A splitting theorem for rank two vector bundles on projective spaces in positive characteristic

A splitting theorem for rank two vector bundles on projective spaces in positive characteristic

Hideyasu Sumihiro and Shigehiro Tagami

(Received April 21, 2000)

Abstract. We shall prove the following splitting theorem for rank two vector bundles Eon then-dimensional projective spacePⁿ …nV4†in positive characteristic. LetPbe a 4- or 5-dimensional projective linear subspace ofPⁿandEˆEjPthe restriction ofE toP. ThenEsplits into line bundles if and only if the ®rst cohomology of the sheaf of endomorphisms of E vanishes.

0. Introduction

Let E be a rank two vector bundle on the n-dimensional projective space P_kⁿ …nV4† de®ned over an algebraically closed ®eld k.

In [4], H. Sumihiro showed the following theorem in the case of char kˆ0.

Theorem 0.1. Let P be a 4- or 5-dimensional projective linear subspace of P_kⁿ and EˆEjP the restriction of E to P. Then E splits into line bundles if and only if H¹…P;End…E†† ˆ0.

The aim of this article is to prove that this theorem holds also true in char kˆp>0. The proof is almost the same as the one for charkˆ0, namely, it is obtained by studying some geometric structures of the Hilbert scheme of P_kⁿ at determinantal subvarieties. In charkˆp>0, however, since we cannotuse the Kodaira vanishing theorem and the Le-Potier vanishing theorem (cf. [1], [3]), we have to observe some vanishings of cohomologies appearing in [4]

carefully.

1. Preliminaries

We ®rst recall the de®nition and some properties of determinantal varieties associated to rank two bundles (cf. [4]).

2000 Mathematics Subject Classi®cation. 14C05, 14F05, 14F17.

Key words and phrases. Vector bundles, Hilbert Schemes, Vanishing Theorems.

1.1. De®nition of determinantal varieties. Let E be a rank two vector bundle on P_kⁿ de®ned over an algebraically closed ®eld k with arbitrary characteris- tic, p:P…E† !P_kⁿ the projective bundle associated to E over P_kⁿ, L_E the tautological line bundle on P…E† and let GˆGrass…H⁰…E†;m‡1† be the Grassmann variety which parametrizes …m‡1†-dimensional linear subspaces of H⁰…P_kⁿ;E†, where nˆ2m (resp. nˆ2m‡1). We assume that E is very ample, i.e., L_E is a very ample line bundle. Then we can take sˆ hs₁;s₂;. . .;s_m‡1i AG …s_iAH⁰…P_kⁿ;E†† satisfying the following condition

1) Y_sˆD₁VD₂V VD_m‡1 is a smooth closed subscheme of P…E†

of pure codimension m‡1,

2) W…s₁†VW…s₂†V VW…s_m‡1† ˆq, …†

where D_i is the tautological divisor onP…E†de®ned by s_i andW…s_i†is the zero locus of s_i in P_kⁿ …1UiUm‡1†.

Let Xsˆp…Ys†. Then we can show that Xs is a closed subscheme of P_kⁿ which is isomorphic to Y_s through p with the following de®ning equations:

s_i5s_jˆ0 …1UiUjUm‡1†:

Definition 1.1. We call the closed subscheme Xs of P_kⁿ the determinantal variety associated to E de®ned by sAG.

ThoughX_sdepends on the choice ofsAG, we call a closed subvarietyX_sa determinantal variety associated to E.

As for determinantal varieties, we obtain the following.

Theorem 1.1. Let the notaion be as above.

1) Uˆ fsAGjs satis®es the condition …†g is a Zariski open subset of G.

2) There exists a closed subscheme X of P_kⁿU such that the second projection q:XHP_kⁿU!U is faithfully ¯at and X_sˆq ¹…s† for any sAU. Thus smooth determinantal varieties associated to E form a smooth family over an open subset of G.

When nˆ4 or 5, let IX be the de®ning ideal of a determinantal subvariety X in Pⁿ. Then I_X has the following resolution by vector bundles.

Lemma 1.2. In the above notation, there exists an exact sequence

0!E… c1† !0³ O_Pⁿ… c1† !IX !0;

where c1 is the ®rst Chern number of E and E is the dual bundle of E.

Proof. Let sˆ fs1;s2;s3g be a setof global sections of E which de®nes the determinantal subvariety X. Then we can de®ne homomorphisms

a:0³ O_PⁿCe_i5e_j7!s_i5s_jA 5² E …1Ui<jU3†;

b:ECf 7!f…s3†e15e2 f…s2†e15e3‡f…s1†e25e3 A 0³ O_Pⁿ;

where fe_i5e_jg is a basis of 0³ O_Pⁿ. Then itsu½ces to verify locally on Pⁿ that the following sequence is exact:

0!E !^b 0³ O_Pⁿ !^a I_XnO…c₁† !0:

r 1.2. Tangent bundles and normal bundles of determinantal varieties. In the following subsections, we consider the case nˆ4 or 5, i.e., mˆ2.

Let E be a very ample rank two bundle on P_kⁿ and X a determinantal variety associated to E which is isomorphic through p to the complete in- tersection Y in P…E† of the tautological divisors fDijiˆ1;2;3g.

Let H be the restriction of a hyperplane of Pⁿ to X and Dthe restriction of a tautological divisor of P…E† to X through the isomorphism p.

Then we have the following commutative diagram of exact sequences:

0 0

??

?y

??

?y T_P…E†=P???y ⁿjY ƒƒ!^@ O_X…2D c₁H†

??

?y^a

0 ƒƒ! T_Y ƒƒ! T_P…E†jY ƒƒ! N_Y=P…E† ƒƒ! 0

???y

??

?y

0 ƒƒ! TX ƒƒ! TPⁿ???yjX ƒƒ! N_X=Pⁿ ƒƒ! 0;

??

?y

0 0

where a is the injection induced by the snake lemma. Since N_Y=P…E†F 0³ OX…D†, we obtain the following.

Proposition 1.3. There exists an exact sequence

0!OX…2D c1H† !0³ OX…D† !N_X=Pⁿ!0:

1.3. Hilbert Schemes. LetHilbbe the Hilbert scheme ofPⁿ. Let j:UCs7!

XsAHilb be the morphism induced by Theorem 1.1. Let Aut…E† be the automorphism group of E. Then Aut…E† is a reduced connected linear algebraic group of dimension dimH⁰…End…E††.

For every element gAAut…E† and sˆhs₁;s₂;s₃i AG, we de®ne gsˆhg…s₁†;g…s₂†;g…s₃†i;

where g…si† is the composite ofsi with g. Then itde®nes an action of Aut…E†

on G and we have

gs_i5gs_jˆdet…g†s_i5s_j …1UiUjU3†;

where det:Aut…E†Cg7!det…g†Akˆknf0g is the determinant character.

Hence X_gsˆX_s. Therefore Aut…E† acts on U and j is an orbitmorphism, i.e., j is constant on any orbit O…s† ˆ fgsjgAAut…E†g.

Then we have the following.

Lemma 1.4. The stabilizerStab…s†of sAU coincides with the multiplicative group k.

As a trivial corollary of the above lemma, we observe that every orbit has the same dimension dim Aut…E†=k, i.e., dimO…s† ˆdimH⁰…End…E†† 1 …sAU†. Hence the action of Aut…E† on Uis closed, i.e., every orbitis closed in U.

2. Proof of the theorem

2.1. Since itis well-known thatE splits into line bundles if and only if EˆEjP splits into line bundles, where P is a 4- or 5-dimensional linear subspaces of Pⁿ, we may assume that E is a rank two vector bundle onPⁿ (n being either 4 or 5) (cf. [2]). In addition after multiplying E by a suitable ample line bundle, we may assume that E is a very ample vector bundle enjoying Hⁱ…EnK_Pⁿ† ˆ0 …1UiU4†, where K_Pⁿ is the canonical line bundle of Pⁿ.

By Proposition 1.3, we have the following exact sequence 0!H⁰…O_X…2D c₁H†† !0³ H⁰…O_X…D†† !H⁰…N_X=Pⁿ†

!H¹…OX…2D c1H†† !0³ H¹…OX…D††:

Now we recall Y ˆD₁VD₂VD₃. Consider the canonical exact sequence 0!O_P…E†…D c₁H† !O_P…E†…2D c₁H† !O_D1…2D c₁H† !0;

…†₁

from which we obtain the following exact sequence:

0!H⁰…O_P…E†…D c₁H†† !H⁰…O_P…E†…2D c₁H†† !H⁰…O_D1…2D c₁H††

!H¹…O_P…E†…D c₁H†† !H¹…O_P…E†…2D c₁H†† !H¹…O_D1…2D c₁H††

!H²…OP…E†…D c1H††:

Since Hⁱ…O_P…E†…D c1H†† ˆHⁱ…E† …0UiU4†and we can show thatH⁰…E†

ˆ0 and Hⁱ…E† ˆH^{n i}…EnK_Pⁿ† ˆ0 …iˆ1;2† by our assumption, it turns outthatHⁱ…O_P…E†…2D c1H††FHⁱ…OD1…2D c1H†† …iˆ0;1†.

In addition considering the following exact sequences similarly 0!O_D1…D c₁H† !O_D1…2D c₁H† !O_D1VD2…2D c₁H† !0;

0!O_P…E†… c₁H† !O_P…E†…D c₁H† !O_D1…D c₁H† !0;

…†₂

0!OD1VD2…D c1H† !OD1VD2…2D c1H† !OY…2D c1H† !0;

0!OD1… c1H† !OD1…D c1H† !OD1VD2…D c1H† !0;

0!O_P…E†… D c1H† !O_P…E†… c1H† !OD1… c1H† !0;

…†₃

we obtain isomorphisms Hⁱ…O_D1…2D c₁H††FHⁱ…O_D1VD2…2D c₁H††

and Hⁱ…OD1VD2…2D c1H††FHⁱ…OY…2D c1H†† …iˆ0;1† because Hⁱ…O_P…E†… D c1H†† ˆ0 …0UiU4†. Summing up the above, we con- clude that Hⁱ…O_X…2D c₁H††FHⁱ…O_P…E†…2D c₁H††FHⁱ…Pⁿ;S²…E†… c₁††

…iˆ0;1†.

On the other hand, since there exists an exact sequence 0!O_Pⁿ!End…E† !S²…E†… c₁† !0;

we have a canonical isomorphism H¹…S²…E†… c₁††FH¹…End…E†† and dimH⁰…S²…E†… c1†† ˆdimH⁰…End…E†† 1.

Moreover we easily see that dimH⁰…O_X…D†† ˆdimH⁰…E† 3.

Summarizing the above, we get the following proposition.

Proposition 2.1. With the above assumption, if H¹…End…E†† ˆ0, then dimH⁰…N_X=Pⁿ† ˆ3…dimH⁰…E† 3† dimH⁰…End…E†† ‡1:

Remark 2.1. When char kˆ0, we get Hⁱ…E†FH^{n i}…EnK_Pkⁿ† ˆ0 for 0UiUn 2 by the Le-Potier vanishing theorem. So we do not need the assumption Hⁱ…EnK_Pⁿ† ˆ0 1UiU4in Proposition 2.1. Also the proof itself becomes slightly simpler because we can use the vanishing theorems.

2.2. Let Hilb⁰ be an irreducible componentof Hilb containing the closure j…U† of j…U† in Hilb and TXs;Hilb the Zariski tangent space of Hilb at X_s. Then itis known thatT_Xs;HilbFH⁰…N_Xs=Pⁿ†. So we have the following proposition.

Proposition 2.2. Under the same assumptions in Proposition 2.1, if H¹…End…E†† ˆ0 then

1) Hilb⁰ coincides with j…U†.

2) Hilb⁰ is smooth at the determinantal subvarieties associated to E.

Proof. It is su½cient to prove that dimj…U† ˆdimH⁰…N_Xs=Pⁿ† for any determinantal surfaceX_s. Using the exact sequence in Proposition 1.3, we see that j ¹…j…s†† …sAU† consists of ®nitely many orbits. Hence

dimj…U† ˆdimU dimO…s†

ˆdim Grass…H⁰…E†;3† dimH⁰…End…E†† ‡1

ˆ3…dimH⁰…E† 3† dimH⁰…End…E†† ‡1:

So our assertion follows by Proposition 2.1. r

2.3. LetPGL…n‡1;k† be the automorphism group of Pⁿ and let T_s:PⁿC x7!sxAPⁿ be the transformation of Pⁿ de®ned by sAPGL…n‡1;k†.

Suppose thatH¹…End…E†† ˆ0. Then it follows from Proposition 2.2 that sj…U† ˆj…U† for every element sAPGL…n‡1;k†. Since j…U† is a con- structible set, there exist two elements s;tAU satisfying X_s…s†ˆXt, where X_s…s† is the determinantal subvariey associated to T_s…E† de®ned by s…s† ˆ hT_s…s1†;T_s…s2†;T_s…s3†i. Consider the resolutions of the de®ning ideal sheaves I_Xt of X_t and I_Xs…s† of X_s…s† respectively (cf. Lemma 1.2):

0 ƒƒ! E ƒƒ! 0³ O_Pⁿ ƒƒ! IXtnO…c1† ƒƒ! 0

c

??

?y ^F

??

?y

0 ƒƒ! T_s…E† ƒƒ! 0³ O_Pⁿ ƒƒ! I_Xs…s†nO…c₁† ƒƒ! 0:

…†

Then it is observed that there exists an isomorphism c:0³ OPⁿ!0³ OPⁿ such that c makes the diagram in …† commutative and so we see that T_s…E† is isomorphic to E, i.e., E is a homogeneous vector bundle. Since every homo- geneous bundle on Pⁿ of rank r<n is a directsum of line bundles even if char kˆp>0 (cf. [2]), we can complete the proof of Theorem 0.1.

References

[ 1 ] J. Le-Potier, Annulation de la cohomologie aÁ valeurs dans un ®bre vectoriel holomorphe positif de rang quelconque, Math. Ann. 218 (1975), 35±53.

[ 2 ] C. Okonek, M. Schneider and H. Spindler, Vector Bundles on Complex Projective Spaces, Progress in Math. 3, BirkhaÈuser (1980).

[ 3 ] B. Shi¨man and A. Sommese, Vanishing Theorem on Complex Manifolds, Progress in Math. 56, BirkhaÈuser (1985).

[ 4 ] H. Sumihiro, Determinantal varieties associated to rank two vector bundles on projective spaces and splitting theorems, Hiroshima Math. J. 29 (1999), 371±434.

Department of Mathematics Graduate School of Science

Hiroshima University Higashi-Hiroshima 739-8526, Japan H. Sumihiro: [email protected]

S. Tagami: [email protected]