LINE BUNDLES ON BOTT-SAMELSON VARIETIES

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1 LINE BUNDLES ON BOTT-SAMELSON VARIETIES NIELS LAURITZEN, JESPER FUNCH THOMSEN 1. Introduction Let G be a connected semisimple, simply connected linear algebraic group over an algebraically closed field k and B be a Borel subgroup in G. Ifw =(P 1,...,P n ) is a sequence of minimal parabolic subgroups containing B, we may form the quotient = P w /B n,wherep w = P 1 P n and B n acts on P w from the right via (p 1,...,p n )(b 1,...,b n )=(p 1 b 1,b 1 1 p 2b 2,b 1 2 p 3b 3,...,b 1 n 1 p nb n ). The quotient is inductively a sequence of P 1 -bundles with natural sections starting with the P 1 -bundle P 1 /B (over a point). The product map P w G induces a proper morphism ϕ w : G/B whose image is a Schubert variety in G/B. For reduced sequences w the morphism ϕ w is birational and equal to the celebrated Demazure desingularization of the Schubert variety ϕ w ( ). In general we call the Bott-Samelson variety associated with w.the construction of originates in the papers [1][2][3] of Bott-Samelson, Demazure and Hansen. See also the master s thesis (speciale) [4] by Hansen. We characterize the globally generated, ample and very ample line bundles on. The generators of the ample cone are naturally defined O(1)-bundles for successive P 1 -bundles. They form a basis of Pic ( ). Proving that they account for all ample line bundles originally lead us to some quite involved computer calculations in the Chow ring of. It later turned out that the key point is Lemma 2.1. Using Frobenius splitting [7] and our description of globally generated line bundles we prove the vanishing theorem H i (, L( D)) = 0,i>0 where L is any globally generated line bundle on and D a subdivisor of the boundary of corresponding to a reduced subexpression of w (cf. Theorem 7.4 for a precise description). A special case (D = 0) of this vanishing theorem has been proved in [6] (with no details on the involved line bundles). The vanishing theorem above is a generalization of the crucial vanishing theorem for pull backs of globally generated line bundles on G/B in Kumar s proof [5] of the Borel-Bott-Weil theorem in the Kac-Moody case. Kumar relied heavily on the Grauert-Riemenschneider vanishing theorem available only in characteristic zero. Our approach shows that one may give 1

2 2 NIELS LAURITZEN, JESPER FUNCH THOMSEN a characteristic free generalization using only the theory of Frobenius splitting. 2. Notation Fix a connected semisimple algebraic group G over an algebraically closed field k and let B be a Borel subgroup in G containing the maximal torus T. A simple reflection s (wrt. B) in the Weyl group W = N G (T )/T determines the minimal parabolic subgroup P s = BsB B B. Weletw denote a sequence (s 1,s 2,,s n )ofsimplereflections, w[j] the truncated sequence (s 1,...,s n j ), P w = P 1 P n and = P w /B n the associated Bott-Samelson variety, where B n acts from the right on P w as (p 1,...,p n )(b 1,...,b n )=(p 1 b 1,b 1 1 p 2b 2,b 1 2 p 3b 3,...,b 1 n 1 p nb n ). By convention [n] will denote a 1-point space. We may also write as P 1 B P 2 B B P n /B where B Y is the quotient Y/B with B acting as (x, y)b = (xb, b 1 y). This shows that comes as the sequence [1] [n 2] = P 1 B P 2 /B [n 1] = P 1 /B of successive P 1 -fibrations. In general we let π w[j] denote the natural morphism [j] in the sequence of P 1 -bundles above and use π w to denote the morphism π w[1] : [1]. Let w(j) = (s 1,, ŝ j,,s n ). The natural embedding P w(j) P w induces a closed embedding σ w,j : (j) which makes (j) into a divisor in. The divisor = (1) (n) in has normal crossing. When A {1, 2,...,n} we define (A) = j A (j), and let σ w,a :,A denote the closed embedding given by σ w,j for j A. Finally we let π : G/B denote the natural proper morphism coming from the product map P w G Induced bundles on. We let L w (V ) denote the locally free sheaf of sections of the associated vector bundle P w Bn V on, where V is a finite dimensional B n -representation. We view a B- representation V as a B n -representation by letting B n act on v V as (b 1,b 2,,b n ).v = b n.v. With this convention we get for a B-character λ (B) thatl w (λ) =L w (0,, 0,λ).

3 LINE BUNDLES ON BOTT-SAMELSON VARIETIES Induced bundles on G/B. Let V be a finite dimensional B representation. We let L G/B (V ) denote the locally free sheaf on G/B associated with V. This is the sheaf of sections of the vector bundle G B V G/B, (g, v) gb. WhenV has dimension 1, associated to a B-character λ, thesheafl G/B (λ) =L G/B (V ) is a line bundle. This gives a bijection between B-characters and line bundles on G/B. Itis well known, that L G/B (λ) is globally generated (resp. ample) exactly when λ is dominant (resp. regular) wrt. the Borel subgroup opposite to B (or equivalently λ, α 0(resp. λ, α > 0) for all simple roots α S). The pull back of L G/B (V )to under the morphism π : G/B is given by the formula π (L G/B (V )) L w (V ). Lemma 2.1. Let 1 i n and 0 j n 1 be integers and let λ denote a B-character. Then { π σw,i π w[j] L w[j](λ) w(i)[j 1] L w(i)[j 1] (λ) if i>n j, πw(i)[j] L w(i)[j](λ) if i n j. Proof. When i>n j the claim follows by the commutativity of the diagram (i) π w(i)[j 1] (i)[j 1] σ w,i π w[j] Z π w [j] G/B. Similarly, the case i n j follows from the commutative diagram (i) π w(i)[j] (i)[j] [j](i) σ w,i σ w[j],i π w[j] [j] π G/B. 3. Line bundles on The Picard group Pic( ) is a free abelian group of rank n. This follows easily by induction using the P 1 -fibration π w : [1]. In fact we have a decomposition Pic( )=Pic([1] ) ZL, where L is any line bundle on with degree one along the fibers of π w The O(1)-basis. Recall our notation w =(s 1,...,s n )forasequence of simple reflections defining. Suppose that s n is a reflection in the simple root α. Thenwelet O w (1) = L w (ω α ), where ω α denotes the fundamental dominant weight corresponding to α. Then O w (1) has degree one along the fibres of π w. It is globally

4 4 NIELS LAURITZEN, JESPER FUNCH THOMSEN generated since it is the pull back of the globally generated line bundle L G/B (ω α )ong/b. This gives inductively a basis for Pic ( )which we call the O(1)-basis. Thus Pic ( )=ZO w (1) ZO w[1] (1) ZO w[n 1] (1), wherewewriteo w[j] (1) instead of the pull back πw[j] O w[j](1). The line bundle m 1 O w (1) + + m n O w[n 1] (1) Pic ( ) is denoted O w (m 1,...,m n ), where m 1,...,m n Z. Theorem 3.1. AlinebundleL = O w (m 1,...,m n ) is very ample on if and only if m 1,...,m n > 0. Proof If n = 1 it is well known that L is ample and very ample if and only if m 1 > 0. We proceed using induction on n. In general the P 1 -bundle π w : [1] may by identified with the projective bundle P(V ) [1],whereV is the rank two bundle L w[1] (H 0 (P α /B, ω α )) on [1] and O P(V ) (1) = O w (1). Since V is the pull back of a globally generated vector bundle on G/B it is globally generated. This implies that we have a commutative diagram (for some N N) ϕ P(V ) P N [1], [1] where ϕ is a closed embedding. Since ϕ (O P N (1) O Zw ) = O P(V ) (1) it follows that O P(V ) (n) πw L is very ample if n>0andl is very ample on [1]. Hence by induction, L is very ample if m 1,...,m n > 0. Suppose on the other hand that L is very ample. By induction we get that m 2,...,m n > 0, since σw,1 L O w(1)(m 2,...,m n ) by Lemma 2.1. Furthermore, Lemma 2.1 also gives σw,2l O w(2) (m 1,m 3,...,m n ) πw(2)[n 2](L w(2)[n 2] (m 2 ω β )) where β is the simple root corresponding to s 2. Suppose s 1 is a reflection in the simple root α. Using that w(2)[n 2] = (s 1 ) and hence (2)[n 2] = P α /B P 1,weidentifyL w(2)[n 2] (ω β )witho P 1( ω β,α ). When s 1 s 2 this means that the line bundle πw(2)[n 2] (L w(2)[n 2](ω β )) is trivial, and m 1 > 0 by induction. If s 1 = s 2 and s 1 is a reflection in the simple root α, then = Pα /B (1) P 1 (1). Under this isomorphism L identifies with O P 1(m 1 ) L w(1) (m 2,...,m n ). This proves that m 1 > 0. We obtain the following two corollaries as immediate consequences.

5 LINE BUNDLES ON BOTT-SAMELSON VARIETIES 5 Corollary 3.2. Ample line bundles on are very ample. Corollary 3.3. AlinebundleL = O w (m 1,...,m n ) is globally generated on if and only if m 1,...,m n 0. Proof If m 1,...,m n 0thenL is globally generated being a tensor product of globally generated line bundles. Assume there is a globally generated line bundle L = O w (m 1,...,m n )withsomem i < 0. Since ample tensor globally generated is ample this contradicts Theorem The divisor basis. The P 1 -bundle π w : [1] comes with a natural section σ w,n : [1] = (n). So the line bundle O((n) ) defined by the divisor (n) has degree one along the fibres of π w. Inductively this shows that the line bundles O Zw ((j) ),j =1, 2,...,n, form a basis of the Picard group of. We call this basis the Z-basis Effective line bundles. The line bundles O((j) ) are effective. They do not necessarily generate the cone of effective line bundles unless the expression w is reduced as shown by the following example. Example 3.4. Consider w = (s α,s α ), where α is a simple reflection. Then the corresponding Bott-Samelson variety is isomorphic to P α /B P α /B by the map (p 1 : p 2 ) (p 1 B,p 1 p 2 B). Under this isomorphism the 2 divisors (1) and (2) correspond to the diagonal Pα/B and {eb} P α /B. From this we conclude that the effective line bundle corresponding to the divisor P α /B {eb} is not contained in the cone generated by O((1) ) and O((2) ). Proposition 3.5. Let w =(s 1,s 2,...,s n ) be a reduced sequence and L = O( n m j(j) ) alinebundleon.thenlis effective if and only if m j 0 for all j. Proof. If m j 0thenLis clearly effective. We are hence left with proving that m j 0ifL is effective. So assume that L is effective. Clearly L is B linearized as and all (j) are compatible B-spaces. Hence the global sections L( ) is a non-zero finite dimensional (as is projective) B-representation. This allows us to pick a B-semiinvariant (i.e. invariant up to constants) global section s of L. The zero-scheme Z(s) ofs is then a B-invariant divisor of. As w is reduced the morphism ψ : (s 1 s n ), is known to be birational. In fact, it is known (essentially by the Bruhat decomposition) that ψ is an isomorphism above the dense Bruhat cell C(s 1 s n )of(s 1 s n ). This shows that \ n (j) is a dense B-orbit. Hence, Z(s) n (j) and L O(Z(s)) O( m j(j) ),

6 6 NIELS LAURITZEN, JESPER FUNCH THOMSEN where m j are nonnegative integers. As O((j) ), j =1, 2,...,n is a basis for the Picard group of, we conclude m j = m j Frobenius splitting Let π : Spec (k) be a scheme defined over an algebraically closed field k of positive characteristic p. Theabsolute Frobenius morphism on is the identity on point spaces and raising to the p-th power locally on functions. The absolute Frobenius morphism is not a morphism of k-schemes. Let be the scheme obtained from by base change with the absolute Frobenius morphism on Spec (k), i.e., the underlying topological space of is that of with the same structure sheaf O of rings, only the underlying k-algebra structure on O is twisted as λ f = λ 1/p f,forλ k and f O. Using this description of, the relative Frobenius morphism F : is defined in the same way as the absolute Frobenius morphism and it is a morphism of k-schemes Definition and results. Recall that a variety is called Frobenius split [7] if the homomorphism O F O of O -modules is split. A homomorphism σ : F O O is a splitting of O F O (called a Frobenius splitting) if and only if σ(1) = 1. A Frobenius splitting σ is said to compatibly split asubvarietyz of if σ(i Z ) I Z,whereI Z is the ideal sheaf of Z in. In this case, σ induces a Frobenius splitting of Z. When is a smooth variety with canonical bundle ω,thereisa natural isomorphism of O -modules : F (ω 1 p ) = Hom O (F O, O ). In this way global sections of ω 1 p correspond to homomorphisms F O O. A section of ω 1 p which, up to a non-zero constant, corresponds to a Frobenius splitting in this way, is called a splitting section. One of the first known criteria ([7], Prop. 8) for a smooth projective variety to be Frobenius split was the following Proposition 4.1. Let be a smooth projective variety over k of dimension n. Assume that there exists a global section s of the anti canonical bundle ω 1, with divisor of zeroes of the form div(s) =Z 1 + Z Z n + D, satisfying : (1) The scheme theoretic intersection i Z i is a point P. (2) D is an effective divisor not containing P = j Z j. Then s p 1 is a splitting section of which compatibly splits the subvarieties Z 1,Z 2,...,Z n. This criterion was taylormade to suit the Bott-Samelson varieties.

7 LINE BUNDLES ON BOTT-SAMELSON VARIETIES 7 Proposition 4.2. Let be a smooth variety over k and let s be a global section of the anti-canonical bundle ω 1 such that sp 1 is a Frobenius splitting section of. Write the zero divisor div(s) of s as div(s) = Z j, with Z j, j =1,...,r, being irreducible subvarieties of of codimension 1. Then for any choice of integers 0 m j <p, j =1,...,r,and any line bundle L on, we have, for all integers i, an embedding of cohomology H i (, L) H i (, L p O( m j Z j )). This result is well known, but we have not been able to find an explicit reference. We therefore include a proof. Proof. Let φ : F O F ω 1 p denote the map induced by the global section s p 1 of ω 1 p. Composing this map with the twisted Cartier operator C : F ω 1 p O we get, by the identifications above, the splitting section ψ : F O O defined by s p 1. In particular, the morphism O F ω 1 p defined by sp 1 is split. As a consequence, if ω 1 p M M and s p 1 = t t with t and t global sections of line bundles M and M, then the morphism O F M defined by t is split. Tensoring with the line bundle L on corresponding to L we find, using the projection formula and F L L p, that the morphism L F (L p M) defined by t is split. Now use this on M = O( r m iz i ). As an immediate consequence of Proposition 4.2 we find Corollary 4.3. ([7], Prop. 1) Let be a Frobenius split projective variety and L an ample line bundle on. Then, for each integer j>0, H j (, L) =0. We also have the following result which will be important later. Proposition 4.4. Let be a smooth variety which is Frobenius split by a (p 1)-power of a section s of ω 1. Assume that Z 1,...,Z n are components of the zero divisor of s, andleta 1,...,a n and 1 r n be a collection of nonnegative integers. Then there exists an integer M such that for each line bundle L and integers i and m M, we have an embedding H i (, L O( Z j )) H i (, L pm O( (a j +1)Z j + a j Z j )). j=r+1

8 8 NIELS LAURITZEN, JESPER FUNCH THOMSEN Proof. Notice first of all that if a j =0forallj =1,...,n, then the result follows by successive use of Proposition 4.2 (with m j = p 1, j =1...,r). Assume hence that not all a j are zero and write a j = pa j + a j with a j 0and0 a j <p. By induction we may assume that there exists an integer M and an embedding H i (, L O( Z j )) H i (, L pm O( (a j+1)z j + a jz j )). j=r+1 for each m M. Using Proposition 4.2, with values m j = p 1 a j when j =1,...,r and m j = a j when j = r +1,...,n, we furthermore find an embedding of the right hand side into H i (, L pm+1 O( (a j +1)Z j + a j Z j )). j=r+1 The claim (with M = M + 1) now follows by composing the two embeddings above. 5. Frobenius splitting of Bott-Samelson varieties Let us now return to the situation of a Bott-Samelson variety associated to a sequence w of simple reflections. Recall the following lemma ([8], Proposition 2). Lemma 5.1. The anti-canonical bundle on is isomorphic to the line bundle ω 1 = O( (j) ) L w (ρ). The divisors (j), j = 1, 2,...,n, intersect transversally and as L w (ρ) is globally generated, and hence base point free, Proposition 4.1 and Lemma 5.1 implies Theorem 5.2 ([7], Thm. 1). The Bott-Samelson variety is Frobenius split compatibly with the divisors (j), j =1, 2,...,n. Furthermore, there exists a Frobenius splitting of by a (p 1)-power of a section s of ω 1, such that the zero section of s contains all the divisors (j), j =1, 2,...,n. The following proposition is a consequence of Proposition 4.2. Proposition 5.3. Let L bealinebundleon. Then for any choice of integers 0 m j <p, j =1, 2...,n and 0 m<p, we have, for each integer i, an embedding of cohomology H i (, L) H i (, L p O( m j (j) ) L w (mρ)).

9 LINE BUNDLES ON BOTT-SAMELSON VARIETIES 9 Proof. Let s i be the global section of O((i) ) with divisor of zeroes equal to (i),andlettbeaglobal section of L w (ρ) with divisor of zeroes j D j with D j irreducible divisors not containing the point (i).asaboves = t s i is a global section of ω 1 such that s p 1 is a Frobenius splitting section of. By Propostion 4.2 applied to s, with coefficients m i corresponding to (i) and m corresponding to D j, the result now follows by identifying L w (ρ) with the line bundle O( j D j). 6. Cohomology vanishing of globally generated line bundles In this section we will prove that the higher cohomology of globally generated line bundles on Bott-Samelson varieties vanishes. Lemma 6.1. There exists integers m 1,...,m n > 0 such that the line bundle O( n m j(j) ) is ample. Proof. Inductively there exists nonnegative integers m 1...,m n 1 such that L = O( n 1 m j(j) )isampleon[1].aso((n) ) has degree one along the fibers of the P 1 -bundle σ w : [1] this means that πw (Lm ) O((n) )isampleform sufficiently large. Theorem 6.2. Let L be a globally generated line bundle on a Bott- Samelson variety.then H i (, L) =0, i > 0. Proof. Choose m 1,...,m n according to Lemma 6.1 such that the line bundle O( n m j(j) ) is ample. By Proposition 4.4 there exists an embedding H i (, L) H i (, L pm O( m j (j) )), for some m. Now the result follows from Lemma 4.3 and the fact that a tensorproduct of an ample line bundle with a globally generated line bundle is ample. 7. Kumar vanishing In his proof [5] of the Demazure character formula in the Kac-Moody case S. Kumar in a crucial way uses the following cohomological vanishing result. Theorem 7.1. Let w =(s 1,...,s n ) be an ordered collection of simple reflections and let be the associated Bott-Samelson variety over a

10 10 NIELS LAURITZEN, JESPER FUNCH THOMSEN field of characteristic zero. Assume that the subexpression (s t,,s r ) of w is reduced for integers 1 t r n. Then H i (, L w (λ) O( (j) )) = 0, i > 0, whenever λ is a dominant weight. j=t Using Frobenius splitting techniques one may extend this result to arbitrary globally generated line bundles and positive characteristic (notice L w (λ) is globally generated when λ is a dominant weight). Lemma 7.2. ThelinebundleL = L w (ρ) m O((n) ) π w (L w[1]( ρ)) is globally generated when m is sufficiently large. Proof. Using Lemma 5.1 we may express the relative anti-canonical sheaf of the P 1 -bundle π w : [1] as ω 1 /[1] = ω 1 πw (ω [1] )=L w (ρ) O((n) ) πw (L w[1]( ρ)) Suppose that s n is a reflection in the simple root α. Consider then the fiber product diagram π G/B π w [1] G/P α where the lower horizontal morphism is the map induced by the product map P w[1] G and the morphism G/B G/P α is the natural projection map. From this we conclude that ω 1 /[1] is the pull back through π of the anti-canonical sheaf of the P 1 -bundle G/B G/P α. The latter is isomorphic to L G/B (α), and hence ω 1 /[1] L w (α). In particular, L L w ((m 1)ρ+α). Using that (m 1)ρ+α is dominant for m sufficiently large, the result now follows. Lemma 7.3. Let w =(s 1,...,s n ) be an ordered collection of simple reflections and the associated Bott-Samelson variety. Assume that the subexpression (s t,,s r ) of w is reduced for integers 1 t r n. Then there exists integers m 1,...,m n and m 0 satisfying all of the following proporties (1) m j 0, j/ {t,...,r}. (2) m j 0, j {t,...,r} and m r = 1. (3) The line bundle O( n m j(j) ) L w (ρ) m is globally generated. Proof. Assume first of all that r < n. By induction (in n) we find integers m 1,...,m n 1 and m 0 satisfying the equivalent properties

11 LINE BUNDLES ON BOTT-SAMELSON VARIETIES 11 for [1]. In particular, the line bundle m j (j) ) πw (L w[1](ρ) m ), n 1 O( is globally generated. Now choose a nonnegative integer m n such that the line bundle L = L w (ρ) mn O((n) ) π w (L w[1]( ρ)) is globally generated (Lemma 7.2). Then the line bundle m j (j) ) O(m(n) ) L w (ρ) mnm, n 1 O( is globally generated. We are left with the case r = n. Write O w (1) in the Z-basis as n 1 O w (1) = O( a j (j) ) O((n) ). By Proposition 3.5 we know that a t,...,a n 1 are nonnegative integers. Now n 1 t 1 L w (ρ) O( (n) ) O( a j (j) )=L O( a j (j) ), j=t where L = L w (ρ) O w ( 1) is a globally generated line bundle. It remains to find nonnegative integers m 1,...,m t 1 such that the line bundle O( t 1 (m j + a j )(j) ) is globally generated. That this is possible follows from Lemma 6.1. Theorem 7.4. Let w =(s 1,...,s n ) be an ordered collection of simple reflections and the associated Bott-Samelson. Assume that the subexpression (s t,,s r ) of w is reduced for integers 1 t r n. Then H i (, L O( (j) )) = 0, i > 0, j=t whenever L is a globally generated line bundle. Proof. Choose integers m 1,...,m n and m 0 according to the conditions in Lemma 7.3. By successive use of Proposition 5.3 we may assume that L L w ( ρ) m is globally generated (e.g. if m =1choose m j = p 1 and m = 1 in Proposition 5.3). By Proposition 4.4 (with values a j = m j, j r and a r = 0) the cohomology group H i (, L O( r j=t (j))) embeds into (for some m > 0) : H i (, L pm O( r 1 m j (j) (j) )), j=t

12 12 NIELS LAURITZEN, JESPER FUNCH THOMSEN which we may rewrite as r 1 H i (, L O( (j) )), where j=t L = L pm 1 (L L w ( mρ)) (L w (mρ) O( m j (j) )), is globally generated by choice of m, m 1,...,m n and assumption on L. The claim now follows by induction in r t. As already noted by S. Kumar it is crucial that the subexpression (s t,,s r ) is reduced. References [1] R. Bott and Samelson, H., Application of the theory of Morse to symmetric spaces, Amer.J.Math.80 (1958), [2] M. Demazure, Désingularisation des variétés de Schubert généralisées Ann. Sci. Ec. Norm. Sup. 7 (1974), [3] H. C. Hansen, On cycles on flag manifolds, Math. Scand. 33 (1973), [4] H. C. Hansen Cykler på flagmangfoldigheder, speciale, Aarhus Universitet (1972) [5] Kumar, S., Demazure character formula in arbitrary Kac-Moody setting, Invent. math. 89 (1987), [6] Lakshmibai, V., Littelmann P. and Magyar P., Standard monomial theory for Bott-Samelson varieties (preprint). [7] Mehta, V. and Ramanathan, A., Frobenius splitting and cohomology vanishing for Schubert varieties, Annals of Math. 122 (1985), [8] Ramanathan, A. Schubert varieties are arithmetically Cohen Macaulay, Invent. math. 80 (1985), Institut for Matematiske Fag, Aarhus Universitet, Ny Munkegade, DK-8000 Århus C, Denmark. address: niels@imf.au.dk, funch@imf.au.dk