DECOMPOSABILITY CRITERION FOR LINEAR

DECOMPOSABILITY CRITERION FOR LINEAR SHEAVES MARCOS JARDIM AND VITOR MORETTO FERNANDES DA SILVA Abstract. We establish a decomposability criterion for linear sheaves on Pn . Applying it to instanton bundles, we show, in particular, that every rank 2n instanton bundle of charge 1 on Pn is decomposable. Moreover, we provide an example of an indecomposable instanton bundle of rank 2n − 1 and charge 1, thus showing that our criterion is sharp.

1. Introduction A monad on a projective variety X is a complex of locally free sheaves α

β

M• : M0 −→ M1 −→ M2 such that β is surjective, α is injective. The (coherent) sheaf E := ker β/ im α is called the cohomology of M• . Monads were introduced by Horrocks in the late 1960’s [2] and are a valuable tool in the theory of sheaves over projective varieties and have been studied by many authors over the past four decades. In the present paper, we will be concerned with linear monads on Pn , which are of the form β α ⊕c M• : OPn (−1)⊕a → OP⊕b . n → OPn (1) Furthermore, we will mostly be concerned with the case where supp{coker α∗ } is empty, which is equivalent to say that E is a locally free sheaf. Note that rank(E) = b − a − c and c1 (E) = c − a. Sheaves that arise as cohomologies of linear monads can be characterized intrinsically. A torsion-free sheaf E on Pn is said to be a linear sheaf if it can be represented as the cohomology of a linear monad, and it is said to be an instanton sheaf if in addition it has c1 (E) = 0, i.e. if it can be represented as the cohomology of a linear monad in which a = c. Using the Beilinson spectral sequence [3], one can show that a torsion-free sheaf E on Pn is linear if and only if the following holds: (1) for n ≥ 2, H 0 (E(−1)) = H n (E(−n)) = 0; (2) for n ≥ 3, H 1 (E(−2)) = H n−1 (E(1 − n)) = 0; (3) for n ≥ 4, H p (E(k)) = 0, 2 ≤ p ≤ n − 2 and ∀k. Linear monads have been studied in many contexts in algebraic geometry, see for instance [1, 3, 8] and the references therein. The most relevant case is the case a = c, i.e. when the cohomology sheaf is an instanton sheaf. In this case, c coincides with c2 (E) and it is called the charge of E; we also have that r := rank(E) = b−2c. Let M(Pn ) denote the category of linear monads on Pn , where the morphisms are chain maps, so that M(Pn ) is a full subcategory of the abelian category Kom(Pn ) of complexes of coherent sheaves on Pn . It is not difficult to show that M(Pn ) is closed under extensions and under direct summands, and that a monad is decomposable if and only if its cohomology sheaf is decomposable. 1

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MARCOS JARDIM AND VITOR MORETTO FERNANDES DA SILVA

The main goal of this paper is to establish the following decomposability criterion for linear monads and sheaves. Theorem 1. If a2 + b2 + c2 − b(n + 1)(a + c) > 1, then the linear monad α

β

⊕c M• : OPn (−1)⊕a → OP⊕b n → OPn (1)

decomposes as a sum of other linear monads. Applied to the case of instanton sheaves, the previous result shows that if the rank of an instanton sheaf E is large in comparison with its charge, then E is decomposable. Corollary 2. If E is a rank r instanton bundle on Pn of charge c such that p r > (n − 1)c + c2 ((n + 1)2 − 2) + 1, then E is decomposable. This paper is organized as follows. Since the proof of the Main Theorem is based on the theory of representations of quivers, we begin in Section 2 detailing the relation between monads and quivers, and briefly outlining the relevant aspects of representation theory; see also [6] for closely related results. The proof of the Main Thorem will be completed in Section 3. We then conclude the paper further discussing the case of instanton sheaves in Section 4. Acknowledgments. The first named author is partially supported by the CNPq grant number 302477/2010-1 and FAPESP grant number 2005/04558-0. The second named author is supported by CAPES. 2. Representations of quivers and monads 2.1. Linear representations of quivers. Recall that a quiver Q is given by a finite set of vertices Q0 , a finite set of arrows Q1 and two maps h, t : Q1 → Q0 called head and tail, respectively. A linear representation of a quiver Q is given by V = ({Vi }i∈Q0 ; {fα }α∈Q1 ) where Vi is a C vector space and fα : Vt(α) → Vh(α) is linear. A morphism between two representations V and V 0 is given by φ = {φi }i∈Q0 where φi : Vi → Vi0 is linear and for each arrow α, fα0 φt(α) = φh(α) fα . Given V = ({Vi }i∈Q0 ; {fα }α∈Q1 ) and U = ({Ui }i∈Q0 ; {gα }α∈Q1 ) two linear representation of Q, we define the representation direct sum of U and V as W = V ⊕ U = ({Vi ⊕ Ui }i∈Q0 ; {(fα , gα )}α∈Q1 ). We call RepC (Q) the category of all linear representations of the quiver Q; it is easy to see that RepC (Q) is an abelian category. We say that W ∈ RepC (Q) is decomposable if there are nontrivial representations U and V such that W ∼ = V ⊕ U ; W is indecomposable if it is not decomposable. The following Theorem was proved by Kac. Its proof and all relevant definitions can be found in [7]. Theorem 3. Let Q be a quiver without oriented cycles with n vertices and x = (x1 , . . . , xn ) ∈ Zn , xi ≥ 0. There is an indecomposable representation V of Q with vector dimension dim V = x if and only if x is in the root system associated to Q and xi ≥ 0, ∀i.

DECOMPOSABILITY CRITERION FOR LINEAR SHEAVES

3

The detailed description of the root system associated with a quiver Q is somewhat involved and lies beyond the escope of the present paper; we refer the interested reader to [7]. The one essential fact for us is stated in the following Lemma: Lemma 4. Let q be the Tits form of the quiver Q and x = (x1 , . . . , xn ) ∈ Zn , i.e. X X xt(α) x(h(α)) x2i − (1) q(x) = α∈Q1

i∈Q0

If q(x) > 1, then x is not in the root system associated to Q. In particular, every representation of Q with dimension vector x satisfying q(x) > 1 is decomposable. From now on, Q will denote the following quiver α0 α1

•

−1

.. . αn

/ /

β0

/

β1

.. .

• 0

βn

/ /

/

! • 1

and let Γn := RepC (Q). The associated Tits form q : Z3 → Z is given by (2)

qn (a, b, c) = a2 + b2 + c2 − b(n + 1)(a + c).

A representation V = (V−1 , V0 , V1 ; {fαi }, {gβj }) in Γn is said to satisfy the relation Pij = βi αj + βj αi when gβi fαj + gβj fαi = 0. Definition 5. Let V = (V1 , V2 ; {fαi }) be a linear representation of the (n + 1)Kronecker quiver: /

α0

/

α1

• 1

.. . αn

/

! • 2

Pn V is globally injective if for any choice λ0 , . . . , λn ∈ C, not all zero, 0 λi fi is injective.PSimilarly, V is globally surjective if for any choice λ1 , . . . , λn ∈ C, not n all zero, 0 λi fi surjective. Let Γgis n the full subcategory of Γn whose objects are the linear representations V = (V−1 , V0 , V1 ; {fαi }, {gβj }) of Q satisfying the following conditions: (i) (V−1 , V0 ; {fαi }) is globally injective, and (V0 , V1 ; {gβj }) is globally surjective as representations of the (n + 1)-Kronecker quiver; (ii) satisfy the relation Pij = βi αj + βj αi for all 0 ≤ i, j ≤ n. Clearly, the first is an open condition in the affine space of all representations of Q with a fixed dimension vector, while the second is a closed condition. Lemma 6. The category Γgis n is closed under direct summands, i.e. if V = V1 ⊕ V2 gis is a decomposable object in Γgis n , then its factors V1 , V2 are also objects in Γn . Proof. Clearly, it is enough to show that V1 ∈ Γgis n . Since V1 = (Ca1 , Cb1 , Cc1 ; {fi1 }, {gi1 }) is subrepresentation V = (Ca , Cb , Cc ; {fi }, {gi }), we have the following commutative diagram

of

4

MARCOS JARDIM AND VITOR MORETTO FERNANDES DA SILVA

/

f0

CO

.. .

a

/ CO

fn

f01

? Ca 1

.. . 1 fn

/

g0 b

/ ? b1 /C

.. .

c / CO

gn

g01

/ ? c1 /C

.. . 1 gn

where the vertical arrows are inclusions. Thus, V1 satisfies the relations Pi,j , 0 ≤ i, j ≤ n. Let W1 = (Ca1 , Cb1 ; {fi1 }) subrepresentation of W = (Ca , Cb ; {fi }). Since W is globally injective, then W1 is also. Moreover, U1 = (Cb1 , Cc1 ; {gi1 }) is isomorphic to U/U2 , where U = (Cb , Cc ; {gi }) and U2 = (Cb2 , Cc2 ; {gi2 }) are representations of the Kronecker quiver. Since U is globally surjective and U1 is a quotient of U , then U1 is globally surjective.  Therefore, it follows that V1 is an object in Γgis n , as desired. 2.2. Category of linear monads. Let M(Pn ) denote the category of linear monads on Pn , regarded as a full subcategory of the abelian category Kom(Pn ) of complexes of coherent sheaves on Pn . A monad is decomposable if it is decomposable as a complex of sheaves. Let us now define a functor F : M(Pn ) → Γgis n+1 . First, choose homogeneous coordinates [x0 ; . . . ; xn ] of Pn , and let {x0 , . . . , xn } be the corresponding basis of H 0 (OPn (1)). Note that 0 ∼ Hom(OPn (−1)⊕a , O⊕b n ) = Mb×a ⊗C H (OPn (1)), P

where Mb×a denotes the vector space of b × a matrices of complex numbers. Consider the linear monad α

β

⊕c M• : OPn (−1)⊕a → OP⊕b . n → OPn (1)

By the previous paragraph, one can write α = α0 x0 + . . . + αn xn and β = β0 x0 + . . . + βn xn , a

where αi : C → Cb and βj : Cb → C c are linear transformations. Therefore, we can define F(M• ) := (Ca , Cb , Cc ; {αi }, {βj }). and it is not difficult to see that the monad conditions are precisely what we need to guarantee that F(M• ) is an object of Γgis n . Indeed, note that X β ◦ α = 0 ⇐⇒ (βi αj + βj αi )xi xj = 0. i≤j 0

Since {xi xj ; i ≤ j} is a basis of H (OPn (2)), it follows that β ◦ α = 0 ⇔ βi αj + βj αi = 0∀i, j hence F(M• ) satisfies the relations Pij . Moreover, (Ca , Cb ; {αi }ni=0 ) is globally injective if and only if ker αP = 0 for every P ∈ Pn , and (Cb , Cc ; {βj }nj=0 ) is globally surjective if and only if coker β = 0.

DECOMPOSABILITY CRITERION FOR LINEAR SHEAVES

5

For each i ∈ Z, we have the canonical isomorphism Hom(OPn (i)⊕k , OPn (i)⊕l ) ∼ = Ml×k . Thus, for a given a morphism φ• : M• → N• between monads, we set F(φ) to be the morphism F(M• ) → F(N• ) between representations obtained through the above isomorphism. In summary, a choice of homogeneous coordinates of Pn defines a functor F from the category of linear monads M(Pn ) to the category of representations Γgis n . Proposition 7. The functor F : M(Pn ) → Γgis n is an equivalence of categories. Proof. First, we argue that the functor F : M → Γgis n is essentially surjective: n given an object of Γgis n and a choice of homogeneous coordinates for P , one easily constructs a linear monad with the desired properties. Since F

HomM (M• , N• ) → HomΓgis (F(M• ), F(N• )) n is isomorphism for any monads M• , N• , then F is equivalence of categories.



Lemma 8. The functor F preserves direct sums, i.e., F(M•1 ⊕ M•2 ) = F(M•1 ) ⊕ F(M•2 ) for every M•1 and M•2 in M(Pn ). Proof. Consider the monads (k = 1, 2) αk

βk

k Mk• : OPn (−1)⊕ak → OP⊕b → OPn (−1)⊕ck , n

and let M• = M•1 ⊕ M•2 . It follows that F(Mk• ) := (Cak , Cbk , Cck ; {αik }, {βjk }). On the other hand, F(M• ) := (Ca1 +a2 , Cb1 +b2 , Cc1 +c2 ; {αi }, {βj }), where  αi (z1 , z2 ) =

αi1 (z1 ) 0 0 αi2 (z2 )



 and βi (w1 , w2 ) =

βi1 (w1 ) 0 0 βi2 (w2 )

with z1 ∈ Ca1 and z2 ∈ Ca2 , while w1 ∈ Cb1 and w2 ∈ Cb2 . It is then easy to see that F(M•1 ⊕ M•2 ) = F(M•1 ) ⊕ F(M•2 ).

 ,



Remark 9. The hypotheses on α being injective at every P ∈ Pn and β being surjective are not necessary to define the functor F; they are essential, however, to identify the subcategory Γgis n ⊂ Γn as the image of the category of linear monads with locally free cohomology.

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MARCOS JARDIM AND VITOR MORETTO FERNANDES DA SILVA

3. Proof of the Main Thorem The proof of the Main Theorem is now a simple application of Kac’s Theorem. Let β α ⊕c M• : OPn (−1)⊕a → OP⊕b n → OPn (1) be a linear monad with a2 + b2 + c2 − b(n + 1)(a + c) > 1, and set V = F(M• ). By Kac Theorem, V is decomposable, hence V ∼ = V1 ⊕ V2 , and both V1 and V2 are i objects of Γgis n , by Lemma 6. Hence, by Proposition 7, Vi = F(M• ) for some monad i M• (i = 1, 2). As F preserves direct sums (Lemma 8), we have that F(M• ) ∼ = F(M1• ⊕ M2• ); 1 2 ∼ since F is equivalence of categories, we have that M• = M• ⊕ M• , as desired. 4. Application to instanton sheaves Let us now apply our Main Theorem to the special case of instanton bundles. Let E be an instanton bundle of rank r and charge c on Pn ; so that E is the cohomology of a linear monad of the form α

β

M• : OPn (−1)⊕c → OP⊕r+2c → OPn (1)⊕c . n First, to prove Corollary 2, note that Theorem 1 says that if 2c2 +(2c+r)2 −2c(2c+ r)(n + 1) > 1, then E is decomposable. We can rewrite this inequality as (r − (n − 1)c)2 > c2 ((n + 1)2 − 2) + 1 and the Corollary follows immediately. If we assume that c ≤ n, the inequality in the statement of Corollary 2 can be greatly simplified:pif r ≥ 2nc, then E is decomposable. Indeed, if c > 0 ,we have 2nc > c(n − 1) + c2 ((n + 1)2 − 2) + 1. On the other hand, if 0 < c < n + 1, then 0 < 2c2 < 2(n + 1)c. It follows that (c(n+1)−1)2 = (c(n+1))2 −2(n+1)c+1 < (c(n+1))2 −2c2 +1 = c2 ((n+1)2 −2)+1, p and c(n + 1) − 1 < c2 ((n + 1)2 − 2) + 1. Therefore, 2nc − 1 < c(n − 1) + p c2 ((n + 1)2 − 2) + 1, as desired. Immediately, we have the following Corollary 10. Every rank 2n instanton bundle E of charge 1 on Pn is decomposable. The following example shows that the decomposability criterion of Corollary 2 is sharp. 2n+1 Example 11. Let {ei }2n+1 and consider the i=1 denote the canonical basis of C β0 .. / C2n+1 .. . / .

α0

following linear representation of Q: R = C

αn

βn

/ /

C where

(i) αi (z) = z.e1+1 , ∀0 ≤ i ≤ n; (ii) βj (v) = e∗j+n+1 (v), when 0 ≤ j ≤ n − 1, and βn (v) = (e2n+1 − e1 )∗ (v). It is not difficult to see that R ∈ Γgis n and that R is indecomposable. It follows that the linear monad M• such that F(M• ) = R is indecomposable. Its cohomology is an indecomposable instanton bundle of rank 2n − 1 and charge 1 on Pn ; it is also slope semistable, by [5, Theorem 3].

DECOMPOSABILITY CRITERION FOR LINEAR SHEAVES

7

We complete this paper with a characterization of rank 2n instanton bundles of charge 1 on Pn . Below, TPn and ΩPn denote the tangent and cotangent bundles, respectively. Proposition 12. Let E be a 2n instanton bundle on Pn of charge 1. Then, either E = TPn (−1) ⊕ ΩPn (1) or E = E 0 ⊕ OPkn ,where E 0 is r instanton bundle of charge 1 and rank r, with n ≤ r ≤ 2n − 1 if n is even, or n − 1 ≤ r ≤ 2n − 1 if n is odd. Note that the instanton bundles of the first type (TPn (−1) ⊕ ΩPn (1)) are not semistable and not of trivial split type. On the other hand, all instanton bundles of the second type are semistable, by [5, Theorem 3]. It also follows that the moduli space of rank 2n, charge 1 instanton bundles on Pn is reducible, since instanton bundles of the first type cannot be deformed into instanton bundles of the second type; they form an isolated 0-dimensional component. Proof. Let E be the cohomology of the monad β

α

M• : OPn (−1) → OP⊕2n+2 → OPn (1). n By the Corollary 11 , we have E = E 1 ⊕ E 2 where E 1 and E 2 are cohomologies of submonads M1• and M2• , respectively, such that M• = M1• ⊕ M2• . Since E has charge 1, we can take E 1 nontrivial as the cohomology of a submonad β1

α1

1 +i+1 M1• : OPn (−1) → OP⊕r → OPn (1)⊕i n

of M• , where r1 = rank E 1 and i is either 0 or 1. By the Main Theorem of [1] rank E 1 ≥ n if n is even and rank E 1 ≥ n − 1 if n is odd. 1 If i = 1, then E 2 = OP⊕k is a rank r1 instanton bundle of charge n is trivial and E 1, where n ≤ r1 ≤ 2n − 1 if n is even or n − 1 ≤ r0 ≤ 2n − 1 if n is odd and r1 + k = n. On the other hand, if i = 0, we have α1

0

M1• : OPn (−1) → OP⊕r+1 →0 n and 0

β1

M2• : 0 → OP⊕k+1 → OPn (1)⊕1 n Note that, as F(M1• ) is globally injective, we have that r + 1 ≥ n + 1. Similarly, as F(M2• ) is globally surjective, we have that k + 1 ≥ n + 1. Since r + k = 2n, we must have r = k = n. It follows by comparison with the Euler sequence that the cohomologies of M1• and M2• must be TPn (−1) and ΩPn (1), respectively.  References [1] Fløystad, G.: Monads on projective spaces. Communications in Algebra 28 (2000), 5503– 5516. [2] Horrocks, G.: Vector bundles on the punctured spectrum of a local ring. Proc. London Math. Soc. 14 (1964), 689–713. [3] Jardim, M.: Instanton sheaves on complex projective spaces. Colletanea Mathematica 57 (2006), 69–97. [4] Jardim, M., Martins, R. V.: Linear and Steiner bundles on projective varieties. Comm. Algebra 38 (2010), 2249-2270. [5] Jardim, M., Mir´ o-Roig, R. M.: On the semistability of instanton sheaves over certain projective varieties. Comm. Algebra 36 (2008), 288-298.

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MARCOS JARDIM AND VITOR MORETTO FERNANDES DA SILVA

[6] Jardim, M., Prata, D. M.: Representations of quivers on abelian varieties and monads on projective varieties. So Paulo J. Math. Sci. 4 (2010), 399–423. [7] Kac, V. G.: Infinite root systems, representations of graphs and invariant theory. Inventiones Math. 56 (1980), 57–92. [8] O. Okonek, M. Schneider, and H. Spindler, Vector bundles on complex projective spaces. Boston: Birkhauser (1980). ´ tica, Rua Se ´rgio Buarque de Holanda, IMECC - UNICAMP, Departamento de Matema 651, 13083-859 Campinas-SP, Brazil E-mail address: [email protected] E-mail address: [email protected]