Admissible subcategories in derived categories of moduli of vector bundles on curves

We show that the Poincar\'e bundle gives a fully faithful embedding from the derived category of a curve of sufficiently high genus into the derived category of its moduli space of bundles of rank $r$ with fixed determinant of degree 1. Moreover we show that a twist of the embedding, together with 2 exceptional line bundles, gives the start of a semi-orthogonal decomposition. This generalises results of Narasimhan and Fonarev-Kuznetsov, who embedded the derived category of a single copy of the curve, for rank 2.


Introduction
Let C be a smooth projective curve of genus ≥ 2 and let r ≥ 2 and d be integers such that gcd(r , d) = 1. Then the moduli space M C (r , L) of stable vector bundles of rank r and fixed determinant L of degree d is a smooth and projective Fano variety, such that Pic M C (r , L) ZΘ. Moreover there exists a universal vector bundle W (called the Poincaré bundle) on C × M C (r , L). We can use W to construct a Fourier-Mukai functor Φ W from the derived category of the curve to the derived category of the moduli of vector bundles.
In [14] Narasimhan showed that the Fourier-Mukai functor Φ W is fully faithful, if r = 2, d = 1, ≥ 4 and W is suitably normalized (see equation (6)). His proof uses the Hecke correspondence to check the Bondal-Orlov criterion. Independently, in [9] Fonarev-Kuznetsov showed the fully faithfulness for r = 2, d = 1 and ≥ 2 when C is a generic curve. Their proof involves an explicit model by Desale-Ramanan [7] of M C (2, L) when C is hyperelliptic, and checking the Bondal-Orlov criterion in this special case using the Borel-Weil-Bott theorem.
The first result in this article is a generalisation of these fully faithfulness results: the rank r is now arbitrary, the degree d is 1, and ≥ 0 , where 0 is the smallest integer such that (1) 2r 0 − 2(r + 0 ) ≥ r 2 − 1.
In particular when r = 2, we get that 0 is 4. The vanishing results in theorem 3 together with the Bondal-Orlov criterion then imply the following.
Theorem A. The Fourier-Mukai transform Φ W gives a fully faithful embedding for any smooth curve C of genus ≥ 0 .
In [14, remark 4] Narasimhan explains that the results in op. cit. also show that is the start of a semiorthogonal decomposition of D b (M C (r , L)). The second result in this article is a generalisation and more interestingly an extension of this result to higher rank: we exhibit a second copy of the derived category of the curve. To do this we twist the functor Φ W by Θ ∨ , or equivalently we consider the Fourier-Mukai functor associated to W ⊗ p * 2 (Θ ∨ ). Theorem B. Let C be any smooth curve of genus ≥ 0 , then there exists a semiorthogonal decomoposition of the form where A is the left-orthogonal complement to the admissible subcategory generated by the 2 exceptional objects and the 2 copies of D b (C).

Remark 1.
This result is already new in the case of r = 2. In a work in progress joint with Sergey Galkin [3] we are studying D b (M C (2, L)) from the point of view of mirror symmetry for Fano varieties: 1. quantum cohomology can be used to give expectations on natural semiorthogonal decompositions; 2. there are various conjectures [10] regarding the eigenvalues of the quantum multiplication c 1 (X ) * − that can be checked for M C (2, L).
The suggested decomposition into indecomposable pieces (recall that D b (C) is indecomposable by [17]) will involve symmetric powers Sym i C for i ≤ − 1. For r ≥ 3 the picture becomes more complicated, and it is currently unclear what the systematic description could be.
The Bondal-Orlov criterion To check fully faithfulness of the Fourier-Mukai functor Φ W , we will use following criterion due to Bondal-Orlov [5, theorem 1.1]. We denote the skyscraper at a point x by k(x).
Vanishing results To apply the Bondal-Orlov criterion in the proof of theorem A we need the following generalisation of [14, parts (1), (2) and (3) of theorem 1.2]. For any point z ∈ C, we denote by W z the restriction of W to {z} × C.
Theorem 3. Let C be a smooth projective curve of genus ≥ 0 , and L a line bundle of degree 1. Let W be the normalised Poincaré bundle on C × M C (r , L). Then For the proof of the semiorthogonal decomposition of theorem B we need another set of vanishing results, which generalise [14, parts (4) and (5)  Theorem 4. Let C be a smooth projective curve of genus ≥ 0 , and L a line bundle of degree 1. Let W be the normalised Poincaré bundle on C × M C (r , L). Then for any x, ∈ C and all i ≥ 0. Remark 5. If r = 2 then parts (1) and (2) are related via Serre duality, but in higher rank this is no longer the case. In arbitrary rank, (2) and (2') are related via Serre duality.
Structure of the paper We prove parts (1) and (2) of theorem 3 in section 3. The proof of part (3) is more technical, and occupies sections 5 and 6. The structure of the proof is as in [14], but we highlight the complications arising in the higher rank case, and why one only gets a proof in the case where d = 1.
The proof of parts (1) and (2) (resp. (2')) of theorem 4 is done in section 7, where we prove a more general vanishing statement for exterior powers of W ∨ x , along the same lines as sections 5 and 6. Finally, the proof of part (3) occupies section 8.
In section 9 we explain how theorems A and B follows from theorems 3 and 4, and give some concluding remarks.
Acknowledgements Both authors were supported by the Max Planck Institute for Mathematics in Bonn. They thank Patrick Brosnan and Sergey Galkin for interesting conversations.

Preliminaries and notation
Let k be an algebraically closed field of characteristic 0. Throughout this article we will take C a smooth projective curve over k of genus ≥ 2. Associated to a rank r and degree d such that gcd(r , d) = 1 there exists a smooth projective moduli space M C (r , d) of dimension r 2 ( − 1) + 1, with a determinant morphism to Pic d (C). Fixing a line bundle L of degree d on C we can consider the fibre over the point [L] in Pic d (C), which will be denoted M C (r , L). Now M C (r , L) has Pic(M C (r , L)) ZΘ with Θ the ample generator of the Picard group, such that In particular, M C (r , L) is a (r 2 − 1)( − 1)-dimensional Fano variety of index 2. We will also use the appropriate analogues of these results when gcd(r , d) ≥ 2. For a proof one is referred to [8].
As M C (r , L) is a fine moduli space, there exists a universal family W on C × M C (r , L), the Poincaré bundle. This universal family is unique if we normalise it as in [18, remark 2.9]: if we denote ℓ ≥ 0 the minimal integer such that ℓd ≡ 1 mod r then we can assume Remark 6. The dependence on ℓ is the reason why we have to restrict to d ≡ 1 mod r . Remark that by Serre duality and tensoring with O C (x) we can always assume d ∈ [0, . . . , r /2], as one can identify the moduli spaces for different d, so we will use d = 1. We expect that for other residue classes the result still holds.
Using W and the projections we can then construct the Fourier-Mukai functor

Generalisation of a result by Narasimhan-Ramanan
In this section we prove parts (1) and (2) of theorem 3. At this point it is not required that d = 1, it is enough that gcd(r , d) = 1.
For (1) it suffices to observe that the result cited in [14] (i.e. [15, theorem 2(b)]) is already valid for all ranks. However for the proof of (2) [15, proposition 7.7] is used, which is only given for = 2. We will show that the expectation expressed in [15, remark 7.2] is correct, and that the vanishing of cohomology of ad x W (i.e. the restriction to {x } × M C (r , L) of the traceless endomorphisms of W) is indeed valid in arbitrary rank, using some new information on Hodge numbers which was not available when op. cit. was published.
Proposition 7. Let C be a smooth projective curve of genus ≥ 2, and L a line bundle of degree d on C such that gcd(r , d) = 1. Let W be the normalised Poincaré bundle on C × M C (r , L), where gcd(r , d) = 1. For all x ∈ C and i ≥ 2 we have that The proof of [15, proposition 7.7] can be generalised to arbitrary rank, provided one has control over the Hodge numbers h 1,i (M C (r , L)). In order to do this, we will use the closed formula for the Hodge-Poincaré polynomial as obtained in del Baño [2, corollary 5.1]. Recall that the Hodge-Poincaré polynomial HP(X , x, ) of a smooth projective variety X is given by p,q ≥0 h p,q (X )x p q .
The formula in loc. cit. gives the Hodge-Poincaré polynomial of M C (r , d), and fixing the determinant changes the Hodge-Poincaré polynomial by removing a factor (1+x) (1+ ) arising from the Jacobian of C 1 . Taking these observations into account, and denoting α the decimal part of a real number α, we have where we sum over all compositions of r .
From this we can read off the following dimensions.
Lemma 8. We have that Proof. We observe that the only composition r = r 1 + . . . + r ℓ contributing to the coefficient of x i is the composition with ℓ = 1. Indeed, developing all the denominators as a power series in x and multiplying them, we see that factor on the fourth line of (10) is zero for ℓ = 1 and bounded below by 2 for ℓ ≥ 2. Hence for the equalities in the lemma, it suffices to understand the contribution with ℓ = 1.
In this case the only non-trivial factor in (10) is the second one. It now suffices to observe that there will not be a monomial , that x appears with coefficient 1 by the factor (1 − x ) in the denominator for i = 1, that x 2 appears with coefficient by the factor (1 + x 2 ) for i = 2 and that there cannot be any monomials of the form x i for i ≥ 3.
Proof of part (2) of theorem 3. It suffices to combine proposition 7 with the defining short exact sequence and the vanishing of the higher cohomology of the structure sheaf as M C (r , L) is Fano.
Remark 9. One can actually prove more. Recall that the level of a Hodge structure is defined as the maximum value of |p − q| as (p, q) ranges over h p,q 0. In [6] it was shown that for M C (2, L) the level of H i (M C (2, L), Q) was bounded above by ⌊i/3⌋. For r ≥ 3 the same is true, which immediately implies lemma 8. For i ≫ 0 this bound can even be improved, but we haven't found a nice closed formula for it.

Determinant of cohomology
One of the main ingredients in the proof of part (3) gives an isomorphism Let us denote the ample generators of Pic(M C (r , L)) (resp. Pic(M C (r , L ∨ ))) by Θ (resp. Θ ′ ). By [14, proposition 2.1] we get that Θ ℓ as explained in section 2, where ℓ is the minimal non-negative integer such that ℓd ≡ 1 mod r . Hence we obtain This proves the proposition.
In appendix A we give an alternative proof of this result using the Grothendieck-Riemann-Roch formula.

Cohomology vanishing for W x ⊗ W ∨ in high degrees
From this point on we will impose the condition that d = 1. As mentioned before, we expect the result is still valid for any degree coprime to the rank, but leave this for future work.
We will split the proof of the vanishing as stated in part (3) of theorem 3 into two parts: first we show it for i ≥ r 2 , and then we deal with the vanishing in low degrees in section 6. For both parts we use the Hecke correspondence, which we will recall now.
Hecke correspondence Let x ∈ C be a closed point. If we restrict the Poincaré bundle W to {x }×M C (r , L), we can consider the projective bundle π : P(W x ) → M C (r , L). We will denote the projective bundle by Q (W, x).
A point q on Q(W, x) corresponds to the vector bundle W π (q) together with a morphism W π (q) ։ k(x). We can take the dual of the kernel, which is a vector bundle of rank r and determinant L ∨ ⊗ O C (x) of degree 1 − d. The variety Q(W, x) can be seen to parametrise a family of such bundles. However in general the family of vector bundles may not be semistable.
Consider the short exact sequence on a curve C, where E 1 and E are vector bundles of rank r and deg(E) = 1. Then we have the following: But this is impossible since 0 < r ′ < r and deg(F ′ ) > 0.
The above lemma shows that Hecke transforms for degree one bundles preserves semistability. Thus we get a diagram Remark 12. When d 1 we don't have a well-defined morphism ψ in (18), there is only a rational morphism. It is conceivable that using parabolic bundles it is possible to resolve the indeterminacy, and continue the proof in this way. We leave this for future work.
With the above considerations, the statement and proof of [14, proposition 3.3] generalises to the following.
Lemma 13. Let Θ ′ be the ample generator of Pic(M C (r , L ∨ ⊗ O C (x)). Then The following proposition is a generalisation of [14, proposition 3.2].
Proposition 14. Let C be a smooth projective curve of genus ≥ 2, and L a line bundle of degree 1 on C. Let W be the normalised Poincaré bundle on C × M C (r , L).
For all x, ∈ C such that x the vector bundle W x ⊗ W ∨ ⊗ ω ∨ M C (r, L) is ample.
Proof. By lemma 13 we get that O Q (W, x ) (1) is nef, hence W x is nef. By [13, proposition 6.2.12(iv)] the exterior power r −1 W x is again nef, and so i W x ⊗ Θ is ample.
Using the ampleness of (20) we can conclude that is again the tensor product of a nef bundle with an ample bundle, so is ample itself.
From this we get the vanishing of W x ⊗ W ∨ in high degree.
Corollary 15. We have that for i ≥ r 2 .
Proof. It suffices to apply Le Potier vanishing [13, theorem 7.3.5] to the ample vector bundle W x ⊗ W ∨ ⊗ ω ∨ M C (r, L) of rank r 2 .

Cohomology vanishing for W x ⊗W ∨ in low degrees
To finish the proof of part (3) theorem 3 we need to show vanishing in degrees i ≤ r 2 −1.
For this we will use the morphism ψ from (18).
of stable vector bundles, the complement being the strictly semistable locus.
Remark 16. If we take a stable bundle V ∈ M C (r , L ∨ ⊗ O C (x)), then by [16, remark 5.2(v) and lemma 5.6(i)] the Hecke transform of V is also stable.
Using remark 16, the proof of [14, proposition 3.4] also gives a proof of the following generalisation.
Proposition 17. The restriction of ψ to M s C (r , L ∨ ⊗ O C (x)) is a P r −1 -fibration. Moreover, for every point ∈ C \ {x } the restriction of π * (W ) to a fibre of ψ in the stable locus is isomorphic to O P r −1 (1) ⊕r .
For completeness' sake, we give the proof.
Proof. Consider a point m ∈ M s C (r , L ∨ ⊗ O C (x)), which corresponds to a stable vector bundle V on C of rank r and determinant L ∨ ⊗ O C (x). We wish to show that Let us denote V x = V x ⊗ k(x) the fibre of V at x. Then as in [15, §4] we obtain a family of vector bundles K(V) on C parametrised by P(V x ) P r −1 , where T = Spec k is just a point. It should be mentioned that we will also use results from [16, §5], and it is important to highlight remark 5.7 of op. cit., which explains how the results in §5 of op. cit. are to be interpreted in the context of [15, §4], in particular we have x ∈ C fixed.
As K(V) is a family of stable vector bundles of rank r and determinant L, the universal property of M C (r , L) gives us a morphism which is a closed immersion by [16, lemma 5.9], where in the setting of loc. cit. we restricted the morphism to the point x ∈ C. The same universal property gives us an isomorphism is the projection onto the first factor. We wish to determine the line bundle ξ more explicitly, and we do this by computing the restriction f * (Θ). On the other hand, using (26) we get [15, remark 4.7], we get Hence ξ O P(V x ) (1), and (26) gives Finally, similar to [15, §5] there exists a commutative diagram (32) such that the image of F is identified with ψ −1 (m). But then (24) follows.
We will need the following codimension estimate for the preimage under ψ for the strictly semistable locus. The proof is similar to that in [19, §2.V.A], but we will now fix the determinant.
Proof. Let M be a line bundle of degree m. Recall that polystable bundles in are direct sums of stable bundles with the same slope. If we denote a = gcd(r , m) ≥ 2 and r 0 = r /a we get where we can choose the determinant for one of the summands freely.
In the situation of (18) we have d = 1, and hence r 0 = 1. By a generalisation of the proof of [19, lemma 5], it follows that the fibers of the map ψ are at most (r −1)-dimensional. Thus we see that the dimension ofψ −1 (K) is at most (2+r 2 −2r −1)( −1)+1+(r −1). Hence we get that Remark 19. If r = 2, it is known that ψ is surjective by [16, lemma 7.3] and the inequality in the statement of lemma 18 is an equality.
In corollary 15 we have shown the vanishing of H i (M C (r , L), W x ⊗ W ∨ ) for i ≥ r 2 .
We can now show vanishing for i ≤ r 2 − 1.
Proof of part (3) of theorem 3. By the Leray spectral sequence for ψ and (19) we get that Let 0 be the smallest integer such that where we have used the codimension estimate from lemma 18. Then by [11,§III.3] we have an inclusion for i ≤ r 2 − 1.
By proposition 17 we have that ψ restricted to U is a P r −1 -fibration. Applying the projection formula and using that the fibers of π * (W ∨ ) restricted to ψ −1 (p) for p ∈ M s C (r , L ∨ ⊗ O C (x)) are isomorphic to O P r −1 (−1) ⊕r we are done.

Cohomology vanishing for W ∨
x and W x ⊗ Θ ∨ In this section we prove parts (1), (2) and (2') of theorem 4. The proof goes along the same lines as the proof of part (3) of theorem 3, and uses ingredients from sections 5 and 6.
To prove cohomology vanishing for W ∨ x and W x ⊗ Θ ∨ we prove the following more general vanishing statement for j W ∨ x with j = 1, . . . , r −1. Setting j = 1 and j = r −1 then implies the result, as Proof. The strategy of the proof is similar to the proof of part (3) of theorem 3. First we claim that j W ∨ x ⊗ ω M C (r, L) is ample. This follows from the isomorphism and the fact that W x is nef, using lemma 13, as in the proof of proposition 14 Thus by Le Potier vanishing [13, theorem 7.3.5] we get that for i ≥ r j , as in section 5. Thus now we only have to prove vanishing when i < r j . As in section 6 the Leray spectral sequence tells us where z is any point on C \ {x }.
Assuming that ≥ j and using the codimension estimate of lemma 18 together with [11,§III.3], we obtain an inclusion for any i < r j and U as in lemma 18. Now by proposition 17, we get that ψ restricted to U is a P r −1 -fibration.
Moreover by proposition 17, we see that the fiber of π * (W ∨ x ) restricted to ψ −1 (p) for p ∈ M s C (r , L ∨ ⊗ O C (z)) is isomorphic to O P r −1 (−1) ⊕r . In particular, the fibers of j π * (W ∨ x ) restricted to ψ −1 (p) are direct sums of line bundles of degree −j. This implies that R k ψ * ( j π * (W ∨ x )) is zero, hence we are done.

Cohomology vanishing for
In this section we discuss the vanishing result of part (3) of theorem 4 required for the proof of theorem B. We will recall the notion of k-ample vector bundles, as the role of Le Potier vanishing in the proof of theorem 3 will be replaced by Sommese vanishing.
Definition 21. Let X be a projective variety. A line bundle L is said to be k-ample if 1. some power of L is globally generated, i.e. L is semi-ample; 2. the fibers of the morphism have dimension at most k.
A vector bundle E on X is said to be k-ample if the line bundle O P(E) (1) is k-ample.
Proposition 22. Let C be a smooth projective curve of genus ≥ 2 and L a line bundle of degree 1. Let W be the normalised Poincaré bundle on C × M C (r , L). Then W x is (r − 1)-ample.
Proof. By the generalisation of [19, lemma 5] we know that the fibres of the morphism ψ in the Hecke correspondence of (18) are at most (r − 1)-dimensional. Moreover, we have already (see 19) shown that where Θ ′ is the ample generator of the Picard group of M C (r , L ∨ ⊗ O C (x ) ). Since Θ ′ is (0-)ample, we get by [13, example 6.2.19(ii) In particular this implies that W x is (r − 1)-ample.
We will use this to prove part (3) of theorem 4.
Proof of part (3) of theorem 4. Let x and be any two points of C, not necessarily distinct, and denote By proposition 22 we know that W x and W are (r − 1)-ample. Now by [12, theorem 3.5] we get that both r −1 W x and W ⊗ r −1 W x are (r − 1)-ample. By our choice of normalisation of W there is a natural isomorphism Thus, we can conclude that the vector bundle V is (r −1)-ample. It follows by Sommese vanishing [20, proposition 1.14], that for all i ≥ r 2 + (r − 1).
Since ω ∨ M C (r, L) is isomorphic to Θ ⊗2 we get by Serre duality that The same argument shows that Combining the above with (52), we get that if (r 2 − 1)( − 1) ≥ 2(r 2 + r − 1), then the cohomology H i (M C (r , L), V) vanishes for all i. But this is satisfied for all ≥ 4.

Concluding remarks
The proofs of theorems A and B We can now explain how the vanishing theorems imply the fully faithfulness of the Fourier-Mukai functor, and how they give the semiorthogonal decomposition of (4).
Proof of theorem A. The Bondal-Orlov criterion from proposition 2 can be applied using theorem 3.
For the proof of theorem B we need to use the following lemma. For sake of notational simplicity a spanning class will be closed under shifts.
Proof. We need to check that for all A ∈ A and B ∈ B. Applying the adjunction G ⊣ G R this is equivalent to G R •F (A) being isomorphic to 0 for all A. This in turn is implied by where T runs over a spanning class for B. Now applying G ⊣ G R and F L ⊣ F , this is equivalent to which is equivalent to F L • G(T ) being isomorphic to zero for all T . This in turn is implied by where S runs over a spanning class for A.
Proof of theorem B. The orthogonality criterion from lemma 23 can be applied to the images of D b (C) under Φ W and Φ W ⊗ Θ ∨ by using the spanning class given by the skyscrapers, and using part (3) of theorem 4.
The other orthogonality checks (there are 5 more) follow from Kodaira vanishing, and parts (1) and (2) of theorem 4.
On theorem B for rank 2 and genus 2 It is expected that in theorem A the condition on the genus is not essential, and combining [14, remark 5] with [9, theorem 1.1] we know that the functor Φ W is fully faithful for r = 2 and all ≥ 2, so a posteriori we can conclude the vanishing results in theorem 3.
In particular, theorem B cannot hold for r = 2 and = 2, so we cannot have the vanishing result in part 3 of theorem 4. We leave it to the interested reader to compute directly on the intersection Q 1 ∩ Q 2 of 2 smooth quadrics in P 5 that the sheaf cohomology of the tensor product of the restriction of (dual) spinor bundles twisted by O Q 1 ∩Q 2 (−1) is non-zero.
On the other hand, for all other combinations of rank and genus (and degree) it is expected that theorem B holds.
Generalised Picard bundles The fully faithfulness result from theorem A allows us to reprove known results on the inversion of generalised Picard bundles, and their deformation theory, originally proven in [1,4]. This is remarked upon in [14, remark 1] in the case when r = 2. We will now give some details in the more general case here.
Definition 24. Let E be a semistable vector bundle of rank n and degree e, such that (61) re + n > rn(2 − 2).
Then the Fourier-Mukai transform Φ W (E) is again a vector bundle, of rank re+n+rn(1− ), called a generalised Picard bundle.
The first result that follows from theorem A is [4, theorem 19]. Recall that the kernel W R for the right adjoint of Φ W is given by W ∨ ⊗ p * 1 (ω C ) [1]. By fully faithfulness we have a natural equivalence Proposition 25 (Inversion formula). Let E be a vector bundle as in definition 24. Then there exists an isomorphism (62) E R 1 p 1, * p * 2 (p 2, * (p * 1 (E) ⊗ W) ⊗ W ∨ ⊗ p * 1 (ω C ) .
Similarly we can describe the deformation theory of generalised Picard bundles as in [4, theorem 22]. As the infinitesimal deformation theory of any sheaf E on a smooth projective variety X is described by H i (X , End(E)) for i = 0, 1, 2, the fully faithfulness of Φ W gives the isomorphism in the next theorem. So the (infinitesimal) deformation theory of a generalised Picard bundle agrees with that of the original bundle. In particular, if E is simple, then by Riemann-Roch the deformation space has dimension n 2 ( − 1) + 1.

A An alternative proof of proposition 10
Because M C (r , L) is Fano, we have an isomorphism Pic(C × M C (r , L)) Pic C ⊕ Z.
We start with a preliminary lemma.
Restricting ourselves to the terms that contribute to the part in degree 2 we obtain (71) p 2, * (ch(End(W)) ⊗ p * 1 td C ) deg=2 We will now apply the normalisation for the Poincaré bundle W. where in the last step we used lemma 27. Now proposition 10 follows from (6).