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    快乐十分精确公式:Admissible sheaves on P^3

    北京十一选五开奖结果 www.frdg.net arXiv:math/0408196v1 [math.AG] 14 Aug 2004

    Admissible sheaves on P3
    Marcos Jardim IMECC - UNICAMP Departamento de Matem? atica Caixa Postal 6065 13083-970 Campinas-SP, Brazil February 1, 2008
    Abstract Admissible locally-free sheaves on P3 , also known in the literature as mathematical instanton bundles, arise in twistor theory, and are in 1-1 correspondence with instantons on R4 . In this paper, we study admissible sheaves on P3 from the algebraic geometric point of view. We discuss examples and compare the admissibility condition with semistability and splitting type.

    Contents
    1 Monads 2 Examples of admissible sheaves 3 Semistability of torsion-free admissible sheaves 4 Trivial splitting type 5 Open problems 3 7 9 12 15

    1

    Introduction
    An intense interest on the construction and classi?cation of locally-free sheaves on the 3-dimensional complex projective space started on the late 70’s, when twistor theory yielded a 1-1 correspondence between instantons (i.e. antiself-dual connections of ?nite L2 norm) on R4 and certain holomorphic vector bundles on P3 ; this is the celebrated Penrose-Ward correspondence [7, 10]. This fact was later used by Atiyah, Drinfeld, Hitchin and Manin to construct and classify all instantons [1]. Since then, many authors have studied the so-called mathematical (or complex) instanton bundles, de?ned in the literature as rank 2 locally-free sheaves E on P3 with c1 (E ) = c3 (E ) = 0 and c2 (E ) = c > 0 satisfying H 0 (P3 , E (?1)) = H 1 (P3 , E (?2)) = 0; see [2] for a recent brief survey of this topic. These correspond to SL(2, C) instantons on R4 of charge c. The correct generalization for higher rank sheaves is given by Manin and Drinfeld (see [7]), and leads us to the key de?nition of this paper: De?nition. An admissible sheaf on P3 is a coherent sheaf E satisfying: H p (P3 , E (k )) = 0 for p ≤ 1 , p + k ≤ ?1, and p ≥ 2 , p + k ≥ 0 Admissible locally-free sheaves of rank r and vanishing ?rst Chern class are in 1-1 correspondence with SL(r, C) instantons on R4 [7]. In this paper, we study mainly torsion-free admissible sheaves with vanishing ?rst Chern class, which can be regarded as a generalization of instantons. Our focus is on the algebraic geometric properties of such objects, like semistability and splitting type. The paper is organized as follows. In Section 1 we remark that admissible sheaves are in 1-1 correspondence with certain monads, exploring a few properties and some examples in Section 2. We then discuss how admissibility and semistability compare with one another in Section 3, and conclude with an analysis of the splitting type of torsion-free admissible sheaves with vanishing ?rst Chern class in the last section. 2

    Acknowledgment. Some of the results presented here were obtained in joint work with Igor Frenkel [4]; we thank him for his continued support. We also thank the organizers and participants of the XVIII Brazilian Algebra Meeting.

    1

    Monads

    Let X be a smooth projective variety. A monad on X is a sequence V? of the following form: V? : 0 → V?1 ?→ V0 ?→ V1 → 0
    α β

    (1)

    which is exact on the ?rst and last terms. Here, Vk are locally free sheaves on X . The sheaf E = ker β/Im α is called the cohomology of the monad V? , also denoted by H 1 (V? ). In this paper, we will focus on the so-called special monads on P3 , which are of the form: 0 → V ? OP3 (?1) ?→ W ? OP3 ?→ V ′ ? OP3 (1) → 0 , where α is injective and β is surjective. The existence of such objects has been completely classi?ed by Floystad in [3]; let v = dim V , w = dim W and v ′ = dim V ′ . Theorem 1. There exists a special monad on P3 as above if and only if at least one of the following conditions hold: ? w ≥ 2v ′ + 2 and w ≥ v + v ′ ; ? w ≥ v + v ′ + 3. Monads appeared in a wide variety of contexts within algebraic geometry, like the construction of locally free sheaves on complex projective spaces, the study of curves in P3 and surfaces in P4 . In this section, we will see how they are related to admissible sheaves on P3 . 3
    α β

    Theorem 2. Every admissible torsion-free sheaf E on P3 can be obtained as the cohomology of a special monad 0 → V ? OP3 (?1) ?→ W ? OP3 ?→ V ′ ? OP3 (1) → 0 ,
    α β

    (2)

    1 3 1 ′ 1 3 where V = H 1 (P3 , E ??2 P3 (1)), W = H (P , E ??P3 ) and V = H (P , E (?1)).

    Proof. Manin proves the case E being locally-free in [7, p. 91], using the Beilinson spectral sequence. However, the argument generalizes word by word for E being torsion-free; just note that the projection formula
    i 3 ? Ri p1? (p? 1 OP3 (k ) ? p2 F ) = OP3 (k ) ? H (P , F )

    holds for every torsion-free sheaf F , where p1 and p2 are the natural projections of P3 × P3 onto the ?rst and second factors. Clearly, the cohomology sheaf E is always coherent, but more can be said in particular situations. Note that α ∈ Hom(V, W ) ? OP1 and β ∈ Hom(W, V ′ ) ? OP1 . Clearly, the surjectivity of β as a sheaf map implies that the localized map βx is surjective for all x ∈ P3 , while the injectivity of α as a sheaf map implies that the localized map αx is injective only for generic x ∈ P3 . Theorem 3. The cohomology E of the monad 0 → V ? OP2 (?1) ?→ W ? OP2 ?→ V ′ ? OP2 (1) → 0 is a coherent admissible sheaf with: rank(E ) = dim W ? dim V ? dim V ′ , c1 (E ) = dim V ′ ? dim V 1 1 ch2 (E ) = (dim V + dim V ′ ) and ch3 (E ) = (dim V ? dim V ′ ) . 2 6 Moreover: ? E is torsion-free if and only if the localized maps αx are injective away from a subset of codimension 2; 4
    α β

    (3)

    ? E is re?exive if and only if the localized maps αx are injective away from ?nitely many points; ? E is locally-free if and only if the localized maps αx are injective for all x ∈ P3 . Proof. The kernel sheaf K = ker β is locally-free, and one has the sequence: 0 → V ? OP3 (?1) ?→ K → E → 0 so E is clearly coherent. Notice also that: ch(E ) = dim W ? dim V · ch(OP3 (1)) ? dim V ′ · ch(OP3 (1)) from which the calculation of the Chern classes of E follows easily. Taking the dual of the sequence (4), we obtain: 0 → E ? → K? ?→ V ? ? OP3 (1) → Ext1 (E, OP3 ) → 0
    α? α

    (4)

    (5)

    since K is locally-free. In particular, Ext2 (E, OP3 ) = Ext3 (E, OP3 ) = 0 and I = supp Ext1 (E, OP3 ) = {x ∈ P3 | αx is not injective } So it is now enough to argue that C is torsion-free if and only if dim I = 1 and that C is re?exive if and only if dim I = 0; the third statement is clear. Recall that the mth -singularity set of a coherent sheaf F is given by: Sm (F ) = {X ∈ P3 | dh(Fx ) ≥ 3 ? m} where dh(Fx ) stands for the homological dimension of Fx as an Ox -module: dh(Fx ) = d ?? Extd Ox (Fx , Ox ) = 0 Extp Ox (Fx , Ox ) = 0 ?p > d

    In the case at hand, we have that dh(Fx ) = 1 if X ∈ I , and dh(Fx ) = 0 if X ∈ / I . Therefore S0 (C ) = S1 (C ) = ?, while S2 (C ) = I . It follows that [9, Proposition 1.20] : 5

    ? if dim I = 1, then dim Sm (C ) ≤ m ? 1 for all m < 3, hence C is a locally 1st -syzygy sheaf; ? if dim I = 0, then dim Sm (C ) ≤ m ? 2 for all m < 3, hence C is a locally 2nd -syzygy sheaf. The desired statements follow from the observation that C is torsion-free if and only if it is a locally 1st -syzygy sheaf, while C is re?exive if and only if it is a locally 2nd -syzygy sheaf [8, p. 148-149]. As part of the proof above, it is worth emphasizing that if E is admissible then Ext2 (E, OP3 ) = Ext3 (E, OP3 ) = 0. It follows from Theorems 2 and 3 that there exists a (set theoretical) 1-1 correspondence between special monads and admissible sheaves. As shown by Manin and Drinfeld (see [7]), such correspondence is categorical. Theorem 4. The functor that associates a special monad on P3 to its cohomology sheaf de?nes an equivalence between the categories of special monads and admissible sheaves. We complete this section with an important fact: Proposition 5. If E is an admissible sheaf, then H 0 (P2 , E ? (k )) = 0 for all k ≤ ?1. Proof. E is the cohomology of the monad (2); setting V = H 1(P2 , E (?2)), ′ 1 2 W = H 1 (P2 , E ? ?1 P2 (?1)) and V = H (P , E (?1)), one had the sequences 0 → K(k ) → W ? OP2 (k ) → V ′ ? OP2 (k + 1) → 0 and 0 → V ? OP2 (k ? 1) → K(k ) → E (k ) → 0 . where K = ker{W ? OP2 → V ′ ? OP2 } is a locally-free sheaf. It follows from the ?rst sequence that: H 0 (P2 , K(k )) = 0 ?k ≤ ?1 , H 2 (P2 , K(k )) = 0 ?k ≥ ?2 and H 0 (P2 , K? (k )) = 0 ? k ≤ ?1 , by Serre duality. The proposition then follows easily from the dual of the second sequence. 6

    2

    Examples of admissible sheaves

    Let us now study various examples of admissible sheaves on P3 . Theorem 1 implies that there are admissible coherent sheaves in rank 0 and 1, but there can be no admissible sheaves with zero ?rst Chern class in these ranks, apart from the trivial ones. Examples of admissible sheaves with vanishing ?rst Chern class start in rank 2. The basic one is an admissible torsion-free sheaf E which is not locally-free; it arises as the cohomology E of the monad:
    ⊕4 OP3 (?1) → OP 3 → OP3 (1) ? ? x ? y ? ? α=? ? 0 ? and β = (?y x z w ) 0 α β

    (6)

    It is easy to see that β is surjective for all [x : y : z : w ] ∈ P3 , while α is injective provided x, y = 0. It then follows from Theorem 3 that E is torsion-free, but not locally-free. In particular, the singularity set of E (i.e. the support of E ?? /E ) consists of the line {x = y = 0} ? P3 . Note also that c2 (E ) = 1 and c1 (E ) = c3 (E ) = 0. Re?exive sheaves on P3 have been extensively studied in a series of papers by Hartshorne [6], among other authors. In particular, it was show that a rank 2 re?exive sheaf F on P3 is locally-free if and only if c3 (F ) = 0. Therefore, we conclude: Proposition 6. (Hartshorne [6]) There are no rank 2 admissible sheaves on P3 which are re?exive but not locally-free. The situation for higher rank is quite di?erent, though, and it is easy to construct a rank 3 admissible sheaf which is re?exive but not locally-free. Setting w = 5 and v = v ′ = 1, consider the monad:
    ⊕5 OP3 (?1) → OP 3 → OP3 (1) α β

    (7)

    7

    Again, it is easy to see that β is surjective for all [x : y : z : w ] ∈ P3 , while α is injective provided x, y, z = 0. It then follows from Theorem 3 that EE is re?exive, but not locally-free; its singularity set is just the point [0 : 0 : 0 : 1] ∈ P3 . Note also that c2 (E ) = 1 and c1 (E ) = c3 (E ) = 0. Finally, we give an example of a rank 2 admissible locally-free sheaf. Setting w = 4 and v = v ′ = 1, consider the monad:
    ⊕4 OP3 (?1) → OP 3 → OP3 (1) ? ? x ? y ? ? α=? ? ?w ? and β = (?y x z w ) z α β

    ? ? α=? ? ?

    ?

    x y 0 0 z

    ?

    ? ? ? and β = (?y x z w 0) ? ?

    (8)

    It is easy to see that β is surjective and α is injective for all [x : y : z : w ] ∈ P3 , so E is indeed locally-free; note that c2 (E ) = 1 and c1 (E ) = c3 (E ) = 0. With these simple examples in low rank, we can produce high rank admissible sheaves using the following: Proposition 7. If F ′ and F ′′ are coherent admissible sheaves, its extention E: 0 → F ′ → E → F ′′ → 0 is also admissible. The proof is an easy consequence of the associated long exact sequence in cohomology, and it is left to the reader. As a consequence of Serre duality, we have: Proposition 8. If E is a locally-free admissible sheaf, then E ? is also admissible. 8

    3

    Semistability of torsion-free admissible sheaves

    Recall that a torsion-free sheaf E on P3 is said to be semistable if for every coherent subsheaf 0 = F ?→ E we have ? (F ) = c 1 (E ) c 1 (F ) ≤ = ? (E ) . rk(F ) rk(E )

    Furthermore, if for every coherent subsheaf 0 = F ?→ E with 0 < rk(F ) < rk(E ) we have c1 (F ) c 1 (E ) < , rk(F ) rk(E ) then E is said to be stable. It is also important to remember that: ? E is (semi)stable if and only if E ? is; ? E is (semi)stable if and only if ?(F ) < ?(E ) (?(F ) ≤ ?(E )) for all coherent subsheaves F ?→ E whose quotient E/F is torsion-free; ? E is (semi)stable if and only if ?(Q) > ?(E ) (?(Q) ≥ ?(E )) for all torsion-free quotients E → Q → 0 with 0 < rk(Q) < rk(E ). Furthermore, if E is locally-free, it is enough to test the locally-free subsheaves F ?→ E with 0 < rk(F ) < rk(E ) to conclude that E is stable. The goal of this section is to compare the semistability and admissibility conditions. We provide to positive results for admissible sheaves of rank 2 and 3. Theorem 9. Let E be a semistable torsion-free sheaf with c1 (E ) = 0. E is admissible if and only if H 1 (P3 , E (?2)) = H 2 (P3 , E (?2)) = 0. Furthermore, an admissible torsion-free sheaf is stable if and only if H 0 (P3 , E ) = 0. In other words, if E is a semistable torsion-free sheaf with c1 (E ) = 0 and H 1 (P3 , E (?2)) = H 2 (P3 , E (?2)) = 0, then E is admissible.

    9

    Proof. Semistability implies immediately that H 0 (P3 , E (k )) = 0, for all k ≤ 1. If E is locally-free, then by Serre duality we have H 3 (P3 , E (k )) = 0 for all k ≥ ?3, since E ? is also semistable. If E is torsion-free, we can use the semistability of E ? ? and the sequence 0 → E → E ?? → Q → 0 , Q = E ?? /E to conclude that 3 (P3 , E (k )) =3 (P3 , E ?? (k )) = 0, since Q is supported in dimension less or equal to 1. Now we assume that H 1 (P3 , E (?2)) = H 2 (P3 , E (?2)) = 0, and let ? be a plane in P3 . From he sequence: 0 → E (k ? 1) → E (k ) → E |? (k ) → 0 we conclude that H 0 (?, E |? (?1)) = H 2 (?, E |? (?2)) = 0. Claim. If V is a torsion-free sheaf on P2 with H 0 (P2 , V (?1)) = H 2 (P2 , V (?2)) = 0, then H 0 (P2 , V (k )) = 0 for k ≤ ?1 and H 2 (P2 , V (k )) = 0 for k ≥ ?2. Proof of the claim: For any line ? ? P2 , we have the sequence 0 → V (k ? 1) → V (k ) → V |? (k ) → 0 , so that 0 → H 0(P2 , V (k ?1)) → H 0 (P2 , V (k )) and H 2 (P2 , V (k ?1)) → H 2 (P2 , V (k )) → 0 . The claim follows easily by induction. Returning to (9), we also have: H 0(?, E |? (k )) → H 1 (P3 , E (k ? 1)) → H 1 (P3 , E (k )) so, for k ≤ ?1, if H 1 (P3 , E (k )) = 0, then also H 1 (P3 , E (k ? 1)) = 0. Thus by induction we conclude that H 1 (P3 , E (k )) = 0 for all k ≤ ?2. Similarly, we have: H 2(P3 , E (k ? 1)) → H 2 (P3 , E (k )) → H 2 (?, E |? (k )) 10 (9)

    and again by induction we conclude that H 2 (P3 , E (k )) = 0 for all k ≥ ?2, as desired. The converse statement seems to depend on the rank, as we will see in the two results below. Theorem 10. Every rank 2 admissible torsion-free sheaf E with c1 (E ) = 0 is semistable. Moreover, if H 0 (P3 , E ? ) = 0, then E is stable. Proof. First, assume that L is a rank 2 re?exive sheaf with c1 (L) = 0 and H 0 (L(k )) = 0 for all k ≤ ?1; we show that L is semistable. Indeed, let F ?→ L be a torsion-free subsheaf of rank 1, with torsion-free quotient Q = L/F . By Lemma 1.1.16 in [8, p. 158], it follows that F is also re?exive; but every rank 1 re?exive sheaf is locally-free, thus F = OP3 (d). Any map F → L yields a section in H 0 (P3 , L(?d)), c1 (F ) = d ≤ 0 and E is semistable, being stable if H 0 (P3 , E ? ) = 0 Now if E is a rank 2 admissible torsion-free sheaf with c1 (E ) = 0, then L = E ? is a rank 2 re?exive sheaf with c1 (L) = 0 and H 0 (L(k )) = 0 for all k ≤ ?1, by Proposition 5 and since the dual of any coherent sheaf is always re?exive. Thus E ? is semistable, so E is as well. Clearly, E is stable if H 0 (P3 , E ? ) = 0, as desired. A similar result for rank 3 sheaves requires a stronger hypothesis: re?exivity, rather than torsion-freeness. Theorem 11. Every rank 3 admissible re?exive sheaf E with c1 (E ) = 0 is semistable. Moreover, if H 0(P3 , E ) = H 0 (P3 , E ? ) = 0, then E is stable. Proof. In fact, one can show that every rank 3 re?exive sheaf with c1 (E ) = 0 and H 0 (E (k )) = H 0 (E ? (k )) = 0 is semistable. The desired theorem follows easily from this fact. Indeed, let F ?→ E be a torsion-free subsheaf, with torsion-free quotient Q = E/F , so that c1 (F ) = ?c1 (Q). As in the proof of Theorem 10, it follows that F is re?exive. There are two possibilities: 11

    (i) rank F = 1. In this case, F is locally-free, so a map F → E yields a section in H 0 (P3 , E (?d)), where d = c1 (F ). Hence c1 (F ) ≤ 0. (ii) rank F = 2, so rank Q = 1. Now Q? is a re?exive (hence locally-free) subsheaf of E ? , which gives a section in H 0 (P3 , E ? (?d)), where d = c1 (Q? ) = c1 (F ). Hence c1 (F ) ≤ 0. It follows that E is semistable, being stable if H 0 (P3 , E ) = H 0 (P3 , E ? ) = 0. Together with Theorem 9, we conclude that: ? a rank 2 torsion-free sheaf on P3 with c1 (E ) = 0 is admissible if and only if it is semistable and H 1 (P3 , E (?2)) = H 2 (P3 , E (?2)) = 0; ? a rank 3 re?exive sheaf on P3 with c1 (E ) = 0 is admissible if and only if it is semistable and H 1 (P3 , E (?2)) = H 2 (P3 , E (?2)) = 0.

    4

    Trivial splitting type

    Since every locally-free sheaf on a projective line splits as a sum of line bundles, one can study sheaves on projective spaces by looking into the behavior of restriction to a line [8]. De?nition. A torsion-free sheaf E on P3 is said to be of trivial splitting type if there is a line ? ? P3 such that E |? is the trivial locally-free sheaf, i.e.
    ⊕rkE E |? ? O? .

    A sheaf of trivial splitting type necessarily has vanishing ?rst Chern class. Note that, by semicontinuity, if E is of trivial splitting type then E |? is trivial for a generic line in P3 . Torsion-free sheaves of trivial splitting type where completely classi?ed in [4], and they were shown to be closely related with a complex version of the celebrated Atiyah-Drinfeld-Hitchin-Manin matrix equations.

    12

    Furthermore, every torsion-free sheaf of trivial splitting type is semistable; indeed, assume that E has rank r , and let F ?→ E be a coherent subsheaf of rank s, with torsion-free quotient E/F . Then on a generic line ? ? P3 we have:
    ⊕r F? = O? (a1 ) ⊕ · · · ⊕ O? (as ) ?→ E |? ? O? ,

    where c1 (F ) = a1 + · · · + as . It follows that c1 (F ) ≤ 0, since we must have ak ≤ 0, k = 1, . . . , s. Theorem 12. Let E be a torsion-free sheaf of trivial splitting type. E is admissible if and only if H 1 (P3 , E (?2)) = H 2 (P3 , E (?2)) = 0. Of course, this is an easy consequence of Theorem 9 and the observation above, but here is a direct proof. Proof. Let E be an admissible torsion-free sheaf. Without loss of generality, we can assume that E |?∞ is trivial for ?∞ = {z = w = 0}. Let ? be a plane containing ?∞ , e.g. ? = {z = 0}. Then E |? is a torsion-free sheaf on ? which is trivial at ?∞ . From the proof of Theorem 9 we know that: H 0 (?, E |? (k )) = 0 ?k ≤ ?1 , H 2 (?, E |? (k )) = 0 ?k ≥ ?2 Now consider the sheaf sequence: 0 → E (k ? 1) ?→ E (k ) ?→ E |? (k ) → 0 Using (10), we conclude that: H 3 (P3 , E (k )) = H 3(P3 , E (k ? 1)) ?k ≥ ?2 But, by Serre’s vanishing theorem, H 3 (P3 , E (N )) = 0 for su?ciently large N , thus H 3(P3 , E (k )) = 0 for all k ≥ ?3. Similarly, we have: H 0 (P3 , E (k ? 1)) = H 0 (P3 , E (k )) ?k ≤ ?1 13
    ·z

    (10)

    (11)

    Since E ?→ E ?? , we have via Serre duality: H 0 (P3 , E (k )) ?→ H 0 (P3 , E ?? (k )) = H 3 (P3 , E ??? (?k ? 4))? . Thus, again by Serre’s vanishing theorem, H 0 (P3 , E (?N )) = 0 for for su?ciently large N , so that H 0 (P3 , E (k )) = 0 for all k ≤ ?1. We also have that: 0 → H 1 (P3 , E (k ? 1)) → H 1 (P3 , E (k )) ?k ≤ ?1 hence H 1 (P3 , E (?2)) = 0 implies that H 1 (P3 , E (k )) = 0 for all k ≤ ?2. Furthermore, H 2 (P3 , E (k ? 1)) → H 2 (P3 , E (k )) → 0 ?k ≥ ?2 forces H 2 (P3 , E (k )) = 0 for all k ≥ ?2 once H 2 (P3 , E (?2)) = 0. As in the previous section, the converse statement seems to depend on the rank. The generic splitting type of a semistable locally-free sheaf with vanishing ?rst Chern class is determined by Theorem 2.1.4 in [8, p. 205-206]. In particular, it follows that every semistable rank 2 locally-free sheaf is of trivial splitting type. Thus, from Theorem 10, we conclude: Theorem 13. Every rank 2 admissible locally-free sheaf E with c1 (E ) = 0 is of trivial splitting type. To explore a few easy consequences of the classical theory of locally-free sheaves on complex projective spaces, let G denote the Grasmannian of lines in P3 . De?nition. Let E be a locally-free sheaf of trivial splitting type. The set JE = {? ∈ G | E? is not trivial} is called the set of jumping lines of E ; it is always a closed subvariety of G. Moreover, E is said to be uniform if JE is empty, i.e. if E? is independent of ? ∈ G. 14

    Theorem 14. Every rank 2 uniform, admissible locally-free sheaf E with c1 (E ) = 0 is trivial. Proof. By Theorem 13, E is of trivial splitting type. Since E is uniform, E? must be trivial for all ? ∈ G. It then follows from Theorem 3.2.1 in [8, p. 51] that E is trivial. Our last result, regarding the set of jumping lines, follows from Theorem 2.2.3 in [8, p. 228]. Theorem 15. If E is an rank 2 admissible locally-free sheaf with c1 (E ) = 0, then its set of jumping lines JE is a subvariety of pure codimension 1 in G.

    5

    Open problems

    The results proved in this paper point to a number of quite interesting questions and possible generalizations. First of all, we expect that if E is a properly torsion-free or properly re?exive admissible sheaf, then its dual E ? is not admissible, but we have not been able to construct any examples. We would also like to see whether the results in Section 3 can be generalized to higher rank. It seems too much to expect every admissible sheaf to be semistable; but the correspondence between instantons and locally-free admissible sheaves makes the statement ”every admissible locally-free sheaf is semistable” an attractive conjecture. On the other hand, is Theorem 11 optimal, i.e. is there a rank 3 torsion-free admissible sheaf which is not semistable? It would also be interesting to study the connection between admissibility and Gieseker stability. Since every Gieseker semistable sheaf on a projective space is also semistable, we conclude from Theorem 9 that every Gieseker semistable torsion-free sheaf is admissible; one would like to determine to what extent the converse is also true.

    15

    Theorems 13 and 14 point to interesting properties of higher rank admissible sheaves: is every admissible locally-free sheaf with vanishing ?rst Chern class of trivial splitting type? Is every uniform, admissible locally-free sheaf with vanishing ?rst Chern class trivial? We’ve also seen that if E is an admissible torsion-free and ? is a plane in P , then the restriction E |? satis?es the following cohomological condition:
    3

    H 0(?, E |? (k )) = 0 ?k ≤ ?1 , H 2(?, E |? (k )) = 0 ?k ≥ ?2 . A sheaf on P2 satisfying the above conditions are called instanton sheaves, and are very interesting on their own right, also being closely related to instantons. The analysis of how the instanton condition compares with semistability and trivial splitting type is work in progress [5], but many of the results proved here have their analogs for instanton sheaves in P2 . In particular, it is shown in [5] that every instanton sheaf is the cohomology of a special monad, and that every rank 2 torsion-free instanton sheaf is semistable. We can then conjecture that if a (rank 2) torsion-free sheaf E on Pk is the cohomology of a special monad, then E is semistable; this is true for k = 2, 3.

    References
    [1] Atiyah, M., Drinfeld, V., Hitchin, N., Manin, Yu.: Construction of instantons. Phys. Lett. 65A, 185-187 (1978) [2] Coand? a, I., Tikhomirov, A., Trautmann, G.: Irreducibility and smoothness of the moduli space of mathematical 5-instantons over P3 . Int. J. Math. 14, 1-45 (2003) [3] Floystad, G.: Monads on projective spaces. Comm. Algebra 28, 55035516 (2000) [4] Frenkel, I., Jardim, M.: Complex ADHM equations, sheaves on P3 and quantum instantons. Preprint math.RT/0408027. 16

    [5] Jardim, M.: Instanton sheaves on P2 . In preparation. [6] Hartshorne, R.: Stable re?exive sheaves I, II, III. Math. Ann. 254, 121176 (1980) Invent. Math. 66, 165-190 (1982) Math. Ann. 279, 517–534 (1988) [7] Manin, Yu.: Gauge ?eld theory and complex geometry. Berlin: SpringerVerlag, 1997 (second edition) [8] Okonek, O., Schneider, M., Spindler, H.: Vector bundles on complex projective spaces. Boston: Birkhauser (1980) [9] Siu, Y.-T., Trautmann, G.: Gap-sheaves and extension of coherent analytic subsheaves. Lec. Notes Math, 172. Berlin: Springer-Verlag (1971) [10] Ward, R., Wells, R.: Twistor geometry and ?eld theory. Cambridge: Cambridge University Press (1990)

    17


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