) . Take the Young diagram (i) corresponding to N (i) ; it consists of n - i boxes. Clearly, (i) differs from (i-1) by one corner box. So we obtain a chain of decreasing Young diagrams, which is equivalent to a standard Young tableau (f ). Let be a standard Young tableau of shape (N ). Define
0 X = {f FN | (f ) = }.

The following theorem is due to Spaltenstein [Spa82].
0 Theorem 2.3. For each standard tableau , the subset X is a smooth irreducible subvariety 0 0 of FN . Moreover, dim X = dim FN , so X = X is an irreducible component of FN . Any irreducible component of FN is obtained in this way.

3


2.2

Two columns case

In this paper, we focus on the case of nilpotent elements N such that N 2 = 0. This is equivalent to saying that the Young diagram (N ) consists of (at most) two columns. Denote by r the rank of N or, equivalently, the number of boxes in the second column. Let X = X be the irreducible component of the Springer fiber over N corresponding to a standard tableau . Denote the increasing sequence of labels in the second column of the standard tableau by (pi )i[1,r] . Set p0 = 0 and pr+1 = n + 1. According to F.Y.C. Fung [Fun03], the previous Theorem can be reformulated as follows. Prop osition 2.4. The irreducible component X is the closure of the variety X0 = (Vi )i
[0,n]

FN

Vi V Vi V

i- 1 i- 1

+ ImN + ImN

for i {p1 , · · · , pr } otherwise

.

An easy interpretation of this result is the following Corollary 2.5. The irreducible component X is the closure of the variety X 0 = (Vi )i
[0,n]

FN | dim(ImN Vi ) = k for al l k [0, r] and al l i [pk , pk

+1

).

Proof. We prove this by induction on i. We have dim(ImN V0 ) = 0. The result is implied by the following equivalence: (Vi+1 Vi + ImN ) (dim(ImN Vi+1 ) = dim(ImN Vi ) + 1).

2.3

A birational transformation of the Springer fib er

The above description gives a natural way to construct a resolution of singularities for X . We start with the following simple birational transformation of X . Define the variety X as follows: X = {((Fk )k
[0,r]

, (Vi )

i[0,n]

) F(ImN ) â F | Fk V

p

k

N

-1

(Fk

-1

), k [1, r]},

where F(ImN ) denotes the variety of complete flags in ImN . The natural pro jections of the product F(ImN ) â F on its two factors induce two maps pX : X F and qX : X F(ImN ). One of the main features of the two columns case that we will use is the following easy observation: ImN KerN . In particular, for any flag (Fk )k[0,r] F(ImN ), the equalities Fr = ImN and N -1 (F0 ) = KerN imply the following inclusions: F0 · · · Fr N Fixing subspaces (Fi )
i[r,n-r] -1

(F0 ) · · · N

-1

(Fr ).

with dim(Fi ) = i such that ImN Fr · · · Fn-r KerN

gives for any choice of (Fk )k

[0,r]

F(ImN ) a complete flag
+1

F0 · · · Fr Fr in N
-1

· · · Fn-

r -1

N

-1

(F0 ) · · · N

-1

(Fr )

(Fr ) = Kn . We denote this complete flag by F· .

Prop osition 2.6. (i) The map qX is dominant and is a local ly trivial fibration over F(ImN ). Its fiber over (Fk )k[0,r] is isomorphic to the fol lowing Schubert variety associated to F· : Fw = {(Vi )
i[0,n]

F | Fk Vpk N

-1

(Fk

-1

), k [1, r]}.

(ii) The map pX is birational onto X . 4


Proof. (i) The first part is clear from the definition of X . (ii) Let (Vi )i[0,n] be in X 0 . We may define Fk = ImN Vpk for k [0, r]. We have dim Fk = k . Since (Vi )i[0.n] is in the Springer fiber, we also have the inclusion N (Vpk ) Vpk -1 . But N (Vpk ) ImN , thus N (Vpk ) ImN Vpk -1 . Since (Vi )
i[0,n]

is in X 0 , we have ImN V Vpk N
-1

pk -1

= ImN Vp
p
k-1

k-1

. Therefore we have the inclusion:
-1

(ImN V

)=N

-1

(Fk

).

In particular X 0 is contained in the image of pX . Conversely, let (Fk )k[0,r] F(ImN ) and (Vi )i[0,n] in the Schubert variety Fw associated to F· . It is easy to check that for (Vi )i[0,n] general in the Schubert variety, we have ImN Vi = Fk for i [pk , pk+1 ). Furthermore, for i [pk , pk+1 ) we have the inclusions N (V therefore (Vi )i
[0,n] i+1

) N (Vp

k+1

) ImN V

p

k

= Fk V

p

k

Vi ,

is in X 0 .

2.4

A Schub ert variety containing X

Let us consider the following subvariety of F(ImN ) â F containing X : Y = {((Fk )k
[0,r]

, (Vi )

i[0,n]

) F(ImN ) â F | Fk Vpk , k [1, r]}.

As for X , the natural pro jections of the product F(ImN ) â F on its two factors induce two maps pY : Y F and qY : Y F(ImN ). Prop osition 2.7. (i) The map qY is dominant and is a local ly trivial fibration with fiber over (Fk )k[0,r] isomorphic to the fol lowing Schubert variety associated to F· : Fv = {(Vi )i (ii) The map p
Y [0,n]

F | Fk Vpk , k [1, r]}.

is birational onto the Schubert variety
[0,n]

Y = {(Vi )i

F | dim(ImN Vpk ) k , k [1, r]}.

Proof. (i) The first part is clear from the definition of Y . (ii) The image of pY is contained in Y . Conversely, let (Vi )i[0,n] be general in Y . We then have dim(ImN Vpk ) = k and we may define Fk = ImN Vpk for k [0, r]. We have dim Fk = k and ((Fk )k[0,r] , (Vi )i[0,n] ) is in the fiber of pY over (Vi )i[0,n] .

3
3.1

Irreducible comp onents of Springer fib ers in typ e D
Preliminaries on orthogonal groups and Springer fib ers

Let V be a 2n-dimensional vector space. Consider the group SO(V ) of unimodular linear operators preserving a symmetric nondegenerate bilinear form . Let B be a Borel subgroup in SO(V ). The flag variety SO(V )/B is the variety OF of orthogonal flags defined by OF = {V0 V1 · · · V
n-1

V

n+1

· · · V2n | V2n 5

-i

= Vi and dim Vi = i for i n - 1}.


We will consider elements in OF as n-tuples of nested isotropic vector spaces ((Vi )i[0,n-1] ). We recover the usual notion of orthogonal flags because there are exactly two maximal isotropic subspaces between Vn-1 and its orthogonal Vn+1 = (Vn-1 ) . Let N gl(V ) be a nilpotent element. N is said to be orthogonalizable if there exists a symmetric nondegenerate bilinear form on V such that N is -invariant; that is, (N v , w) + (v , N w) = 0. This means that N so(V ), where so(V ) is the set of elements of gl(V ) leaving invariant. The following easy consequence of the Jacobson­Morozov Theorem can be found, for instance, in [VGO90, Chap. 6, 2.3]. Prop osition 3.1. A nilpotent element N is orthogonalizable if in the corresponding partition Y (N ) each even term occurs with even multiplicity (such partitions wil l be cal led admissible). Definition 3.2. Given a nilpotent element N so(V ), we define a Springer fiber of type D in the usual way: namely, as the set of all orthogonal flags stabilized by N : OFN = {((Vi )i
[0,n-1]

) OF | N (Vi ) Vi

-1

for i [0, n - 1] and N (V

n-1

) Vn-1 }.

Remark that the orthogonalizability condition on N implies that N (Vi ) Vi for i n - 2. +1 A description of irreducible components of Springer fibers in types B , C , and D was given in M. van Leeuwen's Ph.D. thesis [vLe89]. We briefly recall this description here for the type D. Definition 3.3. Let be a Young diagram with 2n boxes. A map from the boxes of to [1, n] is called a standard domino tableau, if the following conditions hold: (i) For each i, the pre-image -1 (i) consists exactly of two adjacent boxes (adjacent either by horizontal or by vertical); (ii) For each i, the set of boxes (i) (N ) := -1 ([i + 1, n]) corresponding to the numbers greater than i forms a Young diagram. Moreover, a standard domino tableau is said to be admissible, if all the diagrams (i) (N ) are admissible (in the sense of Prop. 3.1). We will think of the pair of boxes -1 (i) as of a domino tile indexed by the number i. Each of these tiles can be either horizontal or vertical. Example 3.4. Let = (3, 3). Then there are three standard domino tableaux of shape (see below), but only the first two of them are admissible. Indeed, for the third diagram -1 (3) corresponds to the Young diagram with one row of length 2, which is not admissible. 321 321 322 311 331 221 of shape (N ). We assign to it 0 the set X of flags (Vi )i[0,n-1] 1, n]) for each i < n. i , so this makes sense.

Definition 3.5. Let be an admissible standard domino tableau a subset X of the Springer fiber OFN obtained as the closure of in OFN such that N |V /Vi corresponds to the partition -1 ([i + i By definition of OFN , N is well defined on Vi /Vi = V2n-i /V

The following theorem is due to M. van Leeuwen [vLe89, Sec. 3.2]. Theorem 3.6. X is an irreducible component of OFN ; al l its irreducible components are obtained in this way. In particular, there is a bijection between the admissible standard domino tableaux of shape (N ) and the irreducible components of OFN . 6


3.2

Description of comp onents in the two columns case

Throughout this subsection we fix a nilpotent element N so(V ) such that N 2 = 0, and an admissible standard domino tableau of shape (N ). We begin with the following combinatorial observation. Prop osition 3.7. The Young diagram (N ) has at most two columns. Each admissible standard domino tableau of shape (N ) contains only vertical tiles. Note that in particular, the rank of N has to be even. Let rk(N ) = 2r, and let (pi )i[1,r] be the numbers of domino tiles forming the second column of the diagram Y (N ). We formally set p0 = 0, pr+1 = 2n + 1. Now we endow the subspace ImN with a bilinear form as follows. For u, v ImN , (u, v ) = (u, v ), where v N
-1

(v ).

Prop osition 3.8. The form is a skew-symmetric nondegenerate form on ImN . Proof. We readily see that is well-defined. To show that it is skew-symmetric, take two vectors u, v ImN along with their preimages u N -1 (u) ,v N -1 (v ). Then (u, v ) = (N (u ), v ) = - (u , N (v )) = - (N (v ), u ) = -(v , u). The non-degeneracy of is also obvious. Remark 3.9. This is a particular case of the construction of a family of nondegenerate bilinear forms on (KerN ImN i )/(KerN ImN i+1 ), which works for arbitrary nilpotent N so(V ). See [vLe89, Section 2.3] for details. We shall denote by the orthogonality relation for the form . We consider the symplectic flag variety SpF(ImN ), defined as follows: SpF(ImN ) = {(0 = F0 F1 F2 · · · F2r = ImN ) | F2
r -k = Fk }.

Similarly as for OF, an element of SpF can also be seen as a sequence (Fk )k[0,r] of nested -isotropic subspaces. As in the type A case (see Corollary 2.5), the description of irreducible components can be reformulated as follows. Prop osition 3.10. (i) Let f = (Vi )i[0,n-1] OFN be an orthogonal N -stable flag. There is a unique partial flag (Ui )i[0,n-1] of ImN such that for al l i, we have Ui Vi N (Ui ), with Ui maximal with this property. Moreover, Ui is given by Ui = Vi N (Ui 1 ) for al l i. - (ii) The subspaces Ui are -isotropic. 0 (iii) f X if and only if dim Ui = #({p1 , . . . , pr } [1, i]). In that case we have the equality i Ui = j =0 Vj N (Vj ). Note that the indices i do not correspond to the dimension of Ui and that some of these subspaces may coincide. Proof. (i) Let us begin with the following lemma. Lemma 3.11. (i) it satisfies Wi = Vi (ii) If W W dim W {dim W , A subspace W Vi is maximal for the property W N (W ) if and only if N (W ). are maximal such that W Vi-1 N (W ) and W Vi N (W ), then dim W + 1}. 7


Proof. (i) If W is maximal, let U = Vi N (W ), we have W U thus U Vi N (U ) and W = U by maximality. Conversely, if W = Vi N (W ) and if we have W U with U Vi N (U ), then we have the inclusions W U Vi N (U ) Vi N (W ). We thus have equalities for all these inclusions. (ii) By (i), we have the equalities W = Vi-1 N (W ) and W = Vi N (W ). We thus have the inclusion W Vi N (W ) and W = Vi-1 N (W ) is of codimension at most one in Vi N (W ). Thus its codimension in W is also at most one. We define the subspaces Ui inductively by the rule U0 = 0 and Ui = Vi N (Ui 1 ). Let us check - by induction that they are maximal so suppose that Ui-1 = Vi-1 N (Ui 1 ). Either Ui = Ui-1 , - and the maximality of Ui = Vi N (Ui ) follows from Lemma 3.11 (i). Or Ui = Ui-1 x . Let y Ui 1 with N (y ) = x. We have the equalities (x, y ) = (N (y ), y ) = - (y , N (y )) = - - (N (y ), y ) = - (x, y ) thus (x, y ) = 0. So y Ui and we have the inclusion Ui N (Ui ), and Ui is maximal by Lemma 3.11 (ii). The existence of such a flag is shown. Let (Ui )i[0,n-1] be another partial flag satisfying the same property. We prove by induction that they coincide. Assume that Ui-1 = Ui-1 , we see that Ui Vi N (Ui ) Vi N (Ui-1 ) = Vi N (Ui 1 ) = Ui . Whence the uniqueness by maximality. - (ii) We have the inclusion Ui Vi N (Ui ), so in particular Ui N (Ui ) = Ui , and all Ui are -isotropic. Note in particular that dim Ui r for all i. (iii) Let us first prove one more lemma. Lemma 3.12. Let (Vi )i
p
k

[0,n-1]

0 X , we have the equalities: pk -1

dim
j =0 k

Vj N (V ) = dim
j =0 k-1

j

(Vj N (Vj )) + 1 for al l k [1, r] and

Vj N (V ) =
j =0 j =0

j

(Vj N (Vj )) for al l k {p1 , · · · , pr }.

Proof. Let us consider the following inclusions:
N (Vj ) 1 O 1c / N (V ) O j -1

Vj N (V )

j

hPPP PPP PPP PPP S

V

j -1

N (Vj ).

mm6 mmm mmm m A û mmmm

Vj

-1

N (V

1c

j -1

)

If j is in the interval [pk , pk+1 ), then the left vertical inclusion is a codimension 2r - 2k inclusion. The right vertical inclusion is of the same codimension except for j = pk in which case it is of codimension 2r - 2k + 2. All the other inclusions are of codimension at most one. Assume first that j = pk , then Vpk N (Vp ) has to be of dimension one more than Vpk -1 k N (Vp-1 ) thus we have Vpk -1 N (Vp ) = Vpk -1 N (Vp-1 ) and Vpk N (Vp ) = Vpk -1 N (Vp-1 ) k k k k k x with x in Vpk but not in Vpk -1 . Therefore we have
pk pk -1

Vj N (V ) =
j =0 j =0

j

(Vj N (Vj )) x

8


and the result follows in this case. If j = k {p1 , · · If the top horizontal inclusion is an equality, then Vk-1 N (Vk 1 ) and by dimension count Vk N (Vk - has to be of dimension one less than Vk-1 N (Vk 1 ) - Vk-1 N (Vk 1 ). In any case the result follows. -

· , pr }, we distinguish between the two cases. we also have the equality Vk-1 N (Vk ) = ) = Vk-1 N (Vk 1 ). If not, then Vk N (Vk ) - thus we have Vk N (Vk ) = Vk-1 N (Vk )
N (Uj ) = Uj , we get the inclusion k j = pk , we have p=0 Vj N (Vj ) = j Upk -1 and dim Upk > dim Upk -1 . We following inequalities and equalities:

Let us finish the proof. Since Vj N (Vj ) Vj i j =0 (Vj N (Vj )) Ui for all i. Note also that for pk -1 j =0 (Vj N (Vj )) x with x Vpk -1 therefore x thus have by the previous fact and this observation the
i

dim Ui k and dim
j =0

Vj N (Vj ) = k for i [pk , pk

+1

).

But Un-1 being -isotropic, we have dim Un-1 r therefore we necessarily have the equality i j =0 Vj N (Vj ) = Ui , and (iii) also follows.
0 Remark 3.13. Any complete flag (Vi )i[0,n-1] in the Springer fiber belongs to some X for some admissible standard tableau . In particular, for any such complete flag, we may assign a n-1 Lagrangian subspace in ImN for the skew form , namely, Un-1 = j =0 Vj N (Vj ).

Corollary 3.14. For f = (Vi ) belongs to SpF(ImN ).

i[0,n-1]

0 X OFN , the flag 0 U

p1

··· U

p

r

ImN

3.3

Birational transformation of the Springer fib er

In this subsection we construct a birational transformation of a given irreducible component X = X , analogous to the one described in Section 2.3. We define it as follows: X = {((Fk )k , (Vi ) ) SpF(ImN ) â OF | Fk Vpk N
[0,r] -1

[0,r]

i[0,n-1]

(Fk

-1

)

k [1, r]}. (1)

Let us remark that for a flag (Fk )k

SpF(ImN ), we may consider the partial flag
-1

F0 · · · F2r = ImN KerN = N

(F0 ) N

-1

(F1 ) · · · N

-1

(F2r ),

where Fj := F2r-j for j > r, and since (Fk )k[0,r] is isotropic for the form , we have Fk = N -1 (Fk ) = N -1 (F2r-k ). Therefore the above partial flag is isotropic for the quadratic form and we may complete it to an isotropic complete flag. As for the type A, denote the two natural pro jections by pX : X SpF(ImN ) and qX : X OF respectively.

Prop osition 3.15. (i) The map qX is dominant and a local ly trivial fibration with fiber isomorphic to the fol lowing Schubert variety: OFw = {(Vi )i (ii) The map p
X [0,n-1]

OF | Fk V

pk

N

-1

(Fk

-1

)

k [1, r]};

is birational onto X .

Proof. (i) This is clear from the definition of X . 0 (ii) Let f = (Vi )i[0,n-1] X . With notation as in Prop. 3.10, we may define the subspaces Fi by the condition Fi = Upi . We have Upk = Vpk N (Upk ) thus Upk Vpk . Furthermore, 9


we have N (Vpk ) Vpk -1 and because of the inclusions Vpk Vp Vp-1 Upk -1 , we get k k N (Vpk ) N (Upk -1 ). Therefore we have the inclusions N (Vpk ) Vpk -1 N (Upk -1 ) = Upk -1 thus Vpk N -1 (Upk -1 ) = N -1 (Upk-1 ). The flag (Fi )i[0,r] therefore satisfies the conditions (1). 0 So the map pX is bijective over the dense subset X X , and hence it is birational.

3.4

A Schub ert-like variety containing X

As in type A, let us consider the following subvariety of SpF(ImN ) â OF containing X : Y = {((Fk )k
[0,r]

, (Vi )

i[0,n-1]

) SpF(ImN ) â OF | Fk Vpk , k [1, r]}.

We shall also consider pro jection but with a slight modification. Let L (ImN ) be the variety of all Lagrangian subspaces of ImN for the form (i.e. isotropic subspaces for the form of maximal dimension r). We define the pro jections qY : Y SpF(ImN ) and pY : Y L (ImN ) â OF Prop osition 3.16. (i) The map qY is dominant and is a local ly trivial fibration with fiber over (Fk )k[0,r] isomorphic to the fol lowing Schubert variety OFv = {(Vi )i
[0,n-1]

OF | Fk Vpk , k [1, r]}.

(ii) The map pY is birational onto the variety Y = {(L, (Vi )i
[0,n-1]

) L (ImN ) â OF | dim(L Vpk ) k , k [1, r]}

which is a local ly trivial fibration over L (ImN ) with fiber over L L (ImN ) being the Schubert variety YL = {(Vi )
i[0,n-1]

OF | dim(L Vpk ) k , k [1, r]}.

Proof. (i) The first part is clear from the definition of Y . (ii) The last assertion is also clear from the description of Y so we only need to prove that Y is the image of pY . The image of pY is contained in Y . Conversely, let (L, (Vi )i[0,n] ) be general in Y . Then we have the equalities dim(L Vpk ) = k for all k [0, r]. We may therefore set Fk = L Vpk for k [0, r]. The point ((Fk )k[0,r] , (Vi )i[0,n-1] ) is in the fiber of pY over (L, (Vi )i[0,n-1] ). Remark 3.17. The variety Y is normal with rational singularities as a locally trivial fibration with smooth base and fibers which are normal with rational singularities (Schubert varieties). Prop osition 3.18. There is a closed immersion of X in Y . Proof. Recall n-1 Un-1 = j =0 dim(Vpk Un- (Vi )i[0,n-1] from Remark 3.13 that for any element (Vi )i[0,n-1] in X , the vector space 0 Vj N (Vj ) lies in L (ImN ). Furthermore, for (Vi )i[0,n-1] in X , we have 1 ) = dim Upk = k for all k [0, r ]. Therefore the map X Y defined by ((Vi )i[0,n-1] , L) with L = Un-1 is a closed immersion.

10


4

Frob enius splitting

In this section, we assume Char(K) = p > 0, we shall intensively use the results from the book [BK05]. We refer to this book for the notions of Frobenius splitting of a scheme X and B canonical splitting of a scheme X with an action of a Borel subgroup B of a reductive group G. We will now deal with type A and type D simultaneously and keep the notation of the previous two sections. In particular X will denote a fixed irreducible component of the Springer fiber with two columns in type A or D. To simplify the notation we will denote by Fw both the Schubert variety Fw in type A and the Schubert variety OFw in type D. We denote by n the number n in type A and n - 1 in type D.

4.1

Bott­Samelson resolutions

We will use Bott­Samelson resolutions of the Schubert varieties Fw and Fv to construct resolutions of X and Y and thus of X and Y . Let us fix some notation and recall some basic facts on Bott-Samelson resolutions (for details we refer to [Dem74] or [BK05]). Recall that the Schubert varieties in F are indexed by the elements u of the Weyl group W . The inclusion of Schubert varieties induces an order on W called the Bruhat order. Any element u W can be written as a product si1 · · · sik where sik is the simple reflection with respect to the simple root ik (we shall use the notation of N. Bourbaki here [Bou54]). An expression of minimal length is called reduced, and its length k is called the length of u. Let us also denote, for a simple root, by G(), respectively F(), the Grassmannian (classical in type A, orthogonal in type D) associated to , respectively the partial flag variety of all subspaces except those in G(). Denote by p the pro jection F F() and by q the pro jection F G(). The fiber of p is isomorphic to P1 . Let F· be a fixed complete flag (classical in type A, orthogonal in type D), and let u = (i1 , · · · , ik ) be a sequence of simple roots. We construct a variety Fu from these data. For this we consider the following elementary construction. Elementary construction 4.1. Having a simple root , we first define a variety F = {((Vi )i
[0,n]

, W ) F â G() | W q (p

-1

(p ((Vi )i

[0,n]

)))}.

There are two natural maps and from F to F defined by ((Vi )i[0,n] , W ) = (Vi )i[0,n] and ((Vi )i[0,n] , W ) = (p ((Vi )i[0,n] ), W ). Remark that there is a natural section : F F of given by (Vi )i[0,n] ((Vi )i[0,n] , q ((Vi )i[0,n] )). Let pZ : Z F be a morphism. We define the variety Z as the fiber product Z = Z âF F F



/Z / F.
pZ

We denote the pro jection Z Z by fZ . The section induces a section Z of fZ . We define the map pZ : Z F as the composition of the pro jection Z F with . The Bott­Samelson variety Fu is constructed from the sequence of roots u and the point F· in F. Indeed, we set Z0 = {F· } with the map pZ0 : Z0 F given by the inclusion and we define Z1 = (Z0 )i1 obtained by the elementary construction from pZ0 and i1 . We define by 11


induction Z

j +1

= (Zj )

ij +1

obtained by the elementary construction from p

Zj

and

i

j +1

. By

definition, the Bott­Samelson variety Fu is Zk . The map fZj : Zj Zj -1 is a P1 -bundle for - - all j and therefore Fu is smooth. The sections Zj define divisors Dj = fZk1 · · · fZj1 Zj (Zj -1 ). +1 These divisors intersect transversally, and we define DJ =
j J

D

j

for J [1, k ]. For such a subset J of [1, k ] we can consider the subword uJ = (ij )j J and there is a natural isomorphism FuJ DJ . We will therefore consider the Bott­Samelson varieties Fu for any subword u of u as subvarieties of the Bott­Samelson variety Fu . We shall denote the union of the divisors Dj for j [1, k ] by Fu . Recall that by the construction, there is a map pFu : Fu F. If u is a reduced expression for an element u of the Weyl group W , the natural map pFu is birational onto Fu yielding a resolution of the Schubert variety Fu . Remark 4.2. The choice of a Bott­Samelson resolution Fu for Fu depends on the choice of a reduced expression for u. Recall from [Dem74] that since Fw is a Schubert subvariety of Fv , we may choose a reduced expression v = (i1 · · · ik · · · il ) for v such that w = (ik · · · il ) is a reduced expression for w. In particular in the diagram Fw Fw
1 /F v / Fv

1

the vertical maps are birational and thus simultaneous resolutions of singularities. We choose such a reduced expression v for v to construct Fv and thus Fw .

4.2
4.2.1

Resolutions of X and Y
Two groups we will need to distinguish between the type A and type D cases. G = SL(ImN ) and G = SL(Kn ). We embed G in G as follows. Choose for ImN in KerN and a complement E2 for KerN in Kn . We consider the defined by: f G f stabilizes ImN , KerN , and Ei for i {1, 2}, det(f |ImN ) = det(f |E2 ) = 1, and f |E1 = idE1 .

In this subsection, In type A, set a complement E1 subgroup G0 of G

G0 =

This group is isomorphic to the product SL(ImN ) â SL(E2 ). Furthermore, observe that E2 is identified with Kn /KerN and with ImN via N . The group G0 is thus isomorphic to G â G; let us embed G into G0 diagonally. For any Borel subgroup B of G, we may find a Borel subgroup B of G such that B B in this embedding. We may thus consider the variety Z0 = G/B as a subvariety of G0 /B0 and also of G /B = F. We thus have a map pZ0 : Z0 F. In type D, set G = Sp(ImN ) (recall that we have a non degenerate skew form on ImN ) and G = SO(K2n ). We embed G in G as follows. Choose an orthogonal complement E1 for ImN in KerN and an isotropic subspace E2 in Kn mapping bijectively to K2n /KerN . We consider the subgroup G0 of G defined by: G0 = f G f stabilizes ImN , KerN , and Ei for i {1, 2}, det(f |ImN ) = det(f |E2 ) = 1, and f |E1 = idE1 12 ,


remark that here the conditions f (KerN ) KerN and det(f |E2 ) = 1 are redundant. This group is isomorphic to SL(ImN ). The group G embeds into G0 . For any Borel subgroup B of G, we may find a Borel subgroup B of G such that B B in this embedding. We may thus consider the variety Z0 = G/B as a subvariety of G0 /B0 and also of G /B = F. We thus have a map pZ0 : Z0 F. 4.2.2 Resolutions

We again deal with types A and D simultaneously. Let us take a sequence of simple roots u = (i1 , · · · , ik ) and apply the same construction as for the Bott­Samelson variety Fu , but starting with Z0 = G/B . We get a variety Xu together with a morphism pXu : Xu F. This variety can also be seen as the homogeneous fiber bundle Xu = G âB Fu where the action of B on Fu is induced by the inclusion B B and the natural action of B on Fu . For any subword u of u, the variety Xu can again be realized as a complete intersection in Xu . In particular we have the same description of divisors G âB Dj for j [1, k ] on Xu as on Fu . We shall denote the union of these divisors by Xu . Finally, we have a natural map pXu : Xu F. Using the reduced expression v of v defined in Remark 4.2, we obtain a variety Y = Xv and a subvariety X = Xw of Y . We have natural maps pXv (resp. pXw ) from Y (resp. X ) to F. Since the maps pFv and pFw are B -equivariant and thus B -equivariant, we get a diagram X = G âB Fw
1 / Y = G âB F v / Y = G âB F v

X = G âB Fw R

1

RRR RRR RRR RRR RR(

F.

where the morphisms G âB Fv F and G âB Fw F are given by (g , x) g · x. The maps pY and pX are the vertical compositions in the above diagram. We also have the pro jection maps qY : Y G/B and qX : X G/B . We must again make a distinction between type A and type D cases. Let us define F to be F in type A and L (ImN ) â OF in type D. There are natural maps pY and pX from Y and X to F . In type A they are simply defined by pY = pXv and pX = pXu . In type D, the variety G/B is isomorphic to SpF(ImN ) and there is a pro jection pr : G/B L (ImN ). We thus have maps Y L (ImN ) and X L (ImN ) defined by the composition of qY and qX with pr. The maps pY : Y F = L (ImN ) â F and pX : X F = L (ImN ) â F are the products of these maps with pXv and pXu . Prop osition 4.3. (i) The maps qY and qX are dominant and local ly trivial fibrations with fiber over (Fk )k[0,r] G/B isomorphic to Bott­Samelson varieties Fv and Fw , respectively. (ii) The maps pY : Y F and pX : X F are birational and dominant onto Y and X respectively. Thus they are resolutions of singularities for Y and X . Proof. The first part is clear from the definition of Y and X . The second part follows from the birationality of the Bott­Samelson resolutions Fv Fv and Fw Fw , the smoothness of Y and X , and the first part.

13


Notation 4.4. For a subword u of v, we define Xu to be the subvariety of Y obtained as the image of Xu (seen as a subvariety of Y = Xv ) under the map pY . With this notation X = Xw . The map pX : X X is the resolution in Theorem 1.2.

4.3

Existence of a splitting

We have the following Theorem 4.5. (i) There exists a B -canonical splitting of the Bott-Samelson variety Fv compatibly splitting al l Bott-Samelson subvarieties Fu of Fv for each subword u of v. (ii) This splitting induces a B -canonical splitting of Y compatibly splitting al l the subvarieties Xu for u a subword of v. (iii) The latter splitting induces a splitting of Y compatibly splitting al l the subvarieties Xu , where u is a subword of v. Proof. (i) This is an application of [BK05, Proposition 4.1.17]. (ii) We first observe that the B -canonical splitting in (i) is a B0 -canonical splitting, where B0 = B G0 and G0 was defined in section 4.2. For this, use the following result (see [BK05, Lemma 4.1.6]): let H be a connected and simply connected semisimple group, let H be a Borel subgroup in H , and let H be a maximal torus in H . Let X be a H -scheme and let (n) Hom(F OX , OX ), where F is the Frobenius morphism. Let us denote by e the divided (n) (n) powers, where is a root of H . There exists a natural action e of e on (see [BK05, Definition 4.1.4]). Lemma 4.6. The element is a H -canonical splitting if and only if is H -invariant and (n) e = 0 for al l n p and a simple root. In our situation, we easily check that the divided powers of G0 are divided powers for G . In particular the splitting in (i) is a B0 -canonical splitting and compatibly splits the varieties Fu . To prove that the B0 -canonical splitting induces a B -canonical splitting, we use the results of X. He and J.F. Thomsen [HT10]. More precisely we use Theorem 4.1 in that paper with = G0 . The hypothesis of that theorem are satisfied by Section 8 in [HT10]. We therefore obtain that Y = G âB Fv has a Frobenius spliting which compatibly splits the subvarieties G âB Fu for all subwords u of v.2 We thus have a B -canonical splitting on Fv compatible with all the divisors Dj and therefore with all the subvarieties Fu (by [BK05, Proposition 1.2.1] and the fact that the varieties Fu are intersections of such divisors. Applying [BK05, Theorem 4.1.17], we get a B -canonical splitting on G âB Fv = Y compatible with all subvarieties G âB Fu = Xu . (iii) This is a direct application of Lemma 1.1.8 in [BK05] together with the fact that Y is normal (in type A it is a Schubert variety and in type D see Remark 3.17).

4.4

D-splitting

In this subsection, we prove that the previous splitting is a D-splitting for an explicit divisor D. For this we first need to compute the canonical divisor of the variety Y . Let us first fix some notation. As we have seen, if v = (i1 , · · · , ik , · · · , il ) and if we denote by v[j ] the subword (i1 , · · · , ij ) for j [1, l], then the variety Y can be realized as a sequence
2 Note that in type A, i.e. for a diagonal embeding, the result of X. He and J.F. Thomsen above was already proved by the same authors in [HT08, Theorem 7.2].

14


of P1 -fibrations Y = Xv Xv[l-1] · · · Xv[1] G/B . For all j [1, l], there is a natural map pX : Xv[j ] F and if OF (1) is the line bundle on F defined by the Plucker embedding, ¨ we define Lv
v [j ]

[j ]

=p

Xv

[j ]

(OF (1)). We shall denote by L

Y

and L

X

the line bundles Lv and Lw ,

respectively. The following lemma is an easy modification of a well-known result on the canonical divisor of the Bott­Samelson resolution, see for example [BK05, Proposition 2.2.2] or [Kum02, Proposition 8.1.2]: Lemma 4.7. We have the equality
-1 Y

= OY ( Y ) LY .

Proof. We prove the following formula by induction over j [0, l]:
-1 Xv[
j]

= OX

v [j ]

( Xv[j ] ) L

v[j ]

.

-1 For j = 0, we have Xv[j ] = G/B . The line bundle G/B is the line bundle Lv[j ] which is twice the ample line bundle defined by the Plucker embedding of G/B , since G/B is diagonally embedded ¨ into F = G /B . Let us denote the fibration Xv[j +1] Xv[j ] by f and its section by . The induction follows from the formula:

X

v[j +1]

= f

Xv[

j]

OX

v[j +1]

(-Xv[j ] ) L

-1 v[j +1]

f L

v[j +1]

which is a direct application of [Kum02, Lemma A-18] and the fact that for each j the divisor Lv[j +1] has the relative degree 1 for the fibration f . To prove the induction step we also the the equality Lv[j +1] = Lv[j ] . Theorem 4.8. There exists a D-splitting of Y compatibly splitting the subvarieties Xu , where u is a subword of v and D is an effective divisor such that OY (D) = Lp-1 .
Y

Proof. Recall that in Theorem 4.5 we constructed a splitting of Y compatibly splitting the subvarieties Xu for a subword u of v. In particular, it is compatible with each of the divisors Xv(j ) for j [1, l], where v(j ) = (i1 , · · · , ij , · · · , il ). By [BK05, Theorem 1.4.10], the splitting provides a (p - 1)Xv(j ) -splitting for all j [1, l]. We may thus write
l

div() = (p - 1)
j =1

Xv(j ) + D = Y + D

with OY (D) = L

p-1 Y

(compare with Lemma 4.7). But again by [BK05, Theorem 1.4.10], the

splitting is a div()-splitting. Now using [BK05, Remark 1.4.2 (ii)] we get that is a Dsplitting. Note that by [BK05, Theorem 1.3.8], the multiplicity of a divisor in the subscheme of zeros of the splitting (here in div()) is at most p - 1 thus D has a support disjoint from any of the divisors Xv(j ) . The divisors Xv(j ) are therefore D-compatibly split. Finally, the result follows from the fact that for any subword u of v the variety Xu is the intersection of certain divisors Xv(j ) . Corollary 4.9. There exists a D-splitting of Y compatibly splitting al l the subvarieties Xu for a subword u of v for an effective divisor D such that OY (D) = (OF (1)|Y )p-1 . 15


Proof. This is a direct application of in the previous Theorem, the divisor property. We may apply Lemma 1.4.5 for the splitting of the varieties Xu b under pY .

[BK05, Lemma 1.4.5] to the map pY : Y Y . Indeed, D is the pullback by pY of a divisor D with the above in [BK05] because Y is normal and the conclusion follows ecause these varieties are the images of the varieties Xu

Remark 4.10. The divisor D may not be ample on Y but its restriction to X is ample as X is a subvariety of F.

5

Normality

In this section we prove the results stated in the introduction. The proof will be similar to the proof of the same results for Schubert varieties as given in the book [BK05]. In order to pass from positive characteristic to characteristic zero, we shall use the results in [BK05, Section 1.6]. For this we need to realize the Springer fiber over Z. This can be easily done by choosing a representative of the nilpotent element N in the normal Jordan form in its GL(Kn )-orbit. To simplify notation also also because the variety Y = Xv and some of the varieties Xu for u a subword of v are not contained in F in type D (but in F ), we will only consider in this section the varieties Xu for u a subword of w. These varieties are subvarieties of X = Xw and can therefore be considered as subvarieties of F. In particular, the restriction of the above divisor D is ample on these varieties.

5.1

Some preliminary results

We prove the normality of all the varieties Xw[j ] for j [0, l - k ] by induction over j . For this we need a more precise description of the geometry relating Xw[j ] and Xw[j +1] . Recall the construction of the variety Xw[j +1] from Xw[j ] by the elementary construction 4.1 as the fiber product Xw[j +1] = Xw[j ] âF Fij +1 . For a subvariety Z in F we denote by Z ij +1 its image under the pro jection pij +1 : F F(ij +1 ) (see Subsection 4.1). We have the equality Xw
[j +1]
j = Xw[+1 âF j]

i

(

ij +1

)

F, so we obtain the following commutative diagram Xw
[j +1]

= Xw


[j ]

âF F

a
ij +1

/X w [j ]
p
Xw [ j ]

(2)

/ i Xw[j ] s mm mmm mmm m p b mmm vmmm ij ij Xw[j +1] = Xw[+1 âF(ij +1 ) F a /s i Xw[+1 j] j] m mmm mmm mm p mmm mmm vm

b

Xw

[j +1]

:= Xw

[j ]

âF F

a

ij +1

j Xw[+1 . j +1]

i

Lemma 5.1. With the above notation, (i) the map pX : Xw[j ] Xw[j ] is birational for al l j [0, l];
w[ j ]

(ii) the map p : Xw

[j ]

j Xw[+1 is birational for al l j [0, l - 1]; j]

i

16


j j (iii) we have the equality Xw[+1 = Xw[+1 for al l j [0, l - 1]. j] j +1]

i

i

Proof. (i) We prove this by descending induction on j . For j = l - k the corresponding map is pX : X X , which is birational by Proposition 4.3. Assume that pX is birational. This map is the composition of the top two left vertical
w[j +1]

arrows b and b in the previous diagram. In particular these two maps b and b are also birational. But the topmost right vertical arrow in the above diagram gives b by fiber product. This map is pX and has to be birational.
w[ j ]

(ii) By what we just proved the map b is birational. But it is a fiber product of the map ij p : Xw[j ] Xw[+1 , which has to be birational. j] (iii) Recall that we have two maps ij +1 and ij +1 from Fij +1 to F obtained by forgetting one of the two subspaces corresponding to points in G(ij +1 ) in Fij +1 . The map a corresponds to ij +1 while b corresponds to ij +1 . The composition of the two forgetful maps ij +1 and ij +1 yield a map Fij +1 Fij +1 . The maps p a and p b correspond by fiber product to the maps ij +1 ij +1 and ij +1 ij +1 , respectively. In particular these two maps are equal and the result follows.

5.2

Pro of of Theorem 1.1
[j ]

We prove by ascending induction over j that Xw

is normal. For j = 0, we have Xw

[0]

G/B ,
ij Xw[+1 j]

which is normal. Let j > 0 and assume that Xw[j ] is normal. The map a : Xw[j +1] is 1 -fibration. Thus to prove the normality of X aP w[j +1] we only need to prove the normality of
j j Xw[+1 . But the map p : Xw[j ] Xw[+1 is birational and surjective (Lemma 5.1 (ii) ), and Xw j] j] is normal by the induction hypothesis, so we only need to prove the equality

i

i

[j ]

p OXw

[j ]

=O

j Xw[+1 j]

i

.

This will be done using the following lemma (see [BK05, Lemma 3.3.3]): Lemma 5.2. Let f : X Y be a surjective morphism between projective schemes, and let L be an ample line bund le on Y . Assume that the map H 0 (Y , L ) H 0 (X, f L ) is an isomorphism for large enough. Then f OX = OY .
j Consider L ample on Xw[+1 and the following commutative diagram j]

i

j H 0 (Xw[+1 , L) j +1]

i

/ H 0 (Xw[j +1] , p L) / H 0 (Xw[j ] , p L).

j H 0 (Xw[+1 , L) j]

i

But since Xw[j +1] is D-split compatibly with Xw[j ] and D is ample, we get by [BK05, Theorem 1.4.8] that the right vertical map is surjective (the invertible sheaf p L is semi-ample as the pullij back of an ample line bundle). Moreover, the map p : Xw[j +1] Xw[+1 is a P1 -fibration, so j +1] the top horizontal map is also surjective. We obtain that the lower horizontal map is surjective. ij It is injective since the map p : Xw[j ] Xw[+1 is surjective. We may thus apply the previous j] lemma and deduce the normality. 17


Remark 5.3. This proof works for K of positive characteristic, but relies only on vanishing of cohomology and surjectivity of restrictions on cohomology results which pass, by semi-continuity, to characteristic zero. The same proof therefore works for Char(K) = 0.

5.3

Pro of of Theorem 1.2

Recall the definition of a rational morphism and of rational singularities for Char(K) = 0: Definition 5.4. (i) A morphism of schemes f : X Y is called rational if f OX = OY and all its higher direct images vanish: Ri f OX = 0 for i > 0. (ii) Assume Char(K) = 0. A normal variety X has rational singularities if there exists a rational birational proper morphism : X X with X smooth. We first prove the following Lemma 5.5. For al l j [0, l - k ], the map p
Xw
[j ]

: Xw

[j ]

Xw

[j ]

is a rational morphism. : Xw
[0]

Proof. We prove this lemma by induction over j . For j = 0, we have pX

w[0]

Xw

[0]

is an isomorphism. Let j > 0 and assume that pXw[j ] is rational. Then, since b is obtained by fiber product from pX , we see that b is rational. So, to prove the rationality of pX we w[ j ] w[j +1] only need to prove the rationality of b. Since b is obtained by fiber product from p, we only ij ij need to prove the rationality of p : Xw[j ] Xw[+1 . But p is birational and Xw[+1 is normal (see j] j] the proof of the normality of X ), therefore by the Zariski Main Theorem we obtain the equality p OXw[j ] = O ij +1 . We need to prove the vanishing of the higher direct images. For this we
Xw[
j]

embed Xw

[j ]

j in F and Xw[+1 in F( j]

i

i

j +1

). We have a commutative diagram
/F
p

Xw[j ] 1 ij Xw[+1 1 j]
p

/ F(i



i j +1

j +1

)

and in particular the fibers of both morphisms are at most one-dimensional, thus Ri p OXw[j ] = 0 for i 2. Note that we also have the vanishing R2 (pij +1 ) E for any coherent sheaf E on F. Thus the surjection OF OXw[j ] induces a surjection R1 (pij +1 ) OF R1 p OXw[j ] . But since the second vertical map is a P1 -fibration, we have R1 (pij +1 ) OF = 0 and the result follows. Let us now prove the vanishing Ri pX
w[ j ]

Xw



[j ]

= 0 for i > 0. For this we use the following

direct application of Theorem 1.2.12 from [BK05]: Lemma 5.6. Let f : X Y is a proper birational morphism with X smooth. Assume that is a splitting for X compatibly splitting a divisor Z such that the exceptional locus of f is set-theoretical ly contained in Z . Then we have Ri f OX (-Z ) = 0 for i > 0. We want to apply this lemma to the map p : Xw[j ] Xw[j ] , the splitting constructed above, and the divisor Z = Xw[j ] . For this we only need to check that the exceptional locus of pX
w[ j ]

is contained in Xw[j ] . But the map Xw Xw
[j ]

[j ]

Xw
[j ]

[j ]

decomposes as follows:
[j ]

= G âB Fw

G âB Fw 18

Xw

[j ]


and this map is G-equivariant. Furthermore, the complement to Fw[j ] in Fw[j ] is a B -equivariant dense open subset thus the complement of Xw[j ] in Xw[j ] is a G-equivariant dense open subset. The map is therefore an isomorphism on this open subset, and the exceptional locus is contained in Xw[j ] . By the previous lemma, we get the vanishing Ri p
Xw[j ]

OX

w[ j ]

(- Xw[j ] ) = 0 for i > 0. (- Xw[j ] ) p
Xw

But from Lemma 4.7, we have X formula, we get: Ri p
Xw
[j ]

w[ j ]

= OX OX

w[ j ]

[j ]

(OF (1)|Xw[j ] ). Thus by pro jection

Xw



[j ]

= Ri p

Xw

[j ]



w[ j ]

(- Xw[j ] ) (OF (1)|

Xw[

j]

) = 0 for i > 0.

This completes the proof of Theorem 1.2. Corollary 1.3 follows from general results on rational resolutions, see [BK05, Lemma 3.4.2], and Corollary 1.4 follows from the definition of rational singularities and Lemma 5.5. Remark 5.7. The proof of Lemma 5.5 works for any characteristic (once the normality is proved). For Char(K) = 0 we do not need to prove the above vanishing Ri pX X = 0 for i > 0.
w[j ]



w[ j ]

This result follows automatically from Grauert­Riemenschneider Theorem [GR70].

References
[Bou54] [BK05] [Dem74] [Fre09a] [Fre09b] [FM10] [Ful97] [Fun03] [GR70] [HT08] Bourbaki, N., Groupes et alg` es de Lie. Hermann 1954. ebr Brion, M., Kumar, S., Frobenius splitting methods in geometry and representation theory. Progress in Mathematics, 231. Birkh¨ auser Boston, Inc., Boston, MA, 2005. Demazure, M., D´ esingularisation des vari´ ´ de Schubert g´ ´ alis´ . Ann. Sci. etes ener ees ´ Ecole Norm. Sup. (4) 7 (1974). Fresse, L., Singular components of the Springer fiber in the two columns case. Ann. Inst. Fourier 59 (2009) no. 6. Fresse, L., Composantes singuli` es des fibres de Springer dans le cas deux-colonnes, er C. R. Acad. Sci Paris, Ser. I 347 (2009). Fresse, L., Melnikov, A., On the singularity of the irreducible components of a Springer fiber in sl(n). Selecta Math. 16 (2010) no. 3. Fulton, W., Young tableaux. London Mathematical Society Student Texts, 35. Cambridge University Press, Cambridge, 1997. Fung, F.Y.C., On the topology of components of the Springer fibers and their relation to Kazhdan-Lusztig theory, Adv. Math. 178 (2003). Grauert, H., Riemenschneider, O., Verschwindungss¨ azte fur analytische kohomolo¨ giegruppen auf komplexen r¨ aumen, Invent. Math. 11 (1970). He, X., Thomsen, J.F., Frobenius splitting and geometry of G-Schubert varieties. Adv. in Math. 219 (2008) no. 5. 19


[HT10] [Kum02] [Spa82] [SW10]

He, X., Thomsen, J.F., On Frobenius splitting of orbit closures of spherical subgroups in flag varieties. Preprint arXiv:1006.5175. Kumar, S., Kac-Moody groups, their flag varieties and representation theory, Progress in Mathematics, 204. Birkh¨ auser Boston, Inc., Boston, MA, 2002. Spaltenstein, N., Classes unipotentes et sous-groupes de Borel. Lecture Notes in Mathematics, 946. Springer-Verlag, Berlin-New York, 1982. Stroppel, C., Webster, B., 2-block Springer fibers: convolution algebras, coherent sheaves and embedded TQFT, to appear in Comm. Math. Helvetici (2010). Preprint arXiv:0802.1943. van Leeuwen, M.A.A., A Robinson­Schensted algorithm in the geometry of flags for classical groups, Ph.D. thesis, Rijksuniversiteit Utrecht, 1989. http://www-math.univ-poitiers.fr/~maavl/postscript/thesis.ps.gz Vinberg, E.B., Gorbacevich, V.V., Onishchik, A.L., Structure of Lie groups and Lie algebras, in: Lie groups and Lie algebras 3 Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 41, VINITI, Moscow, 1990, 5­253 (Russian). English transl.: Encyclopaedia of Mathematical Sciences, vol. 41, Springer, 1994.

[vLe89]

[VGO90]

Nicolas Perrin, Hausdorff Center for Mathematics, Universit¨ Bonn, Villa Maria, Endenicher Allee 62, 53115 at Bonn, Germany, and Institut de Math´ ematiques de Jussieu, Universit´ Pierre et Marie Curie, Case 247, 4 place Jussieu, e 75252 Paris Cedex 05, France. email: nicolas.perrin@hcm.uni-bonn.de Evgeny Smirnov, Department of Mathematics, Higher School of Economics, Myasnitskaya ul., 20, 101000 Moscow, Russia, and Laboratoire J.-V. Poncelet, Bolshoi Vlassievskii per., 11, 119002 Moscow, Russia. email: smirnoff@mccme.ru

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