aa r X i v : . [ m a t h . AG ] A ug EKEDAHL-OORT STRATA IN THE SUPERSINGULAR LOCUS
MAARTEN HOEVE
Abstract.
We give a description of the individual Ekedahl-Oort strata con-tained in the supersingular locus in terms of Deligne-Lusztig varieties, refininga result of Harashita. Introduction
Fix a prime p and consider the moduli stack A g of principally polarized abelianvarieties of dimension g over F p . In [Oo01] Ekedahl and Oort defined a stratificationof A g , the Ekedahl-Oort stratification. Harashita showed that certain unions ofstrata in the supersingular locus are isomorphic to Deligne-Lusztig varieties (see[Ha07]). Here we show that individual strata are isomorphic to a finer kind ofDeligne-Lusztig varieties.The difference between these results is clearest in the index sets. Van der Geerand Moonen showed how to index Ekedahl-Oort strata using the Weyl group W g = { w ∈ Aut( { , . . . , g } ) | w (2 g + 1 − i ) = 2 g + 1 − w ( i ) } of the symplectic group Sp g : there is an open stratum S w for each w in I W g = { w ∈ W g | w − (1) < · · · < w − ( g ) } , see [Mo01]. Equivalently, we can use cosets: if W g,I is the subgroup generated bythe permutations ( i, i + 1)(2 g − i, g + 1 − i ) for i = 1 , . . . , g −
1, then the naturalmap I W g → W g,I \ W g is a bijection.We now summarize Harashita’s description of the Ekedahl-Oort strata in thesupersingular locus. The stratum attached to w ∈ I W g is in this locus if and onlyif w is in the subgroup W [ c ] g = { w ∈ W g | w ( i ) = i for i = 1 , . . . , g − c } for some c ≤ g/
2. This subgroup is isomorphic to W c via r : W [ c ] g ∼ → W c , r ( w )( i ) = w ( g − c + i ) − ( g − c ) . Write I W [ c ] g = W [ c ] g ∩ I W g and I W ( c ) g = I W [ c ] g − I W [ c − g . Note that r maps I W [ c ] g to I W c .Fix a c ≤ g/ E over F p and let Λ g,c be theset of isomorphism classes of polarizations on E g with kernel isomorphic to α cp .For any µ ∈ Λ g,c the isogenies from ( E g , µ ) to principally polarized abelian vari-eties correspond one-to-one with maximal isotropic subspaces in a 2 c -dimensionalsymplectic vector space (see section 3). Let X be the variety over F p that param-eterizes both objects.On the one hand we get a morphism i µ : X → A g, F p that sends an isogeny toits target and an action of Aut( E g , µ ) on X by precomposition. The morphism i µ factors through the quotient stack [ X / Aut( E g , µ )]. On the other hand we get Deligne-Lusztig varieties X [ α ] in X . For α ∈ W c,I \ W c /W c,I the variety X [ α ] consists of all parabolic subspaces U such that U and U ( p ) (or rather their stabilizers) are in relative position α . Theorem 1.1. ( [Ha07] , main theorem) For each double coset α = W c,I w ′ W c,I with w ′ ∈ I W [ c ] g the morphism of stacks over F p a µ ∈ Λ g,c [ X [ α ] / Aut( E g , µ )] → [ w ∈ I W g s.t. r ( w ) ∈ α S w induced by the i µ is finite and surjective. If w ′ is in I W ( c ) g , then this morphism isa bijection on geometric points. Unfortunately, this theorem only describes unions of strata. In this paper werefine it to individual strata by looking at a finer kind of Deligne-Lusztig varieties.Inspired by ideas of Moonen and Wedhorn in [MW04], we consider not just therelative position of a subspace U with U ( p ) , but also that of their refinements. Weget varieties X ( w ′ ) indexed by I W c (instead of W c,I \ W c /W c,I ). With these wecan refine theorem 1.1 to the following. Theorem 1.2.
For each w in I W ( c ) g the morphism of stacks over F p a µ ∈ Λ g,c [ X ( r ( w )) / Aut( E g , µ )] → S w , induced by the i µ is an isomorphism. An Ekedahl-Oort stratum is reducible if it is contained in the supersingular locus,at least if p is large enough (see [Ha07] corollary 3.5.3) and irreducible otherwise(see [EG06] theorem 11.5). In section 7 we show that X ( r ( w )) is irreducible for w ∈ I W ( c ) g . So theorem 1.2 gives the exact number of components of the supersingularstrata. Corollary 1.3.
For w ∈ I W ( c ) g the Ekedahl-Oort stratum S w has g,c irreduciblecomponents. Here it is crucial that w is in I W ( c ) g and not just in I W [ c ] g . The number g,c isa class number, see [Ha07] section 3.2 and 3.5.This description of the EO-strata in the supersingular locus bears a strikingresemblance to the description of Kottwitz-Rapoport strata in the supersingularlocus in a recent paper [GY08] by G¨ortz and Yu. It would be interesting to furtherinvestigate the relation between Ekedahl-Oort and Kottwitz-Rapoport strata.1.0.1. Conventions. If S is a scheme over F p and F is an O S -module, then wedenote with F ( p r ) = O S ⊗ O S , Frob r F the pull-back by the r -th power of the absoluteFrobenius. Further, if S → T is a morphism of schemes and X a T -scheme, wewrite X S for S × T X .1.0.2. Acknowledgements.
I thank my advisors Ben Moonen and Gerard van derGeer for their help. I thank Torsten Wedhorn for pointing out a mistake in anearlier version.
KEDAHL-OORT STRATA IN THE SUPERSINGULAR LOCUS 3 Deligne-Lusztig varieties
We can improve Harashita’s theorem, because we use finer Deligne-Lusztig va-rieties. Originally Deligne and Lusztig defined their varieties for Borel subgroups(see [DL76]). We can generalize them to parabolic subgroups in two ways. If weuse exactly the same definition, we get what we will call coarse Deligne-Lusztig va-rieties. By theorem 1.1 they correspond to unions of strata. If we use a more subtledefinition, pioneered by Lusztig and B´edard (see section 1.2 of [Lu03] and [Be85]),then we get fine Deligne-Lusztig varieties. Theorem 1.2 says that they correspondto individual strata.In this section we gather some definitions and results on Deligne-Lusztig varieties.Although we will only need them for Sp c , we give them for more general groups.2.1. Coarse Deligne-Lusztig varieties.
Let k be an algebraic closure of thefinite field with q elements F q . Suppose that G is a reductive connected algebraicgroup over k , obtained by extension of scalars from G over F q . Write F : G → G for the corresponding Frobenius morphism.Let W be the Weyl group of G . The Frobenius F acts on W . Let S ⊂ W be theset of reflections in simple roots. We denote with P I the variety over k of parabolicsubgroups in G of type I ⊂ S . The group G acts on P I by conjugation. So it alsoacts on P I × P J for all I, J ⊂ S .Let W I is the subgroup of W generated by the reflections in the roots in I ⊂ S .The G -orbits in P I × P J are in bijection with W I \ W/W J , see [Be85] II lemma 7.Suppose P and Q are two parabolic subgroups, of types I and J respectively. Thenwe say that they are in relative position w ∈ W I \ W/W J if the point ( P, Q ) ∈ P I ×P J is in the G -orbit corresponding to w . We write relpos( P, Q ) = w .Each double coset in W I \ W/W J contains a unique element of minimal length.Denote the set of such elements with I W J ⊂ W , so that the quotient map I W J → W I \ W/W J is a bijection. We will often see this bijection as an identification. Inparticular, we will speak of parabolic subgroups in relative position w ∈ I W J . Definition 2.1.
The coarse Deligne-Lusztig variety P I [ w ] attached to w ∈ I W F ( I ) is the locally closed subscheme of P I consisting of all parabolic subgroups P suchthat P and F ( P ) are in relative position w .The orbit in P I × P F ( I ) corresponding to w ∈ I W F ( I ) is smooth of dimension l ( w ) + dim( P I ∩ F ( I ) ), where l is the length function. The variety P I [ w ] is theintersection of this orbit with the graph of the Frobenius morphism. Since theintersection is transversal, we get the following (compare [DL76] section 1.3). Lemma 2.2.
The variety P I [ w ] is smooth and purely of dimension l ( w )+dim( P I ∩ F ( I ) ) − dim( P I ) . In particular, if F ( I ) = I , then P I [ w ] has dimension l ( w ). There is also thefollowing result of Bonnaf´e and Rouquier [BR06] on irreducibility. Theorem 2.3.
A variety P I [ w ] is reducible if and only if W I w is contained in aproper subgroup of the form W J for some J ⊂ S that is F -stable. Using this one can determine the number of irreducible components of any P I [ w ],see [GY08] corollary 5.3. MAARTEN HOEVE
Fine Deligne-Lusztig varieties.
Keep the notation from the previous sec-tion. To get finer varieties, we look not just at the relative position of P and F ( P ),but also of their refinements. Given two parabolic subgroups P and Q of G , the refinement of P with respect to Q isRef Q ( P ) = ( P ∩ Q ) U P = U P ( P ∩ Q ) , where U P is the unipotent radical of P . This is again a parabolic subgroup and itis contained in P . If P is of type I and Q of type J and they are in relative position w , then Ref Q ( P ) is of type I ∩ w J .Suppose I is a subset of the set of simple roots. Given a sequence u = ( u , u , . . . )of elements of W , define a sequence of subsets I n ⊂ I by I = I and I n +1 = I n ∩ u n F ( I n ). Let T ( I ) be the set of sequences u , such that u n ∈ I n W F ( I n ) and u n +1 ∈ W I n +1 u n W F ( I n ) . Then we have the following description of T ( I ), see [Be85] I proposition 9. Proposition 2.4.
Each sequence u in T ( I ) stabilizes to some u ∞ , i.e. satisfies u n = u n +1 = · · · = u ∞ for n large enough. The map T ( I ) → W which sends u to u ∞ induces a bijection from T ( I ) to I W . Definition 2.5.
The fine Deligne-Lustig variety P I ( u ) attached to a sequence u in T ( I ) is the locally closed subscheme of P I consisting of all parabolic subgroups P such that if we define P = P and P n +1 = Ref F ( P n ) ( P n )then P n and F ( P n ) are in relative position u n .See [Lu03] 1.3 and 1.4 for some examples of fine Deligne-Lusztig varieties.It follows from proposition 2.4 that we can also index the fine Deligne-Lusztigvarieties by I W . So we will often speak of P I ( w ) for w ∈ I W .Most questions about the fine varieties can be reduced to ones about coarsevarieties, using the following proposition of B´edard, see [Be85] II proposition 12. Proposition 2.6.
The morphism P I ( u , u , . . . ) → P I ( u , u , . . . ) which sends P to Ref F ( P ) ( P ) is an isomorphism. Iterating the above proposition until we get to u ∞ we get the following. Corollary 2.7.
There is an isomorphism P I ( u ) ∼ = P I ∞ [ u ∞ ] , where I ∞ = T I n . Now we combine this with lemmas 2.2 and 2.3.
Corollary 2.8.
The variety P I ( u ) is smooth and purely of dimension l ( u ∞ ) +dim( P I ∞ ∩ F ( I ∞ ) ) − dim( P I ∞ ) . It is reducible if and only if W I ∞ u ∞ is contained ina proper F -stable standard parabolic subgroup of W . When G = Sp c , the Frobenius F is the identity on W , so P I ( u ) has dimension l ( u ∞ ).The action of G on P I by conjugation restricts to an action of G ( F q ). For g ∈ G ( F q ) we have F ( g ) = g . Hence, F ( gP g − ) = gF ( P ) g − . The relativeposition of two parabolic subgroups is unchanged if you conjugate them by thesame element. So both the coarse and fine Deligne-Lusztig varieties are stableunder the G ( F q )-action. KEDAHL-OORT STRATA IN THE SUPERSINGULAR LOCUS 5
An equivalent definition.
It will be convenient to have the following descrip-tion of the fine Deligne-Lusztig varieties.
Proposition 2.9.
The variety P I ( u ) consists of all parabolics P in P I such thatif we set P ′ = P and P ′ n +1 = Ref F ( P ′ n ) ( P ) , then P and F ( P ′∞ ) are in relative position u ∞ . This description differs in two ways from the definition of P I ( u ). First of allit uses P ′ n instead of P n . That P ′ n = P n follows by induction from the followinglemma with P = P , Q ′ = F ( P ′ n ) and Q = F ( P ′ n − ). Lemma 2.10.
Suppose Q ′ ⊂ Q and P are parabolic subgroups. Then Ref Q ′ (Ref Q ( P )) = Ref Q ′ ( P ) . Proof.
The left hand side is U P (( U P ( P ∩ Q ) ∩ Q ′ ). So it is sufficient to showthat ( U P ( P ∩ Q )) ∩ Q ′ = P ∩ Q ′ . On the one hand, P ∩ Q ⊂ U P ( P ∩ Q ), gives P ∩ Q ′ = P ∩ Q ∩ Q ′ ⊂ ( U P ( P ∩ Q )) ∩ Q ′ . On the other hand, P ⊇ U P ( P ∩ Q )gives P ∩ Q ′ ⊇ ( U P ( P ∩ Q )) ∩ Q ′ . (cid:3) Second of all the condition that P and F ( P ′∞ ) are in relative position u ∞ isequivalent to the condition that P ∞ and F ( P ∞ ) are in relative position w by thefollowing lemma with Q = F ( P ∞ ). Lemma 2.11.
For any two parabolic subgroups P and Q one has relpos( P, Q ) = relpos(Ref Q ( P ) , Q ) ∈ W. Proof.
This is the case Z = P in lemma 3.2(c) in [Lu03]. (cid:3) Since the sequence u is determined by u ∞ (lemma 2.4), the fact that P ∞ and F ( P ∞ ) are in relative position u ∞ is equivalent with P being in P I ( u ). This provesthe proposition.2.3. The case that G = Sp c . Let L be a 2 c -dimensional vector space over F q with a symplectic form and let G = Sp( L ) be the symplectic group of L . Apartial flag in L := k ⊗ L is a collection of subspaces C that is totally ordered bythe inclusion. There is a bijection between such flags and parabolic subgroups of G by sending a flag C to its stabilizer stab( C ).The Weyl group of G is the group W c from the introduction. For w ∈ W c wedefine r w ( i, j ) = { a ∈ { , . . . , i } | w ( a ) ≤ j } . Using elementary linear algebra one can prove the following.
Lemma 2.12.
Suppose C and D are two flags in L and let P and Q be theirrespective stabilizers. Let I be the type of P and J that of Q . Then P and Q arein relative position w ∈ I W J if and only if dim( C ∩ D ) = r w (dim( C ) , dim( D )) for all C ∈ C and D ∈ D . If P is the stabilizer of C = { C ⊂ C ⊂ · · · ⊂ C r = L } and Q that of D = { D ⊂ D ⊂ · · · ⊂ D s = L } , then Ref Q ( P ) is the stabilizer of the flagconsisting of ( C i +1 ∩ D j ) + C i for i = 1 , . . . , r and j = 1 , . . . , s. MAARTEN HOEVE
Denote this flag with Ref D ( C ). Also, note that if P is the stabilizer of C , then F ( P )is the stabilizer of C ( q ) = { C ( q ) i } .3. A moduli space of isogenies
In this section we construct an isomorphism between the variety of isogeniesfrom ( E g , µ ) to principally polarized abelian varieties and a variety of maximalisotropic subspaces in a symplectic vector space. This is a direct generalization ofthe construction of the families of abelian surfaces considered by Moret-Bailly in[MB81].3.1. Dieudonn´e modules.
Suppose that S is a scheme over F p . Let ˆCW be theformal group of Witt-covectors over Z , see [Fo77], chapter II 1.5. For a formal p -group scheme G over S let M ( G ) = H om ( G, ˆCW S ) , be the sheaf on S whose sections over U ⊂ S are the homomorphisms G | U → ˆCW U .There is a natural action of the Witt-vectors W on ˆCW. This makes M ( G ) intoa W ( O S )-module. The Frobenius and Verschiebung on G give homomorphisms F : M ( G ) ( p ) → M ( G ) and V : M ( G ) → M ( G ) ( p ) respectively (where M ( G ) ( p ) = W ( O S ) ⊗ W ( O S ) , Frob M ( G )). By abuse of notation we will sometimes write M ( G )for the triple ( M ( G ) , F, V ).Over the spectrum of a perfect field M ( G ) is the classical contravariant Dieudonn´emodule of G as defined by Fontaine in [Fo77]. The functor G M ( G ) is exactand gives an equivalence from the category of formal p -groups to the category ofmodules over a certain ring, see loc. cit. theorems 1 and 2 in the introduction.Over other schemes S , this is no longer true in general, but we will still call M ( G )the Dieudonn´e module of G .If G is annihilated by p , multiplication by p is zero on M ( G ). Hence, M ( G ) isactually on O S -module. If G is annihilated by V , then M ( G ) = H om ( G, ˆ G a ) byproposition III, 3.2 in [Fo77]. It follows from [Jo93] proposition 2.2 that for any S the functor M is an equivalence from the categories of group schemes that areannihilated by V to that of O S -modules M with a homomorphism F : M ( p ) → M .If π : A → S is an abelian scheme over S , then there is an isomorphism M ( A [ p ]) ( p ) ∼ = H ( A/S ) def = R π ∗ (Ω ∗ A/S )by [BBM82] theorem 4.2.14.3.2.
The moduli space.
Like in [LO98] section 1.2, we fix a supersingular ellipticcurve E ′ over F p such that the Frobenius satisfies F + p = 0. All the endomor-phisms of E ′ are defined over F p , so it is convenient to work with E = E ′ F p .Fix a g ≥
1. Every polarization of E g is defined over F p , since E g is isomorphicto its own dual, so a polarizations can be seen as an endomorphism. For c ≤ g/ g,c be the set of isomorphism classes of polarizations µ on E g whose kernel isisomorphic to α cp .Now fix a polarization µ ∈ Λ g,c . If we put M = M ( E g [ p ]), then µ gives amorphism of Dieudonn´e modules µ ∗ : M → M t , where M t is the dual of M .Let L be the cokernel of µ ∗ . Then L is the Dieudonn´e module of the kernel of µ and so it is isomorphic to F cp with F and V both the zero map. The polarization µ induces a pairing e µ on its kernel and this gives a symplectic form on L . KEDAHL-OORT STRATA IN THE SUPERSINGULAR LOCUS 7
Let S be any scheme over F p . We want to study the set I µ ( S ) of isomorphismclasses of isogenies ρ : ( E gS , µ S ) → ( A, λ ) , where A is an abelian scheme over S and λ is a principal polarization on A . Notethat ρ gives a morphism ρ ∗ : M ( A [ p ]) → M ( E gS [ p ]) = O S ⊗ F p M of Dieudonn´e modules on S . Lemma 3.1.
The map that sends an isogeny ρ to the image of the compositionof ρ ∗ with O S ⊗ F p M → O S ⊗ F p L , gives a bijection from I µ ( S ) to the set ofisotropic subbundles of rank c in O S ⊗ F p L .Proof. An isogeny ρ is determined up to isomorphism by its kernel. Because µ S = ρ ∗ λ , we have ker( ρ ) ⊂ ker( µ S ). Since ker( µ S ) ∼ = α cp is annihilated by V , theinclusion ker( ρ ) ⊂ ker( µ S ) is determined by the induced morphism O S ⊗ F p L = M (ker( µ S )) → M (ker( ρ ∗ )) = coker( ρ ∗ )on Dieudonn´e modules. This morphism is again determined by its kernel, which isexactly the image of ρ ∗ composed with M → L .Let us determine which subbundles of O S ⊗ F p L one can get in this way. Anysubbundle of O S ⊗ F p L gives by Dieudonn´e theory a subgroup scheme G ⊂ ker( µ S ).The polarization µ descends to E gS /G if and only if G is isotropic with respect to e µ and if it descends, say to λ , then the kernel of λ is G ⊥ /G . Therefore, one getsall maximal isotropic subspaces of O S ⊗ F p L . (cid:3) Let X be the variety over F p that parameterizes isotropic subspaces of rank c in L . Note that as a space it depends only on c . The lemma shows that X isa fine moduli space for the functor S I ( S ). In particular, there is a universalisogeny ρ X : ( E gX , µ X ) → ( A X , λ X )over X . The principally polarized abelian scheme ( A X , λ X ) induces a morphism i µ : X → A g, F p over F p .Let k be an algebraic closure of F p . Write X for the base change of X to k .Using the bijection between flags and parabolics from section 2.3, we can see X as a variety of parabolics P I for some I (the type of the stabilizer of a maximalisotropic subgroup). In particular there are Deligne-Lusztig varieties X ( w ) in X for w ∈ I W c .3.3. The action of
Aut( E g , µ ) . The automorphism group Γ µ := Aut( E g , µ ) actson I ( S ) by precomposition: γ ∈ Γ µ sends an isogeny ρ : ( E gS , µ S ) → ( A, λ ) to ρ ◦ γ S .The action is functorial in S , so it induces an action of Γ µ on X .Equivalently, we can define this action as follows. The action of Γ µ on E g induces actions on M and M t such that µ ∗ is equivariant. This gives an action ofΓ µ on the cokernel L which respects the symplectic pairing, i.e. a homomorphismΓ µ → Sp( L )( F p ). Via the action of this last group on X , we get an action ofΓ µ on X . Since the Deligne-Lusztig varieties are Sp( L )( F p )-stable, they are alsoΓ µ -stable. MAARTEN HOEVE The EO-stratification on X In this section we pull the EO-stratification back to X by the morphism i µ : X → ( A g ) k . Keep the notation from the previous section. Suppose x is a k -valued pointof X , corresponding to an isogeny ρ : ( E gk , λ k ) → ( A, λ ) . To see in which EO-stratum i µ ( x ) is, we must express the Dieudonn´e module N of A [ p ] in terms of the subspace U ⊂ L := k ⊗ L attached to x . This is done usingthe filtration constructed below.4.1. A filtration of the Dieudonn´e modules.
It will be convenient to workwith the p ∞ Dieudonn´e modules. Let M be the Dieudonn´e module of E g [ p ∞ ]and let N be that of A [ p ∞ ]. These are modules over the Witt vectors W ( F p ) and W ( k ) respectively. Put M = W ( k ) ⊗ W ( F p ) M . Then µ and ρ induce injectivemorphisms µ ∗ : M t ֒ → M and ρ ∗ : N ֒ → M. Since over a perfect field the Dieudonn´e functor is exact, we have M = M /pM and N = N/pN .Define a descending filtration Fil ∗ on M byFil i ( M ) = M for i ≤ , im( µ ∗ ) for i = 1 ,F ((Fil i − ) ( p ) ) for i = 2 , ,p Fil i − for i ≥ . Then gr ( M ) = M / im( µ ∗ ) = L , since the image of µ ∗ contains pM . Forbrevity we write K = gr ( M ) and K = k ⊗ K (this space plays no significantrole). Because of the inherent periodicity of the filtration, the graded modules are(1) gr i +4 j ( M ) ∼ = L for i = 0 ,K for i = 1 ,L ( p )0 for i = 2 ,K ( p )0 for i = 3 , for each j ≥
0. The isomorphisms are induced by multiplication by − p j for i = 0 , − p j F for i = 2 , F + p = 0 for our choice of E , we have F = − V on M and V = − F V = − p . So by construction V (Fil i M ) = Fil i +2 ( M ) ( p ) . The isomorphisms in(1) are chosen so that the graded maps gr i ( M ) → gr i +2 ( M ( p )0 ) induced by V arethe identity. Note that L ( p )0 = L , since it is a vector space over F p .The filtration Fil ∗ ( M ) on M is the main tool in our analysis of the EO-strataon X . By pulling it back or taking it modulo p we get filtrations on other spaces.We denote them with Fil ∗ ( . . . ) with the space in brackets, for instance Fil ∗ ( M ( p )0 )and Fil ∗ ( M ). We now focus on the filtration Fil ∗ ( N ), obtained by pulling Fil ∗ ( M )back by ρ ∗ and taking it modulo p . It allows us to express N in terms of L and U . KEDAHL-OORT STRATA IN THE SUPERSINGULAR LOCUS 9
Lemma 4.1.
The graded modules of N are gr i ( N ) = U for i = 0 ,k ⊗ gr i ( M ) for i = 1 , , ,L/U for i = 4 , otherwise,where U ⊂ L is the subspace corresponding to x .Proof. We see µ ∗ and ρ ∗ as inclusions: M t ⊂ N ⊂ M . Then gr ( N ) = N/M t andthis quotient is U by the construction of the universal isogeny on X in lemma 3.1.Also, gr ( N ) = pM/pN ∼ = M/N = L/U.
Because pN ⊂ pM = Fil the graded modules for i = 1 , , k ). (cid:3) Again we have V (Fil i ( N )) ⊂ Fil i +2 ( N ) ( p ) . So V induces mapsgr i ( V ) : gr i ( N ) → gr i +2 ( N ) ( p ) . Since on M the graded maps induced by V are the identity, these maps are asfollows. Lemma 4.2.
Under the isomorphisms in lemma 4.1: (1) gr ( V ) is the inclusion U ⊂ L ; (2) gr ( V ) is the identity on K ; (3) gr ( V ) is the canonical surjection L → L/U and (4) gr i ( V ) is for all other i . A submodule H ⊂ gr i ( N ) gives a submodule of N by pulling it back via theprojection pr i : Fil i → gr i . For such submodules it is easy to calculate the pull-back by F and V . Lemma 4.3.
For every i and H ⊂ gr i +2 ( N ) we have V − (pr − i +2 ( H ) ( p ) ) = pr − i (gr i ( V ) − ( H ( p ) )) ,F − (pr − i +2 ( H )) = pr − i (gr i ( F ) − ( H )) . Proof.
This follows from the relation gr i − ( V ) ◦ pr i − = pr i ◦ V defining gr i ( V ) andthe same relation for F . (cid:3) In particular this can be applied to the subspace U = 0 of gr ( N ) = L/U to getthe following.
Corollary 4.4.
For N , we have ker( F ) = pr − ( U ) and ker( V ) = pr − ( U ( p ) ) . Calculating the EO-type.
Now we prove the following proposition.
Proposition 4.5.
The reduced underlying subscheme of the pull-back i − µ S w = X × A g,k S w of the (open) Ekedahl-Oort stratum S w by i µ : X → A g,k is the Deligne-Lustigvariety X ( r ( w )) if w ∈ I W [ c ] g . Otherwise it is empty. We must show that i µ ( x ) is in S w if and only if x is in X ( r ( w )). We do this byconstructing flags on N from flags on L . Definition 4.6.
Given symplectic flags D and D ′ in L that contain U , define aflag E ( D , D ′ ) in N as follows. On gr ( N ) = U take all subspaces in D contained in U , on gr = L/U take those containing U and on gr = L ( p ) take the flag ( D ′ ) ( p ) .Finally pull everything back by pr i : Fil i → gr i .Remember the construction of the canonical flag (see [Oo01] section 5, in par-ticular lemma 5.2, 4): start with the flag C = { ker( V ) } and create C i +1 from C i by adding V − ( C ) and V − ( C ) ⊥ for C ∈ C i . The flag C ∞ to which this sequencestabilizes is the canonical flag.Let P be the stabilizer of U and let D i (resp. D ∞ ) be the flag correspondingto the parabolic subgroup P i (resp. P ∞ ) in definition 2.5. The flags C i and D i arethen related as follows. Proposition 4.7.
For all i C i = E ( D i , D i ) and C i +1 = E ( D i +1 , D i ) . Proof.
Use induction on i . The case i = 0 follows from corollary 4.4. Suppose nowthat C i = E ( D i , D i ). To calculate C i +1 , we need to pull back C ( p )2 i by V : N → N ( p ) .By lemma 4.3, we can do this on the graded modules. Only in three cases is gr i ( V )non-zero:(1) gr ( V ) : U → L = L ( p ) is the inclusion. The restriction of C ( p )2 i to (gr ) ( p ) = L ( p ) is D ( p ) i . So the pull-back by gr ( V ) consists of U ∩ D ( p ) for D ∈ D i ,i.e. the subspaces of Ref D ( p i D = D i +1 contained in U .(2) gr ( V ) : K → K is the identity. But the restriction of C ( p )2 i to (gr ) ( p ) = K is empty. Hence, so it its pull-back.(3) gr ( V ) : L ( p ) → L ( p ) /U ( p ) is the quotient map. The restriction of C ( p )2 i to(gr ) ( p ) = L ( p ) /U ( p ) consists of the subspaces in D ( p ) i that contain U ( p ) .The pull-back to L ( p ) consists of the same subspaces.After adding the orthogonal complements, we get E ( D i +1 , D i ). The other case issimilar and we omit the details (cid:3) Corollary 4.8.
The canonical flag is C ∞ = E ( D ∞ , D ∞ ) . Define ψ w ( i ) = i − r w ( g, i ) with r w as in section 2.3. Then i µ ( x ) is in S w if andonly if F ( C ( p ) ) has dimension ψ w (dim( C )) for all C ∈ C ∞ . This is equivalent todim(ker( F ) ∩ C ( p ) ) = dim( C ) − ψ w (dim( C )) = r w ( g, dim( C )) . By corollary 4.4 the kernel of F contains Fil and is contained in Fil . So theintersection with a pull-back of D ( p ) in (gr ) ( p ) , for D ∈ D ∞ , has dimension dim( D ).The intersection with a pull-back of the D ( p ) in (gr ) ( p ) has dimension g . Hence, i µ ( x ) is in S w for some w ∈ I W [ c ] g .Now look at the pull-backs of the D ( p ) in (gr ) ( p ) . We havedim(ker( F ) ∩ ( p − D ( p ) )) = g − c + dim( U ∩ D ( p ) ) . One easily checks that for w ∈ W [ c ] g r w ( g − c + i, g − c + j ) = g − c + r r ( w ) ( i, j ) . KEDAHL-OORT STRATA IN THE SUPERSINGULAR LOCUS 11
So for w ∈ I W [ c ] g i µ ( x ) ∈ S w ⇐⇒ dim(ker( F ) ∩ C ( p ) ) = r w ( g, dim( C )) for all C ∈ C ∞ ⇐⇒ dim( U ∩ D ( p ) ) = r r ( w ) ( c, dim( D )) for all D ∈ D ∞ . By lemma 2.12 the last statement is equivalent with P and P ∞ being in relativeposition r ( w ). This proves proposition 4.5.5. The differential of i µ In this section we calculate the cotangent map of i µ : X → ( A g ) k and show thatit is surjective.We keep the notation of section 4. In particular X is the variety parameterizing c -dimensional isotropic subspaces in a 2 c -dimensional vector space L . Write L = O X ⊗ k L and let U ⊂ L be the universal subbundle on X . Then we have thefollowing well-known result. Lemma 5.1.
The composition U ֒ → L d ⊗ → Ω X ⊗ L → Ω X ⊗ ( L / U ) is O X -linear. It induces an isomorphism Sym ( U ) → Ω X . The universal isogeny ρ X : E gX → A X induces a homomorphism ρ ∗ X : H ( A X /X ) → H ( E gX /X )which is horizontal with respect to the Gauss-Manin connections on both sides.Note that H ( E gX /X ) = O X ⊗ F p H ( E g / F p ) ∼ = O X ⊗ F p M ( p )0 (see section 3.1). With respect to these isomorphisms, the Gauss-Manin connectionis just d ⊗
1. Write H for H ( A X /X ) and let Fil i ( H ) be the pull-back of O X ⊗ Fil i ( M ) ( p ) by ρ ∗ X . Lemma 5.2.
The subspaces
Fil i ( H ) are horizontal with respect to the Gauss-Maninconnection ∇ , i.e. ∇ (Fil i ( H )) ⊂ Ω X ⊗ Fil i ( H ) .Proof. Since Gauss-Manin connection on H ( E gX /X ) is d ⊗
1, the subspaces O X ⊗ Fil i ( M ) ( p ) s are horizontal. Since ρ ∗ X is horizontal, the same for their pull-backsFil i ( H ). (cid:3) Let E = π ∗ Ω A X /X ⊂ H be the Hodge bundle. Then i ∗ µ Ω ( A g ) k is isomorphic toSym ( E ). Like in lemma 5.1, the composition(2) E ֒ → H ∇ → Ω X ⊗ H → Ω X ⊗ ( H / E )gives an O X -module homomorphism Sym ( E ) → Ω X . This is exactly the cotangentmap of i µ : X → ( A g ) k .Note that ρ ∗ X induces an isomorphismgr ( H ) ρ ∗ → O X ⊗ gr ( M ( p )0 ) = O X ⊗ L = L . In particular, we can see U as a submodule of gr ( H ). For a closed point x of X we have k ( x ) ⊗ E = ker( F ) in N ( p ) = H ( A/k ) (notation as in section 4). So itfollows from corollary 4.4 that E is the pull-back of U by pr : Fil ( H ) → gr ( H ). Proposition 5.3.
Under the isomorphisms Ω X ∼ = Sym ( U ) (lemma 5.1) and i ∗ µ Ω ( A g ) k ∼ = Sym ( E ) the cotangent map i ∗ µ Ω A g → Ω X of i µ is just Sym ( E ) → Sym ( E / Fil ( H )) ρ ∗ X ∼ = Sym ( U ) . Proof.
By lemma 5.2 the map (2) factors as
E → E / Fil ( H ) h → Ω X ⊗ (Fil ( H ) / E ) ֒ → Ω X ⊗ ( H / E ) . Now ρ ∗ X map E / Fil ( H ) isomorphically to U and Fil ( H ) / E isomorphically to L / U .Since ρ ∗ X is horizontal and on O X ⊗ M ( p )0 the connection is just d ⊗
1, under ρ ∗ X the map h becomes the same map as the one in lemma 5.1. (cid:3) In particular the cotangent map is surjective.6.
Proof of the main theorem 1.2
To prove the main theorem, we need some facts about the EO-stratification. Fixa w in I W [ c ] g . Oort showed that the open Ekedahl-Oort stratum S w is purely ofdimension l ( w ) (see [Oo01] theorem 1.2; the formulation in terms of lengths is dueto Moonen, [Mo03]). Also it is smooth by [EG06] corollary 8.4 (this is implicit in[Oo01] where the tangent spaces are calculated).Consider the composition j µ : X ( r ( w )) ֒ → i − µ S w i µ → S w . It is proper, since the first map is a closed immersion and the second one comesfrom i µ : X → ( A g ) k . Because both sides are smooth of dimension l ( w ) and i µ issurjective on cotangent spaces, j µ is ´etale. Since j µ is quasi-compact, it is quasi-finite and, hence, finite ´etale.The disjoint union of the j µ a µ ∈ Λ g,c j µ : a µ ∈ Λ g,c X ( r ( w )) → S w . is finite ´etale and also surjective by theorem 1.1. Each j µ is defined over F p .However, σ ∈ Gal( F p / F p ) sends j µ to j σ ( µ ) . So the morphism ` j µ descends to F p .Suppose we have two isogenies ρ : ( E gk , µ k ) → ( A, λ ) and ρ ′ : ( E gk , µ ′ k ) → ( A, λ )with the same target, so that the corresponding points in ` X ( r ( w )) map to thesame point in A g . To show that ` j µ gives an isomorphism a µ ∈ Λ g,c [ X ( r ( w )) / Aut( E g , µ )] ∼ → S w , we must show that there is an automorphism γ of E g such that γ ∗ µ = µ ′ .By [Ha07] proposition 3.1.5 both ρ and ρ ′ are minimal isogenies in the terminol-ogy of [LO98] section 1.8 (here we really use that w is in I W ( c ) g and not just I W [ c ] g ).Minimal isogenies are unique up to isomorphism, so there is an isomorphism γ of E g such that ρ ′ = ρ ◦ γ and γ ∗ µ = γ ∗ ( ρ ∗ λ ) = ( ρ ′ ) ∗ λ = µ ′ as required. KEDAHL-OORT STRATA IN THE SUPERSINGULAR LOCUS 13 Number of components
In this section we show using corollary 2.8 that X ( r ( w )) is irreducible for w in I W ( c ) g . When we combine this with theorem 1.2, we get corollary 1.3.Let W c be the Weyl group of Sp c and let S c = { s , . . . , s c } ⊂ W c be the set ofreflections in simple roots, where s c = ( c, c + 1) and s i = ( i, i + 1)(2 c − i, c + 1 − i ) for i = 1 , . . . , c − . For w ∈ W c let S c ( w ) consist of those elements of S c that occur in a reducedexpression for w . This set is independent of the choice of a reduced expression.Also, w is in the subgroup W c,J generated by J ⊂ S c if and only if S c ( w ) ⊂ J (see[Bo02] chapter 4, section 8, proposition 7 and corollary 1). Lemma 7.1. If w is in I W ( c ) g , then S c ( r ( w )) = S c .Proof. Example 3.6 in [Mo01] tells us that any w ′ ∈ I W c has a reduced expressionof the form w ′ = ( s c s c − . . . s i l )( s c s c − . . . s i l − ) . . . ( s c s c − . . . s i )for some i < · · · < i l . Hence S c ( w ′ ) = { s i , s i +1 , s i +2 , . . . , s c } with i = i . So weare done if we show that s is in S c ( r ( w )).Note that r ( W [ c − g ) = W [ c − c is the subgroup generated by s , s , . . . , s c and r ( w ) is not in this subgroup, since w is in W ( c ) g = W [ c ] g − W [ c − g . So S c ( r ( w )) is notcontained in { s , s , . . . , s c } and it must contain s . (cid:3) Let w be in I W ( c ) g and suppose that W I ∞ r ( w ) is contained in W c,J for some J ⊆ S c (with I ∞ as in section 2.2). Corollary 2.8 says that X ( r ( w )) is irreducibleif this implies that J = S c . But W c,J contains r ( w ). So J contains S c ( r ( w )) andby the lemma S c ( r ( w )) = S c . References [Be85] R.B´edard,
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