Maximal compatible splitting and diagonals of Kempf varieties
aa r X i v : . [ m a t h . AG ] A ug Maximal compatible splitting and diagonalsof Kempf varieties
Niels Lauritzen and Jesper Funch ThomsenNovember 4, 2018
Abstract
Lakshmibai, Mehta and Parameswaran (LMP) introduced the notion of max-imal multiplicity vanishing in Frobenius splitting. In this paper we define thealgebraic analogue of this concept and construct a Frobenius splitting vanishingwith maximal multiplicity on the diagonal of the full flag variety. Our split-ting induces a diagonal Frobenius splitting of maximal multiplicity for a specialclass of smooth Schubert varieties first considered by Kempf. Consequences areFrobenius splitting of tangent bundles, of blow-ups along the diagonal in flagvarieties along with the LMP and Wahl conjectures in positive characteristic forthe special linear group.
In [11], Lakshmibai, Mehta and Parameswaran introduced the notion of multiplic-ities of Frobenius splittings: if X is a smooth projective algebraic variety over analgebraically closed field k of positive characteristic p , duality for the Frobenius mor-phism identifies Frobenius splittings with certain sections of the ( p − ) -th power ofthe anticanonical line bundle ω − X on X . If Y ⊆ X is a compatibly split smooth subva-riety of codimension d under the section s of ω − pX , then s vanishes with multiplicity1 ( p − ) d on Y . The splitting s is said to split Y compatibly with maximal multiplic-ity if s vanishes with multiplicity ( p − ) d on Y (cf. § Y lifts to a Frobenius splitting of the blow-up Bl Y ( X ) splitting the exceptional divisorcompatibly.Let X = G / P , where G is a semisimple linear algebraic group and P ⊂ G aparabolic subgroup. In a beautiful geometric argument Lakshmibai, Mehta andParameswaran proved that a Frobenius splitting of the blow-up Bl ∆ ( X × X ) com-patibly splitting the exceptional divisor implies Wahl’s conjecture in positive charac-teristic. They conjectured the existence of a Frobenius splitting of X × X vanishingwith maximal multiplicity on the diagonal ∆ (we refer to this as the LMP conjecture,cf. § § H ( X × X , I ∆ ⊗ p ∗ L ⊗ p ∗ L ) → H ( X , Ω X ⊗ L ⊗ L ) (1.1)is surjective for L and L ample line bundles on X . This conjecture was proved byKumar [8] for complex semisimple groups using detailed information on the decom-position of tensor products. In positive characteristic the conjecture has been provedfor Grassmannians by Mehta and Parameswaran [13], for symplectic and orthogo-nal Grassmannians by Lakshmibai, Raghavan and Sankaran [12] and by Brown andLakshmibai for minuscule G / P [3]. These positive characteristic results were provedby verifying the LMP conjecture in the specific cases. The LMP conjecture for G / P is implied by the conjecture for the full flag variety G / B (cf. Proposition 2.14 of thispaper). Lakshmibai, Mehta and Parameswaran verified their conjecture for SL n / B and n ≤ n / P by explicitly constructing aFrobenius splitting of SL n / B × SL n / B vanishing with maximal multiplicity on thediagonal for every n ≥
2. Our splitting compatibly splits X × X , where X is a Kempfvariety in SL n / B (Kempf varieties are special smooth Schubert varieties introduced2y Kempf in [7]. See also § -case that the product of theminors from the lower left hand corner in x x x x x x x x x x y x y x y x y x y x y x y x y x y ,where x x x x x x x x x , y y y y y y y y y ∈ SL is a section of the anticanonical bundle on SL / B × SL / B giving a Frobenius split-ting vanishing with maximal multiplicity on the diagonal and compatibly splitting X × X , where X is one of the five Kempf varieties in SL / B (cf. Example 5.2 in thispaper).In the last part ( §
6) of this paper, we enhance the geometric arguments in [11]and show that the Gaussian map (1.1) is surjective, provided that L = L ⊗ M and L = L ⊗ M , where L is ample and M , M globally generated line bundles on X (a projective smooth variety) and the diagonal ∆ ⊂ X × X is maximally compatiblysplit. Here we do not need the underlying field to have odd characteristic (as in[11]). This enables us to prove Wahl’s conjecture also for Kempf varieties, since theyposses unique minimal ample line bundles as Schubert varieties in G / B . We donot know, even over the complex numbers, if Wahl’s conjecture holds for smoothSchubert varieties.We have found it very difficult to prove the LMP conjecture in a general Lie the-oretic context and hope this paper will add to the inspiration for further research in3his direction. We feel nevertheless, that Frobenius splitting of tangent bundles (cf.the already known case of the cotangent bundle [9]), diagonal Frobenius splitting ofKempf varieties along with the LMP and Wahl conjecture for the special linear groupare of some interest.We thank an anonymous referee for careful reading and pointing out severalsharpenings in our manuscript. A scheme will refer to a seperated scheme of finite type over an algebraically closedfield k of characteristic p >
0. A variety will refer to a reduced scheme.
Let X be a smooth variety of dimension n , L a line bundle on X and Y ⊂ X a smoothsubvariety of codimension d . Then the blow-up B = Bl Y ( X ) is a smooth varietyand the exceptional divisor E ⊂ B a prime divisor. Let s be a section of L . Thevanishing multiplicity of s on Y is defined as v E ( π ∗ s ) (in the notation of [5, II.6]),where π : B → X is the projection. Notice that the vanishing multiplicity of s on Y can be computed locally on an open subset U ⊂ X with U ∩ Y = ∅ . Locally thisdefinition is easy to handle: if P ∈ Y , then there exists a regular system of parameters x , . . . , x n in O X , P , such that Y is defined by I = ( x , . . . , x d ) [17, VIII. Theorem 26].The vanishing multiplicity of s is the maximal m ≥ s P ∈ I m L . We recall the crucial definitions and concepts on Frobenius splitting from [2] witha few added generalizations on Frobenius splitting of O X -algebras along with thenotion of maximally compatibly split subschemes .4he absolute Frobenius morphism on a scheme X is the morphism F : X → X , whichis the identity on point spaces and the Frobenius homomorphism on the structuresheaf O X . A Frobenius splitting of X is an O X -linear map σ : F ∗ O X → O X splitting F : O X → F ∗ O X . Another way of saying this, is that σ is a group homomorphism O X → O X satisfying • σ ( f p g ) = f σ ( g ) • σ ( ) = Y ⊂ X is called compatibly split under a Frobenius splitting σ if σ ( F ∗ I Y ) ⊂ I Y .The following very useful results follow (almost) from first principles (cf. [2, Propo-sition 1.2.1 and Lemma 1.1.7]). Proposition 2.1.
Let σ be a Frobenius splitting of a scheme X and let Y and Z be compatiblysplit subschemes of X under σ .(i) The irreducible components of Y are compatibly split under σ .(ii) The scheme theoretic intersection Y ∩ Z given by I Z + I Y is compatibly split under σ .(iii) The scheme theoretic union Y ∪ Z given by I Z ∩ I Y is compatibly split under σ .(iv) If U is a dense open subscheme of a reduced scheme X, and if σ ∈ Hom O X ( F ∗ O X , O X ) restricts to a splitting of U, then σ is a splitting of X. If, in addition, Y is a reducedclosed subscheme of X such that U ∩ Y is dense in Y and compatibly split by σ | U , thenY is compatibly split by σ . .3 Frobenius splitting of O X -algebras The Frobenius homomorphism makes perfect sense for a sheaf A of O X -algebras,where X is a scheme. In analogy with the classical definition we define A to beFrobenius split if there exists a homomorphism σ : F ∗ A → A of A -modules splitting the Frobenius homomorphism A → A . Similarly we call asheaf of ideals J in A compatibly split under σ if σ ( F ∗ J ) ⊂ J .We let R ( I ) = M m ≥ I m t m = O X [ I t ]= { a + a t + · · · + a n t n | a j ∈ I j } ⊂ O X [ t ] denote the Rees algebra corresponding to a sheaf of ideals
I ⊂ O X . The sheaf of ideals I R ( I ) is called the exceptional ideal .A Frobenius splitting σ : F ∗ O X → O X can always be extended to the Frobeniussplitting σ [ t ] : F ∗ O X [ t ] → O X [ t ] given by σ [ t ]( a + a t + · · · ) : = σ ( a ) + σ ( a p ) t + σ ( a p ) t + · · · Definition 2.2.
Let σ : F ∗ O X → O X be a Frobenius splitting of X. A closed subschemeY ⊂ X is called maximally compatibly split under σ if σ ( I np + ) ⊂ I n + for every n ≥ , where I is the ideal sheaf defining Y. Notice that a maximally compatibly split scheme is compatibly split and that σ ( I np ) ⊂ I n for n ≥
0. The following result can be checked explicitly by reducingto the affine case.
Proposition 2.3.
Let Y ⊂ X be a maximally compatibly split closed subscheme under aFrobenius splitting σ : F ∗ O X → O X and I the ideal sheaf defining Y. i) Then σ [ t ] restricts to a Frobenius splitting of the Rees algebra R ( I ) compatibly split-ting the exceptional ideal I R ( I ) .(ii) If furthermore Z is a compatibly split closed subscheme under σ , then the inducedsplitting on Z splits Y ∩ Z maximally, where Y ∩ Z denotes the scheme theoretic inter-section.
The blow-up of a scheme X along a closed subscheme Y given by the ideal sheaf I is defined as Bl Y ( X ) : = Proj R ( I ) . The exceptional ideal identifies with the inverseimage ideal sheaf π − ( I ) , under the canonical morphism π : Bl Y ( X ) → X . It is aninvertible sheaf defining the exceptional divisor of π . In this setting we will prove thefollowing analogue of Proposition 2.3. Proposition 2.4.
Let Y ⊂ X be a maximally compatibly split closed subscheme under aFrobenius splitting σ : F ∗ O X → O X .(i) Then σ extends to a Frobenius splitting of the blow-up Bl Y ( X ) compatibly splitting theexceptional divisor.(ii) If the closed subscheme Z is compatibly split under σ , then the induced splitting on Zextends to a Frobenius splitting of Bl Y ∩ Z ( Z ) splitting the exceptional divisor compati-bly, where Y ∩ Z denotes the scheme theoretic intersection.
Proposition 2.4 is a consequence of the next subsection, where we give the neces-sary details for turning a Frobenius splitting of a homogeneous O X -algebra A into aFrobenius splitting of the scheme Proj A . For a commutative ring R of characteristic p > R -module M with scalarmultiplication ( r , m ) rm , we let F ∗ M denote the R -module coinciding with M asan abelian group but with scalar multiplication ( r , m ) r p m .7et S = S ⊕ S ⊕ · · · be a graded noetherian ring of characteristic p , such that F ∗ S is a finitely generated S -module. If M = M ⊕ M ⊕ · · · is a graded S -module,then we have a direct sum decomposition of F ∗ M into graded S -modules F ∗ M = F ∗ M ( ) ⊕ · · · ⊕ F ∗ M ( p − ) ,where F ∗ M ( j ) = M i ≡ j ( mod p ) M i ,for j =
0, . . . , p −
1. An element m ∈ M np + j ⊂ F ∗ M ( j ) has degree n . Lemma 2.5.
Let X = Proj ( S ) and F : X → X be the absolute Frobenius morphism on X.Then there is a canonical isomorphism ^ F ∗ M ( ) ∼ = F ∗ e M . Proof.
Let f ∈ S be a homogeneous element. Then ϕ f (cid:18) mf n (cid:19) = mf np defines a local isomorphism ( F ∗ M ( ) ) ( f ) → F ∗ ( M ( f ) ) on D + ( f ) . The isomorphisms ϕ f patch up to give the desired global isomorphism. Example 2.6.
Suppose that S = k [ x , x , . . . , x n ] , where k is a field of characteristic p. Thenthere is an isomorphism F ∗ S ( ) ∼ = S ⊕ S ( − ) ℓ ⊕ · · · ⊕ S ( − n ) ℓ n , of graded S-modules for certain ℓ , . . . , ℓ n ∈ N . Lemma 2.5 shows thatF ∗ O X ∼ = O X ⊕ O X ( − ) ℓ ⊕ · · · ⊕ O X ( − n ) ℓ n for X = P nk = Proj ( S ) . In particular, it follows that P nk is Frobenius split. Building amonomial basis for F ∗ S ( ) in degrees p , 2 p , . . . , np we also have the following recursiveformula for ℓ j : ℓ j = (cid:18) jp + nn (cid:19) − j ∑ i = (cid:18) i + nn (cid:19) ℓ j − i , where j =
0, . . . , n. The fact that F ∗ O P n ( m ) splits into a direct sum of line bundles is aclassical result due to Hartshorne (cf. [6, § .4.1 Frobenius splitting of Proj ( S ) For σ ∈ Hom S ( F ∗ S , S ) , we let σ ∈ Hom S ( F ∗ S ( ) , S ) ⊂ Hom S ( F ∗ S ( ) , S ) denote the degree 0 component of σ restricted to F ∗ S ( ) . Then σ : F ∗ S ( ) → S is ahomomorphism of graded S -modules. We may view σ ∈ Hom S ( F ∗ S , S ) satisfying σ ( S np ) ⊂ S n and σ ( S m ) = p ∤ m . Lemma 2.7.
Suppose σ ∈ Hom S ( F ∗ S , S ) , where S = S ⊕ S ⊕ · · · is a graded ring. Then σ is a Frobenius splitting if σ is a Frobenius splitting. If I ⊂ S is a homogeneous ideal, then σ splits I compatibly if σ splits I compatibly.If S is Frobenius split, then X = Proj ( S ) is Frobenius split. If I is a compatibly splithomogeneous ideal, then the closed subscheme Y = Proj ( S / I ) is compatibly split in X.Proof. Let σ : F ∗ S → S be a Frobenius splitting. Clearly σ ( ) = σ ( ) , so that σ is aFrobenius splitting if σ is. Notice that σ ( I ) ⊂ I implies σ ( I ) ⊂ I , since σ ( x ) = σ ( x ) n for x ∈ S np . Now the statements in the first part of the lemma follow. For the secondpart let I ⊂ O X be the ideal sheaf defining Y . Then I = e I and O X = e S . Now Lemma2.5 gives F ∗ I = ^ F ∗ I ( ) , F ∗ O X = ^ F ∗ S ( ) .The graded S -homomorphism σ : F ∗ S ( ) → S then gives a Frobenius splitting e σ : F ∗ O X → O X with e σ ( F ∗ I ) ⊂ I . Corollary 2.8.
Let X be a scheme, S = S ⊕ S ⊕ · · · a sheaf of graded O X -algebrasand I ⊂ S a homogeneous ideal. We will assume that F ∗ S locally is a finitely generated S -module. If S is Frobenius split compatibly with I , then Proj ( S ) is Frobenius split compatiblywith Proj ( S / I ) . roof. Let σ : F ∗ S → S be a Frobenius splitting of S with σ ( F ∗ I ) ⊂ I . The construc-tion of σ globalizes to give a Frobenius splitting σ : F ∗ S ( ) → S (with σ ( S np ) ⊂ S n and σ ( S m ) = p ∤ m ). For an affine open subset U ⊂ X , σ gives by Lemma 2.7a Frobenius splitting σ U : F ∗ O Proj ( S ( U )) → O Proj ( S ( U )) compatibly splitting the closed subscheme Proj ( S ( U ) / I ( U )) . Coming from theglobal splitting σ , these splittings patch up to give the desired global splitting ofProj ( S ) . On a non-singular variety X duality for the Frobenius morphism F : X → X isavailable for the study of Frobenius splitting: there is a functorial isomorphism F ∗ ω − pX → H om O X ( F ∗ O X , O X ) , where ω X is the canonical line bundle on X . In [14],it is shown how geometric properties of the zero divisor of a section of ω − pX trans-late into properties of compatible Frobenius splitting. To recall this powerful resultin more precise terms, we need to introduce some notation.If α ∈ Q \ N and x is a variable, we define x α : =
0. Now let x = ( x , . . . , x n ) denote a regular system of parameters (in a regular local ring) and α = ( α , . . . , α n ) ∈ Q n a rational vector. Then we define x α : = x α · · · x α n n .and x γ : = x γ · · · x γ n for γ ∈ Q . Theorem 2.9 (Mehta and Ramanathan [14]) . Let X be a non-singular variety of dimen-sion n over an algebraically closed field k of characteristic p. Then there is a canonical iso-morphism ∂ : F ∗ ω − pX → H om O X ( F ∗ O X , O X ) of O X -modules whose completion ˆ ∂ P : F ∗ ω − p ˆ R → Hom ˆ R ( F ∗ ˆ R , ˆ R ) t a closed point P ∈ X, is given by ˆ ∂ P (cid:18) x α ( dx ) p − (cid:19) ( x β ) = x ( α + β + ) / p − , where ˆ R = k [[ x , . . . , x n ]] and x , . . . , x n is a regular system of parameters in R : = O X , P with dx = dx ∧ · · · ∧ dx n . Remark 2.10.
Notice that ∂ ( s ) in Theorem 2.9 is a Frobenius splitting if and only if ∂ ( s )( ) = . This translates into a local condition on the section s. Supposes P = ( ∑ α a α x α )( dx ) p − is a local expansion of s at P ∈ X. Let supp ( s P ) denote the exponents of the monomialsoccurring with non-zero coefficient in s P . For ∂ ( s P ) to be a Frobenius splitting we musthave p − ∈ supp ( s P ) and p − + pv supp ( s P ) for v ∈ N n \ { } . If X is complete,then ∂ ( s ) is a Frobenius splitting if and only if p − ∈ supp ( s P ) for some P ∈ X [14,Proposition 6] .
An important consequence of this result is the following [2, Proposition 1.3.11].
Lemma 2.11.
Let X be a complete smooth variety. If σ is a section of ω − X such that ∂ ( σ p − ) is a Frobenius splitting of X, then the subscheme of zeros, Z ( σ ) ⊂ X, is compatibly splitunder ∂ ( σ p − ) . We have the following result analogous to [11, Proposition 2.1]. In the proof weuse the notation | α | : = α + · · · + α n for a vector α = ( α , . . . , α n ) ∈ Q n . Lemma 2.12.
Let Z be a non-singular variety of dimension n and W ⊂ Z a non-singularsubvariety of codimension d. Let s be a section of ω − pZ , such that ∂ ( s ) is a Frobenius splittingof Z. Then s vanishes with multiplicity ≤ d ( p − ) on W. The section s vanishes withmaximal multiplicity d ( p − ) on W if and only if W is maximally compatibly split under ∂ ( s ) . roof. Let z , . . . , z n be a regular system of parameters in R : = O Z , P , where P ∈ W .We may assume that the ideal I ⊂ R defining W at P is given by x : = ( z , . . . , z d ) .Define y : = ( z d + , . . . , z n ) and let t = ( ∑ a α , β x α y β ) (cid:18) dx ∧ dy (cid:19) p − (2.1)be the local expansion of s at P in the completion k [[ z , . . . , z n ]] of R . If s vanisheswith multiplicity > d ( p − ) on W , then the term x p − y p − cannot occur with non-zero coefficient in (2.1) contradicting that ∂ ( s ) is a Frobenius splitting. Therefore s vanishes with multiplicity ≤ d ( p − ) on W .Assume that t vanishes with multiplicity d ( p − ) on W . This means that | α | ≥ d ( p − ) for every α with a α , β = ∂ ( t )( I mp + ) ⊂ I m + for m ≥
0. For this we assume that w = ∑ c γ , δ x γ y δ ∈ I mp + i.e. | γ | ≥ mp + γ with c γ , δ =
0. Now we have | ( α + γ + ) / p − | ≥ d ( p − ) + mp + + dp − d = m + p .So if the vector ( α + γ + ) / p is integral, then | ( α + γ + ) / p − | ≥ m +
1. Thisshows that ∂ ( t )( w ) ∈ I m + recalling the definition of ∂ ( t ) in Theorem 2.9.Now assume that ∂ ( t )( I mp + ) ⊂ I m + for m ≥
0. We will prove that t has tovanish with multiplicity d ( p − ) on W . Suppose that | α | < d ( p − ) for some non-zero a α , β in (2.1). Let m i ∈ N be given by m i ( p − ) ≤ α i < ( m i + )( p − ) for i =
1, . . . , d and similarly m j ( p − ) ≤ β j < ( m j + )( p − ) for j = d +
1, . . . , n .Define the monomial x γ y δ ∈ I by γ = (( m + ) p − α −
1, . . . , ( m d + ) p − α d − ) and similarly δ = (( m d + + ) p − β d + −
1, . . . , ( m n + ) p − β n − ) . Then ∂ ( x α y β ( dx ∧ dy ) p − )( x γ y δ ) ∈ I m + ··· + m d \ I D ,12here D = m + · · · + m d +
1. But x γ y δ ∈ I ( D − ) p + , since d ∑ i = (( m i + ) p − α i − ) > d ∑ i = ( m i + ) p − d ( p − ) − d = d ∑ i = m i p = ( D − ) p .This contradicts our assumption and we must have | α | ≥ d ( p − ) for every non-zero a α , β in (2.1).The following remark relates to the issue of Frobenius splitting of the tangentbundle on a Frobenius split variety (cf. our remarks in the end of the introduction). Remark 2.13.
If X is smooth and X × X is Frobenius split with the diagonal ∆ X ⊂ X × Xmaximally compatibly split, then the tangent bundle T X on X is Frobenius split, since theexceptional divisor in Bl ∆ X ( X × X ) is isomorphic to P ( T X ) [2, Lemma 1.1.11]. We also need the following ([11, Proposition 2.3] and [2, Exercises 1.3.E.(13)]).
Proposition 2.14.
Let f : X → Y be a proper morphism of smooth varieties with f ∗ O X = O Y . Let Z ⊂ X be a smooth subvariety such that f is smooth at some point of Z. If X isFrobenius split and Z compatibly split with maximal multiplicity, then the induced splittingof Y has maximal multiplicity along the non-singular locus of f ( Z ) . In this section we recall a very important concept introduced by Mehta, Lakshmibaiand Parameswaran [11, Definition 1.6].
Definition 2.15.
A power series f ∈ k [[ x , . . . ., x n ]] is said to have residual normal cross-ings if either • n = and f = or • n > x | f and f / x + ( x ) ∈ k [[ x , . . . , x n ]] / ( x ) ≃ k [[ x , . . . ., x n ]] has residualnormal crossing in k [[ x , . . . , x n ]] . f ∈ k [[ x , . . . , x n ]] has residual normal crossing, it isimplicitly assumed that the variables x , . . . , x n are ordered. Example 2.16.
The polynomial f = x ( zy − x )( w − y ) ∈ k [[ x , y , z , w ]] has residual nor-mal crossing. However, if the variables are ordered x , w , z , y, then f does not have residualnormal crossing i.e. f ∈ k [[ x , w , z , y ]] does not have residual normal crossing. The minimal term in a residual normal crossing power series f ∈ k [[ x , . . . , x n ]] isprecisely x · · · x n , when the monomials are ordered according to the lexicographicalordering < given by x n < x n − < · · · < x . This implies the following result byRemark 2.10. Proposition 2.17.
Let X be a complete smooth variety, P ∈ X and x , . . . , x n a system ofparameters of O X , P . If s ∈ Γ ( X , ω − X ) , such that s P ∈ ˆ O X , P = k [[ x , . . . , x n ]] has residualnormal crossing, then ∂ ( s p − ) is a Frobenius splitting of X. Let G be a semisimple algebraic group, B a Borel subgroup of G and P ⊃ B aparabolic subgroup. A Schubert variety is defined as the closure of a B -orbit in thegeneralized flag variety G / P . The singular locus of a Schubert variety is B -stable.The map π : G → G / B is a locally trivial principal B -fibration and G × B E → G / B is a vector bundle of rank dim k E , where E is a finite dimensional representationof B . We let Γ ( E ) denote the global sections of this vector bundle i.e. Γ ( E ) = { f : G → E | f ( xb ) = b − f ( x ) , for every x ∈ G , b ∈ B } .For a one dimensional representation χ of B we get the following explicit descriptionof the global sections of the line bundle G × B χ on G / B : Γ ( χ ) = { f ∈ k [ G ] | f ( gb ) = χ ( b ) − f ( g ) , for every g ∈ G , b ∈ B } . (3.1)14or the rest of this paper we will assume that G = SL n ( k ) with B equal to theupper triangular matrices containing the diagonal matrices T = { diag ( t , . . . , t n ) | t i ∈ k , t · · · t n = } , U the unipotent upper triangular matrices and U − the unipotent lower triangularmatrices. The canonical map π : U − → π ( U − ) ⊂ G / B identifies U − with an (affine) open subset of G / B , since U − ∩ B = { e } . This opensubset is isomorphic to affine n ( n − ) /2 – space.Furthermore, B = TU and X ( B ) = X ( T ) , where X ( B ) denotes the one-dimensionalrepresentations (characters) of B . Let ǫ i ( t ) = t i for t ∈ T and ω i = ǫ + · · · + ǫ i for1 ≤ i ≤ n − T . Then X ( T ) is a free abelian group of rank n − ω , . . . , ω n − . The canonical line bundle on G / B can be identified with G × B ( ω + · · · + ω n − ) .In the next section we give an example showing the explicit nature of residualnormal crossings in constructing Frobenius splittings for G / B vanishing with differ-ent multiplicities on B / B . G / B by residual normal crossings For an n × n -matrix g ∈ G we let δ i ( g ) denote the i × i minor from the lower left handcorner i.e. the minor corresponding to the columns {
1, . . . , i } and rows { n , . . . , n − i + } of g . Similarly we let δ ′ i ( g ) denote the (principal) i × i minor from the upperleft hand corner i.e. the minor corresponding to the columns {
1, . . . , i } and rows {
1, . . . , i } .We let δ ( g ) = δ ( g ) · · · δ n − ( g ) and similarly δ ′ ( g ) = δ ′ ( g ) · · · δ ′ n − ( g ) . Then δ , δ ′ ∈ Γ ( − ω − · · · − ω n − ) and s ( g ) = δ ( g ) δ ′ ( g ) (3.2)15s a section of the anticanonical line bundle. Example 3.1.
As a global section of the anticanonical line bundle (3.2) identifies by (3.1) with the regular functionf = x ( x x − x x ) x ( x x − x x ) ∈ k [ SL ] (3.3) for g = x x x x x x x x x ∈ SL and restricts to the function x ( x x − x ) (3.4) on U − . This polynomial has residual normal crossing with respect to x , x , x provingthat f p − defines a Frobenius splitting of SL / B by Proposition 2.1 (iv) and Proposition2.17.However, (3.4) does not vanish with maximal multiplicity on the point B / B, since ( x x − x ) ( x , x , x ) . There is, however, a section with this maximal vanishing property:s = x x ( x x − x x )( x x − x x ) , Specializing, it follows that s restricts to the (residual) normal crossing polynomialx x x on U − . This idea can be generalized from SL to SL n for n > . See [11] for this and astandard monomial approach to constructing Frobenius splittings of maximal multiplicity. In [7], Kempf inspired many subsequent developments in algebraic groups prov-ing his celebrated vanishing theorem first for the general linear group. Kempf con-sidered a very natural class of (smooth) Schubert varieties as stepping stones in aninductive proof. Here we review the definition of these Schubert varieties from [7].16e let A = { ( a , . . . , a n ) ∈ N n | a ≥ a ≥ · · · ≥ a n = n − a j ≥ j for j =
1, . . . , n } .For a ∈ A we let M ( a ) denote the closed subset of G given by ( x · · · x n ... . . . ... x n · · · x nn ∈ G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ij = i > n − a j ) .This subset is B × B -stable, as it is stable with respect to row operations adding a mul-tiple of a higher index row to a lower index one and similarly adding a multiple of alower index column to a higher index one. The Schubert variety K ( a ) = π ( M ( a )) ⊂ G / B is called a Kempf variety (see also [10]). Notice that K ((
0, . . . , 0 )) = G / B and K (( n − n −
2, . . . , 1, 0 )) = B / B . The codimension of K ( a ) is a + · · · a n . In par-ticular the unique codimension one Kempf variety is given by the vanishing of thelower left hand corner i.e. a = (
1, 0, . . . , 0 ) . Kempf varieties are smooth as U − ∩ K ( a ) is a linear subspace of U − ∼ = A n ( n − ) /2 and U − ∩ K ( a ) is an open subset of K ( a ) containing B / B . Example 3.2.
The Kempf varieties corresponding to (
1, 0, 0, 0 ) , (
2, 1, 0, 0 ) and (
2, 1, 1, 0 ) inG = SL are depicted below. ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ . Informally a placement of a lower triangular zero implies zeros below and to the left of thezero.
Every Kempf variety arises as the scheme-theoretic intersection of distinguishedKempf varieties, which we call rectangular Kempf varieties . A rectangular Kempf va-17iety K ( r ) of height t ≤ n − r ∈ { a ∈ A \ { } | a i ∈ { t } for i =
1, . . . , n } .The width of a Kempf variety K ( r ) is the number of non-zero entries in r . Example 3.3.
The rectangular Kempf varieties of heights one and two for SL are depictedbelow: ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ , ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ , ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ , ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ , ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ They correspond to the defining vectors (
1, 0, 0, 0 ) , (
1, 1, 0, 0 ) , (
1, 1, 1, 0 ) , (
2, 0, 0, 0 ) , (
2, 2, 0, 0 ) and widths
1, 2, 3, 1, 2 respectively.
Lemma 3.4.
For G = SL n there are n ( n − ) /2 rectangular Kempf varieties. Every Kempfvariety is the scheme-theoretic intersection of rectangular Kempf varieties. In this section we outline the rather explicit linear algebra which is the basis of ourdiagonal Frobenius splitting of SL n / B × SL n / B .We let δ i ( M ) denote the i × i minor from the lower left hand corner in a matrix M . For two n × n matrices g = x · · · x n ... . . . ... x n · · · x nn and h = y · · · y n ... . . . ... y n · · · y nn G we define the 2 n × n matrix M ( g , h ) = x n x n · · · x nn x x · · · x n x x · · · x n x y x y · · · x n y n x y x y · · · x n y n ... ... ... ... . . . ... ... x n y n x n y n · · · x nn y nn with determinant ±
1. Notice that δ i ( M ( g , h )) is invariant under right translation by U × U for 1 ≤ i ≤ n . We are interested in the lower n × n submatrix L ( g , h ) = · · · x y · · · · · · ... ... x n − y n − x n − y n − · · · x n y n x n y n · · · of M ( g , h ) for g , h ∈ U − . Definition 4.1.
The following definitions are necessary to introduce our Frobenius splitting.(i) For ≤ i ≤ n we define L i ( g , h ) = δ i ( L ( g , h )) .(ii) When n ≤ i ≤ n + we define L i ( g , h ) to be the ( n − i ) × ( n − i ) -submatrix ofL ( g , h ) obtained by deleting the first ( i − n ) columns and the first ( i − n ) rows fromthe first i columns of L ( g , h ) .(iii) For ≤ i ≤ n − , we let V i denote the variables in the diagonal of L i , M i themonomial ideal generated by them and m i the monomial given by their product. Fori = , we define V = ∅ and M = ( ) . Example 4.2.
For G = SL and g , h ∈ U − ,L ( g , h ) = x y x y x y x y x y x y . HereL = (cid:16) x (cid:17) , L = x y x y , L = x y x y x x y x , L = x y x y x y x y x y and L = x y x y x , L = x y , L = (cid:16) (cid:17) . Notice that V = { x } V = { x , y } V = { x , y , x } V = { y , x , y } V = { y , x } V = { y } V = ∅ and that det L i ≡ m i mod M + · · · + M i − or i =
1, . . . , 7 . Notice also that the columns in L i are pairwise identical in the set ofvariables { x ij } and { y ij } . This ensures that the determinants of the L i ’s will vanish withhigh multiplicity on the diagonal in U − × U − . To prepare for showing that δ ( M ( g , h )) p − is a Frobenius splitting section of theanticanonical bundle on G / B × G / B we need the following result when restrictingto the open affine subset U − × U − . Proposition 4.3.
For ≤ i ≤ n − and g , h ∈ U − , we have(i) δ i ( M ( g , h )) = det L i ( g , h ) (4.1) (ii) det L i ( g , h ) ∈ I µ ( i ) ∆ , where I ∆ ⊆ k [ U − × U − ] is the ideal defining the diagonal, µ ( i ) = min $ i % , $ n − i %! and ⌊ x ⌋ denotes the largest integer ≤ x(iii) V ∪ · · · ∪ V n − = { x n , x n − , y n , . . . , x n , n − , y n , n − } . (iv) det L i ( g , h ) ≡ m i mod M + · · · + M i − . Proof.
For 1 ≤ i ≤ n , (4.1) is clear. When i > n and g , h ∈ U − the i × i -submatrix of M ( g , h ) in the lower left hand corner will have a lower triangular unipotent structurein the top 2 ( i − n ) rows (up to row permutation of these rows). In particular, whencomputing the determinant δ i ( M ( g , h )) one might as well start by deleting the first2 ( i − n ) columns and rows. The connection with det ( L i ( g , h )) is then clear.21he proof of (ii) follows from pairwise subtraction of columns, before computingthe determinant, using the fact that µ ( i ) is the number of identical x -columns and y -columns in L i ( g , h ) .Let ∆ r ( g , h ) = { L ( g , h ) ij | i − j = n − r } for r =
1, . . . , n { L ( g , h ) ij | j − i = n − r } for r = n +
1, . . . , 2 n − n − L ( g , h ) starting with the lower left hand corner.Then (iii) follows from the fact that V i picks up the variables in ∆ i ( g , h ) for i =
1, . . . , 2 n − L i ( g , h ) , a term different from the product of thediagonal elements always involves a variable in V ∪ · · · ∪ V i − for i =
1, . . . , 2 n − SL n / B × SL n / B The following simple lemma is the fundamental tool for showing compatible split-ting for Kempf varieties.
Lemma 5.1.
Let f , g ∈ k [ x m + , . . . , x n ] be relatively prime polynomials. Then ( x , . . . , x m , f g ) = ( x , . . . , x m , f ) ∩ ( x , . . . , x m , g ) in k [ x , . . . , x n ] . To get an initial grasp of our diagonal Frobenius splitting, the reader is encour-aged to look at the following example. 22 xample 5.2.
For G = SL and g , h ∈ U − , f : = δ ( M ( g , h )) isf = δ x x x x y x y x y ! Here f = det L ( g , h ) det L ( g , h ) det L ( g , h ) det L ( g , h )= x ( x y − x y )( y x − y − x x + x )( y − x ) and f ∈ k [ x , x , y , y , x , y ] has residual normal crossing. Furthermore f vanisheswith multiplicity three on the diagonal V ( y − x , y − x , y − x ) as µ ( ) + µ ( ) + µ ( ) + µ ( ) = + + + = (cf. Proposition 4.3 (ii) ). Therefore f p − is a Frobeniussplitting of SL / B × SL / B by Proposition 2.17 vanishing with maximal multiplicity onthe diagonal. Lemma 2.11 and Proposition 2.1 (i) show that the ideals ( x ) , ( x y − x y ) , ( y x − y − x x + x ) and ( y − x ) are compatibly split by f p − . Consequently ( x , x y ) = ( x ) + ( x y − x y ) is compatibly split by Proposition 2.1 (ii) and ( x , x ) , ( x , y ) are compatibly split by Lemma 5.1. Similarly ( x , y , x ) + ( y x − y − x x + x ) =( x , y , x , y x ) =( x , y , x , y ) ∩ ( x , y , x , x ) howing that ( x , y , x , y ) is compatibly split. Along the same lines we get that ( x , y ) + ( y x − y − x x + x ) =( x , y , x ( y − x )) =( x , y , x ) ∩ ( x , y , y − x ) and ( x , y , x ) is compatibly split showing that ( x , y , x ) + ( y − x ) = ( x , y , x , y ) is compatibly split.We have verified that X × X ⊂ SL / B × SL / B is compatibly split, where X is anyrectangular Kempf variety.
With this example in mind, we state and prove our main result.
Theorem 5.3.
For g , h ∈ U − ⊂ SL n , letf = δ ( M ( g , h )) ∈ k [ U − × U − ] ∼ = k [ V ∪ · · · ∪ V n − ] . Then(i) f is a residual normal crossing polynomial when the variables are ordered respectingV , V , . . . , V n − : if x ∈ V i and y ∈ V j are variables and i < j, then x must precede yin the ordering of the variables.(ii) f vanishes with multiplicity ≥ n ( n − ) /2 on the diagonal ∆ U − .(iii) Let ω denote the canonical line bundle on SL n / B × SL n / B. Then δ ( M ( g , h )) p − ∈ k [ G × G ] is a Frobenius splitting section of ω − p vanishing with maximal multiplicity on ∆ G / B compatibly splitting X × X, where X is a Kempf variety. roof. Proposition 4.3(iv) shows (i). Since n − ∑ i = µ ( i ) = n ( n − ) ( ii ) .Let us prove (iii). The regular function δ ( M ( g , h )) ∈ k [ G × G ] is invariant underright translation by U × U . This amounts to observing that the column operationson g and h coming from right multiplication by U × U do not change δ i ( M ( g , h )) for1 ≤ i ≤ n − ω = ω n =
0. Then δ i ( M ( g , h )) ∈ Γ ( − ω i , − ω i ) and δ i − ( M ( g , h )) ∈ Γ ( − ω i , − ω i − ) for 1 ≤ i ≤ n . This shows that δ ( M ( g , h )) ∈ k [ G × G ] is a sectionof the anticanonical line bundle G × B ( ω + · · · + ω n ) on G / B × G / B . Now (i)and (ii) show after restricting to U − × U − that δ ( M ( g , h )) p − is a Frobenius splittingvanishing with (maximal) multiplicity ( p − ) n ( n − ) /2 on ∆ G / B . We have silentlyapplied Proposition 2.1(iv), Proposition 2.17 and the fact that vanishing multiplicitycan be checked on an open subset (cf. Section 2.1)It remains to show that X × X is compatibly split, where X ⊂ SL n / B is a Kempfvariety. We can assume by Lemma 3.4 that X is a rectangular Kempf variety (theargument works for general Kempf varieties, but is slightly less clear).Suppose that X is of height r and width s . Then we must show that the monomialideal generated by the variables V X = n x ij , y ij (cid:12)(cid:12)(cid:12) n − r < i ≤ n , 1 ≤ j ≤ s o is compatibly split under f p − . We will prove that the monomial ideal generated bythe variables V X ∩ ( V ∪ · · · ∪ V m ) is compatibly split by induction on m . Since V X ∩ V = { x n } and x p − n is the firstfactor in f p − , compatible splitting holds for m =
1. Suppose now that the monomialideal generated by W : = V X ∩ ( V ∪ · · · ∪ V m ) ( V X
25s compatibly split. Then ( W , δ m + ( M ( g , h ))) = ( W , D ) , where D is a monomial ofthe form dm , where m is the product of the variables V X ∩ V m + and d is a monomial.This is a consequence of the formuladet A B C = det ( A ) det ( C ) ,where A , B and C are compatible block matrices.It follows by Lemma 5.1 that the ideal generated by V X ∩ ( V ∪ · · · ∪ V m ∪ V m + ) is compatibly split. Since V X ⊂ V ∪ · · · ∪ V N for N ≥ r + s − Let Z denote a smooth projective variety. The sheaf of differentials on Z is definedby Ω Z = I ∆ / I ∆ ,where I ∆ denotes the sheaf of ideals defining the diagonal within Z × Z . In thissetup we may consider the quotient morphism I ∆ → I ∆ / I ∆ = Ω Z .Fixing line bundles L and L on Z we obtain an induced restriction morphismH (cid:0) Z × Z , I ∆ ⊗ ( L ⊠ L ) (cid:1) → H (cid:0) Z , Ω Z ⊗ L ⊗ L (cid:1) , (6.1)where L ⊠ L : = p ∗ L ⊗ p ∗ L and p , p : X × X → X are the projections on thefirst and second factors. In case Z is a flag variety and L and L are ample it hasbeen conjectured by J. Wahl [16] that the map (6.1) is surjective. In characteristiczero this is now a theorem proved by S. Kumar [8]. In positive characteristic onlysporadic cases are known as outlined in the introduction.26he aim of the last part of this paper is to obtain the following related and seem-ingly stronger result Theorem 6.1.
Assume that the blow-up Bl ∆ ( Z × Z ) admits a Frobenius splitting which iscompatible with E Z . Let L denote a very ample line bundle on Z and let M and M denoteglobally generated line bundles on Z. Let j > denote an integer. Then the natural mapfrom H (cid:0) Z × Z , I j ∆ ⊗ (( L j ⊗ M ) ⊠ ( L j ⊗ M )) (cid:1) , to H (cid:0) Z , S j Ω Z ⊗ L j ⊗ M ⊗ M (cid:1) , induced by the identification I j ∆ / I j + ∆ = S j Ω Z , is surjective. Notice that when Z admits a minimal ample line bundle L ; i.e. an ample linebundle on Z such that every line bundle of the form M ⊗ L − , with M ample, isglobally generated, then Wahl’s conjecture is a consequence of Theorem 6.1. Schu-bert varieties are examples of varieties admitting minimal ample line bundle. Whenthe Schubert variety is a flag variety this is well known; e.g. in the notation of theprevious sections the minimal ample line bundle on G / B is defined by the weight − ρ = − ( ω + · · · ω n − ) .For a general Schubert variety the claim follows by the fact that any ample line bun-dle on a Schubert variety may be lifted to an ample line bundle on the flag varietycontaining the Schubert variety [1, Prop.2.2.8]With these remarks in place the following corollary now follows from Proposition2.4 and Theorem 5.3 Corollary 6.2.
The conjecture of Wahl on the surjectivity of the map (6.1) is satisfied forKempf varieties Z and ample line bundles L and L . The rest of this paper is concerned with the proof of Theorem 6.1. The proofis highly inspired by the discussion in Section 3 of [11]. As a side result we obtain27ertain cohomological vanishing results for smooth varieties admitting various typesof Frobenius splitting (cf. Prop. 6.8 and Prop. 6.9); e.g. for Kempf varieties. We startby collecting a number of well known results about blow-ups along diagonals. P N × P N along the diagonal Consider the variety P N = P ( V ) with homogeneous coordinates X , . . . , X N . Thehomogeneous ideal defining the diagonal within the product P N × P N is generatedby the elements X i , j = X i ⊗ X j − X j ⊗ X i , 0 ≤ i < j ≤ N ;all of the same multidegree (
1, 1 ) . Applying the Rees algebra description of the blow-up this leads to an embedding of Bl ∆ ( P N × P N ) as a closed subvariety of the product P N × P N × P ( N + ) − . (6.2)Alternatively one could also obtain this embedding by considering Bl ∆ ( P N × P N ) asthe graph of the rational morphism P N × P N P ( N + ) − , (6.3)defined by the generators X i , j of the diagonal ideal (cf. [4, Ex. 7.18]). The latterdescription makes it evident that Bl ∆ ( P N × P N ) is contained within P N × P N × Gr ( V ) , (6.4)where Gr ( V ) denotes the Grassmannian of planes in V with the Pl ¨ucker embeddingin P ( N + ) − . This also explains the following setwise description of the blow-upBl ∆ ( P N × P N ) = { ( l , l , b ) ∈ P N × P N × Gr ( V ) : l , l ⊂ b } . (6.5)In this setting the exceptional divisor E is determined as the set of points E = { ( l , b ) ∈ P N × Gr ( V ) : l ⊂ b } ⊂ P N × Gr ( V ) ,28here we consider P N as being diagonally embedded in P N × P N .The projection on the first two coordinates π : Bl ∆ ( P N × P N ) → P N × P N ,is the blow-up map. Restricting π to the exceptional divisor E defines the map π E : E → P N ,coinciding with the projectivized tangent bundle on P N . Finally we let τ : Bl ∆ ( P N × P N ) → Gr ( V ) ,denote the map induced by projection on the third coordinate, while τ E denotes itsrestriction to E . Lemma 6.3.
Let O V ( ) (resp. O ( ) ) denote the ample generator of the Picard group of Gr ( V ) (resp. P N ). Then as locally free sheaves τ ∗ ( O V ( )) ≃ O ( − E ) ⊗ π ∗ (cid:0) O ( ) ⊠ O ( ) (cid:1) . (6.6) Proof.
This follows from a local calculation but can also be obtained in the followingmore abstract way : assume, first of all, that N ≥ (cid:0) Bl ∆ ( P N × P N ) (cid:1) ≃ Pic ( P N × P N ) ⊕ Z .In particular, we may find unique integers c , c and c such that τ ∗ ( O V ( )) ≃ O ( − c E ) ⊗ π ∗ (cid:0) O ( c ) ⊠ O ( c ) (cid:1) .Restricting to the open subset Bl ∆ ( P N × P N ) \ E ≃ ( P N × P N ) \ ∆ we determine ( c , c ) as the bidegree of the rational morphism (6.3). In particular, we find that c = c =
1. To find c we fix some line P inside P N and consider P × P asa closed subset of Bl ∆ ( P N × P N ) by identifying it with its strict transform. As the29ational morphism (6.3) is constant on an open dense subset of P × P , the sameis true for the restriction of τ to P × P . In particular, the restriction of the sheaf τ ∗ ( O V ( )) to P × P is trivial. Now as the sheaf of ideals of the diagonal in P × P equals O ( − ) ⊠ O ( − ) we conclude − c + c = − c + c = c =
1. This ends the proof in case N ≥
2. For N = τ is constant andthe claimed isomorphism (6.6) is trivial.We claim that τ is a P × P -bundle. More precisely, let b ∈ Gr ( V ) denote anyplane in V and let P denote the stabilizer of b in the group SL ( V ) . Then Gr ( V ) isisomorphic to the quotient SL ( V ) / P while Bl ∆ ( P N × P N ) may be described asBl ∆ ( P N × P N ) = SL ( V ) × P ( P ( b ) × P ( b )) , (6.7)where P acts by the diagonal action on P ( b ) × P ( b ) . Thus τ is just the natural map τ : SL ( V ) × P ( P ( b ) × P ( b )) → SL ( V ) / P .In this notation we may describe the exceptional divisor as E = SL ( V ) × P P ( b ) ,where we think of E as a subset of (6.7) by embedding P ( b ) diagonally in the prod-uct P ( b ) × P ( b ) . It follows that the restriction τ E : SL ( V ) × P P ( b ) → SL ( V ) / P ,is a P -bundle over Gr ( V ) . Returning to the general case of a smooth projective subvariety Z in P ( V ) we mayconsider the blow-up Bl ∆ ( Z × Z ) as the strict transform of Z × Z in Bl ∆ ( P N × P N ) .In particular, we obtain a closed embeddingBl ∆ ( Z × Z ) ⊂ Z × Z × Gr ( V ) . (6.8)30he exceptional divisor E Z is thus embedded as E Z ⊂ Z × Gr ( V ) . (6.9)In this setting the blow-up morphism π Z : Bl ∆ ( Z × Z ) → Z × Z ,coincides with the projection on the first two coordinates, while its restriction π E Z : E Z → Z coincides with the projectivized tangent bundle on Z . Thus if we consider Gr ( V ) asthe set of lines in P N = P ( V ) , then E Z consists of the set of pairs ( l , b ) ∈ Z × Gr ( V ) such that b is a line tangent to the point l in Z .The projection on the third coordinate is denoted by τ Z : Bl ∆ ( Z × Z ) → Gr ( V ) ,while its restriction to E Z is denoted by τ E Z . τ E Z By the discussion above the fibre of τ E Z over a line b in P ( V ) consists of the set ofpoints l in Z such that b is tangent to Z at l . Thus the following result is now easy toprove Lemma 6.4.
If every nonempty fibre of τ E Z has dimension then Z coincides with P ( V ′ ) for some vector subspace V ′ of V.Proof. The assumptions means that every tangent line of Z is contained in Z . Inparticular, Z contains all of its tangent planes. But any tangent plane of Z is of thesame dimension as Z and consequently Z , and all of its tangent planes, must coincide(this simple argument was suggested by the referee).31 .4 Technical results For technical reasons we will need the following setup : let Z denote a projectivevariety and let f : Z × Z → Z ,denote a morphism. The projective morphism τ f = ( τ Z , f ◦ π Z ) : Bl ∆ ( Z × Z ) → Gr ( V ) × Z , (6.10)has a Stein factorization for which we use the notationBl ∆ ( Z × Z ) µ f −→ B f → Gr ( V ) × Z . (6.11)The restriction of τ f to E Z is denoted by τ E , f : E Z → Gr ( V ) × Z .More important is the map µ E , f : E Z → S f : = µ f (cid:0) E Z (cid:1) ,induced by the restriction of µ f . We claim Lemma 6.5.
The derived direct images R i ( µ E , f ) ∗ O E Z are zero when i > .Proof. As the second map B f → Gr ( V ) × Z of the Stein factorization (6.11) is afinite map it suffices to prove that R i ( τ E , f ) ∗ O E Z = i >
0. Consider an openaffine subset U of Gr ( V ) such that τ E : E → Gr ( V ) ,is a trivial P -bundle over U . Then we may consider τ − E Z ( U ) as a closed subvarietyof P × U . Embedding τ − E Z ( U ) by the graph of f ◦ π Z defines a closed embedding ι : τ − E Z ( U ) ֒ → Y : = P × U × Z .32he map τ U : τ − E Z ( U ) = τ − E , f ( U × Z ) → U × Z ,induced by the projection p of Y on the second and third coordinate, coincideswith the restriction of τ E , f to the inverse image of U × Z . It thus suffices to provethat R i ( τ U ) ∗ O τ − EZ ( U ) = i >
0. Now apply the identity R i ( τ U ) ∗ O τ − EZ ( U ) = R i ( p ) ∗ ( ι ∗ O τ − EZ ( U ) ) ,and the long exact sequence · · · → R ( p ) ∗ I → R ( p ) ∗ O Y = → R ( p ) ∗ ( ι ∗ O τ − EZ ( U ) ) → → · · · associated to the trivial P -bundle p , and the short exacts sequence0 → I → O Y → ι ∗ O τ − EZ ( U ) → τ − E Z ( U ) as a closed subvariety in Y . Lemma 6.6.
Assume that Z does not coincide with a closed subvariety of P ( V ) of the form P ( V ′ ) , for some vector subspace V ′ of V. Then µ E , f is birationalProof. Let Y ⊂ Gr ( V ) × Z denote the image of τ f . We claim that there exists apoint y ∈ Y such that the fibre τ − f ( y ) is nonempty and finite. To see this we useLemma 6.4 to obtain a point b ∈ Gr ( V ) such that the fibre τ − E Z ( b ) is nonempty andfinite. Assume, for a moment, that τ − Z ( b ) is infinite : then π Z ( τ − Z ( b )) is an infiniteclosed subvariety of P ( b ) × P ( b ) = P × P and thus P ( b ) is contained in Z . As aconsequence ( P ( b ) × P ( b )) \ ∆ ( P ( b )) × { b } ,is a subset of Bl ∆ ( Z × Z ) and thus, by taking the closure, we find that P ( b ) × { b } ⊂ E Z .But then P ( b ) × { b } is a subset of the finite set τ − E Z ( b ) , which is a contradiction. Itfollows that τ − Z ( b ) is finite and nonempty. Choose an element y in τ f ( τ − Z ( b )) . As asubset of τ − Z ( b ) the set τ − f ( y ) is then finite.33et now Y denote the nonempty set of points in Y where the associated fibre of τ f is finite. Then Y is an open subset of Y ([15, Cor. I.8.3]). It follows that µ f inducesan isomorphism between B f and Bl ∆ ( Z × Z ) over Y µ f ,0 : τ f − ( Y ) ≃ −→ µ f ( τ f − ( Y )) .It thus suffices to prove that the intersection of E Z and τ f − ( Y ) is nonempty. Butthis is clear as τ − E Z ( b ) is a nonempty subset of τ f − ( Y ) .From now on we will assume that f : Z × Z → Z is the product ( f , f ) of twomorphisms f i : Z → Z i , i =
1, 2.We can then prove.
Lemma 6.7.
The fibres of µ E , f are connected.Proof. Let z denote an element in S f and let ( b , x ) denote the image of z in under thesecond morphism B f → Gr ( V ) × Z (6.12)of the Stein factorization (6.11). As µ − E , f ( z ) ⊂ µ − f ( z ) and µ − f ( z ) is connected wemay assume that µ − f ( z ) is infinite. Consequently the intersection Z ∩ P ( b ) mustalso be infinite and thus equal to P ( b ) . It follows that P ( b ) × P ( b ) × { b } ⊂ Bl ∆ ( Z × Z ) .This leads to the inclusion µ − f ( z ) ⊂ τ − f ( b , x ) = ( f − ( x ) ∩ P ( b )) × ( f − ( x ) ∩ P ( b )) × { b } , (6.13)where we have used the notation x = ( x , x ) ∈ Z , with x i ∈ Z i for i =
1, 2. As µ f and τ f only differ by a finite morphism it follows that τ − f ( b , x ) is a disjoint union of µ − f ( z ) with another closed (possibly empty) subset of τ − f ( b , x ) . At the same time µ − f ( z ) is connected and thus (6.13) implies that µ − f ( z ) is of one of the forms P ( b ) × P ( b ) × { b } , { l } × P ( b ) × { b } , P ( b ) × { l } × { b } ,34or some line l contained in b . We conclude that µ − E , f ( z ) is either equal to P ( b ) × { b } ⊂ E Z ,or of the form { l } × { b } ⊂ E Z .In both cases µ − E , f ( z ) is connected. We continue the notation of Section 6.4. The proof of Theorem 6.1 is built from thefollowing two results.
Proposition 6.8.
Assume that E Z admits a Frobenius splitting. Let L (resp. M ) denote avery ample (resp. globally generated) line bundle on Z and let j > denote an integer. Then H i (cid:0) Z , S j Ω Z ⊗ L j ⊗ M (cid:1) = for i >
0. (6.14)
Proof.
We assume that the embedding Z ⊂ P N is defined by the very ample linebundle L , and that the map f , of Section 6.4, is the composition f : Z × Z → Z → Z : = P (cid:0) H ( M ) ∨ (cid:1) ,where the first map is projection on the first coordinate while the second map is theprojective morphism defined by the globally generated line bundle M . Let O M ( ) denote the ample generator of the Picard group of P (cid:0) H ( M ) ∨ (cid:1) . By (6.6) the pull-back of O V ( j ) ⊠ O M ( ) by τ E , f : E Z → Gr ( V ) × Z ,is then the line bundle L j = O ( − jE Z ) | E Z ⊗ π ∗ E Z (cid:0) L j ⊗ M (cid:1) ,35n E Z . Consider the Stein factorization E Z ˜ µ E , f −−→ ˜ S f → S f ,of µ E , f . By Lemma 6.5 and the definition of the Stein factorization, the map ˜ µ E , f is arational morphism, i.e. R i ( ˜ µ E , f ) ∗ O E Z = O ˜ S f if i = i > L j of O V ( j ) ⊠ O M ( ) by the finite morphism˜ S f → S f → B f → Gr ( V ) × Z , (6.15)is an ample line bundle on ˜ S f whose pull back by ˜ µ E , f coincides with L j . As ˜ S f isFrobenius split (by push-down of the Frobenius splitting on E Z [2, Lemma 1.1.8] ) itfollows that the higher cohomology of ˜ L j , and hence of L j , is trivial [2, Thm.1.2.8].Notice finally that by [5, Ex. III.8.4] the cohomology of L j and the direct image ( π E Z ) ∗ L j = S j Ω Z ⊗ L j ⊗ M ,coincide. Here we use that the identification ( π E Z ) ∗ O E Z ( − jE Z ) = S j Ω Z . This endsthe proof. Proposition 6.9.
Assume that the blow-up Bl ∆ ( Z × Z ) admits a Frobenius splitting whichis compatible with E Z . Let L denote a very ample line bundle on Z and let M and M denote globally generated line bundles on Z. Let j > denote an integer. Then H i (cid:0) Z × Z , I j + ∆ ⊗ (( L j ⊗ M ) ⊠ ( L j ⊗ M )) (cid:1) = for i >
0, (6.16) where I ∆ denotes the sheaf of ideals defining the diagonal in Z × Z.Proof.
We will assume that L is the line bundle defining the embedding Z ⊂ P N ,and that f i : Z → Z i : = P (cid:0) H ( M i ) ∨ (cid:1) , i =
1, 2,36re the maps defined by the globally generated line bundles M and M . Let L j denote the line bundle L j = O ( − jE Z ) ⊗ π ∗ Z (cid:0) ( L j ⊗ M ) ⊠ ( L j ⊗ M ) (cid:1) ,on Bl ∆ ( Z × Z ) . We claim that the restriction morphismH (cid:0) Bl ∆ ( Z × Z ) , L j (cid:1) → H (cid:0) E Z , L j (cid:1) , (6.17)is surjective. To see this let O i ( ) , for i =
1, 2, denote the ample generator of thePicard group of Z i . Consider the ample line bundle˜ M j = O V ( j ) ⊠ O ( ) ⊠ O ( ) ,on Gr ( V ) × Z and let ˜ L j denote the ample pull back of ˜ M j to B f by the finitemorphism in (6.11). Then by (6.6) the line bundle L j is the pull back of ˜ L j by µ f . Inparticular, as µ f is part of a Stein factorization we obtain an identificationH (cid:0) Bl ∆ ( Z × Z ) , L j (cid:1) = H (cid:0) B f , ˜ L j (cid:1) .Assume, for a moment, that Z is not of the form P ( V ′ ) as in the assumptions ofLemma 6.6. Then µ E , f is a birational morphism with connected fibres by Lemma 6.6and Lemma 6.7. Moreover, by push-forward of the Frobenius splitting on Bl ∆ ( Z × Z ) we know that B f is Frobenius split compatibly with S f [2, Lemma 1.1.8]. Thus by [2,Ex. 1.2.E(3)] the variety S f is normal, and henceH (cid:0) E Z , L j (cid:1) = H (cid:0) S f , ˜ L j (cid:1) , (6.18)by Zariski’s main theorem. Thus to prove (6.17) it suffices to prove that the restrictionmap H (cid:0) B f , ˜ L j (cid:1) → H (cid:0) S f , ˜ L j (cid:1) ,is surjective. As S f is compatibly Frobenius split in B f and as ˜ L j is ample the latterfollows by general theory of Frobenius splitting [2, Thm.1.2.8] . Consider next thecase Z = P ( V ′ ) . If either M or M are ample then µ E , f is easily seen to be an37somorphism and we may argue as above. This leaves us with the case M = M = O Z . Then Z is just a 1-point space and thus S f = Gr ( V ′ ) while µ E , f coincides with τ E Z which is a P -bundle over Gr ( V ′ ) . So again we obtain the identification (6.18).This proves the claim about the surjectivity of (6.17).As the blow-up map satisfies R i ( π Z ) ∗ O ( − jE Z ) = ( I ∆ ) j if i = i > i (cid:0) Bl ∆ ( Z × Z ) , L j ⊗ O ( − E Z ) (cid:1) =
0, for i > → O ( − E Z ) → O Bl ∆ ( Z × Z ) → O E Z → i (cid:0) Bl ∆ ( Z × Z ) , L j (cid:1) =
0, for i > E Z is compatibly Frobenius split divisor in Bl ∆ ( Z × Z ) we have by [2, Lemma1.4.11] an inclusion (of abelian groups)H i (cid:0) Bl ∆ ( Z × Z ) , L j (cid:1) ⊂ H i (cid:0) Bl ∆ ( Z × Z ) , L pj ⊗ O (( p − ) E Z ) (cid:1) .Thus, as Bl ∆ ( Z × Z ) is Frobenius split, it suffices to show that the line bundle L pj ⊗ O (( p − ) E Z ) = O (( p ( − j ) − ) E Z ) ⊗ π ∗ Z (cid:0) ( L pj ⊗ M p ) ⊠ ( L pj ⊗ M p ) (cid:1) ,is ample on Bl ∆ ( Z × Z ) . But the latter line bundle is by (6.6) isomorphic to the re-striction to Bl ∆ ( Z × Z ) of the line bundle (cid:0) M p ⊗ L ( p − ) (cid:1) ⊠ (cid:0) M p ⊗ L ( p − ) (cid:1) ⊠ O V (cid:0) p ( j − ) + (cid:1) , (6.19)on Z × Z × Gr ( V ) . Here O V (cid:0) (cid:1) denotes the ample generator of the Picard groupof Gr ( V ) . As the line bundle (6.19) is ample this ends the proof.Theorem 6.1 is now a direct consequence of Proposition 6.9.38 eferences [1] Michel Brion, Lectures on the geometry of flag varieties , Topics in cohomologicalstudies of algebraic varieties , 33–85, Trend. Math., Birkh¨auser, 2005. MR 2143072(2006f:14058)[2] Michel Brion and Shrawan Kumar,
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