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A FOURIER TRANSFORM FOR HIGGS BUNDLES

arXiv:math/0104253v1 [math.AG] 26 Apr 2001

J. BONSDORFF A BSTRACT. We de?ne a Fourier-Mukai transform for Higgs bundles on smooth curves (over C or another algebraically closed ?eld) and study its properties. The transform of a stable degree-0 Higgs bundle is an algebraic vector bundle on the cotangent bundle of the Jacobian of the curve. We show that the transform admits a natural extension to an algebraic vector bundle over projective compacti?cation of the base. The main result is that the original Higgs bundle can be reconstructed from this extension.

C ONTENTS 1. Introduction Notation and conventions 2. Fourier-Mukai transforms 2.1. A base change result 2.2. Integral transforms 2.3. WIT complexes 2.4. Fourier transform for curves 3. Transforms of Higgs bundles 3.1. De?nitions and basic properties 3.2. Invertibility References 1 3 3 4 6 8 10 12 12 17 19

1. I NTRODUCTION Higgs bundles on a compact Riemann surface X are pairs (E , θ ) consisting of a holomorphic vector bundle E and a holomorphic one-form θ with values in End(E ) on X . They originated essentially simultaneously in Nigel Hitchin’s study [13] of dimensionally reduced self-duality equations of Yang-Mills gauge theory and, over general K?hler manifolds, in Carlos Simpson’s work [26] on Hodge theory. To explain Hitchin’s point of view, we consider solutions of the self-dual YangMills equations on R4 that are invariant under translations in one or more directions in R4 . Invariance in one direction reduces the SDYM equations to the Bogomolnyi equations describing magnetic monopoles in R3 , while invariance in three directions leads to Nahm’s equation on R. Invariance in two directions leads to the conformally invariant Hitchin’s equations (or Higgs bundle equations), which the conformal invariance allows to be considered on compact Riemann surfaces. A

2000 Mathematics Subject Classi?cation. Primary 14F05, 14H60; Secondary 14H40. Key words and phrases. Higgs bundles, Mukai transform, Fourier transform. The author was supported by The Finnish Cultural Foundation, The V?is?l? Foundation, and The Scatcherd European Scholarship.

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2

J. BONSDORFF

solution to Hitchin’s equations has an interpretation as a triple (E , θ , h) with E a holomorphic vector bundle, θ a holomorphic section of End(E ) ? ωX and h a Hermitian metric on E , satisfying F + [θ , θ ? ] = 0, where F is the curvature of the connection determined by the metric. It was shown in Hitchin [13] for rank-2 bundles on Riemann surfaces, and in Simpson [26] in general, that a pair (E , θ ) admits a unique such metric precisely when E has vanishing Chern classes and the pair (E , θ ) is stable in a sense which generalises the usual stability for vector bundles. For an excellent overview of Simpson’s viewpoint of non-Abelian cohomology, see Simpson [28]. An important class of transforms in Yang-Mills theory, including the ADHM construction and the Fourier transform for instantons (Donaldson-Kronheimer [6]) and the Nahm transform for monopoles, is based on using the kernel of the Dirac operator coupled to the (twisted) connection. A recent work of Marcos Jardim [16, 17] uses a version of the Nahm transform to establish a link between singular Higgs bundles on a 2-torus and doubly-periodic instantons. Our goal is to generalise this work to Riemann surfaces of genus g ≥ 2. In this paper we shall consider the purely holomorphic aspect of the transform; we plan to return to the properly gauge-theoretic questions in a future paper. The holomorphic side of the Nahm transform is captured by the (generalised) Fourier-Mukai transform, which originated in the work of Shigeru Mukai [22] on ? its dual, and let D(X ) and Abelian varieties. Let X be a complex torus and X ? ) denote the derived categories of the categories of coherent sheaves on X and D(X ? ? , Mukai de?ned a functor X respectively. Using the Poincaré sheaf P on X × X ? M : D(X ) → D(X ) by

? M (?) = RprX ? ? (prX (?) ? P ),

and showed that it is a category equivalence. This construction can be generalised to any varieties X and Y together with a sheaf P on X × Y . The properties of these generalisations have been studied by A. Bondal and D. Orlov [1, 2], A. Maciocia [19], T. Bridgeland [4, 5], and others. We interpret the endomorphism-valued one-form θ as a bundle map, making a Higgs bundle (E , θ ) into a sheaf complex E → E ? ωX , where ωX is the canonical sheaf of X . Hence a Higgs bundle gives us an object of the derived category D(X ), and we can use the general machinery of Fourier-Mukai transforms. The key observation is that it is necessary to consider relative transforms of families of Higgs bundles twisted by adding a scalar term to the Higgs ?eld θ . Our transform produces sheaves on J(X ) × H 0 (X , ωX ), where J(X ) is the Jacobian of X . This base can be identi?ed with the cotangent bundle of the Jacobian. While the motivation for the present work comes from differential and complex analytic geometry, we are actually working within the framework of algebraic geometry, noting that on an algebraic curve the Higgs bundle data is purely algebraic. Translation between these categories is provided by Serre’s GAGA [25]. Our approach has the advantage that all constructions are automatically algebraic (or holomorphic), while the fact that we are dealing with rather high-dimensional base spaces would make some of the analytic techniques of Jardim hard to use. The ?rst part of this paper develops the machinery of generalised Fourier-Mukai transforms. While some of the material presented in section 2 cannot be found in the literature, it is mostly well known. The main new contributions are the

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de?nition of a relative Fourier transform for curves and the reduction of it to the original Mukai transform. The transform for Higgs bundles is developed in section 3. The ?rst interesting application is that our Fourier transform takes stable Higgs bundles of degree zero to vector bundles on J(X ) × H 0(X , ωX ). Our approach has the advantage of giving directly an algebraic (holomorphic) extension of this bundle over the projective completion J(X ) × P(H 0(X , ωX ) ⊕ C) of the base space, without a need to separately compactify a bundle on J(X ) × H 0 (X , ωX ). Denote this extension of the transform of a Higgs bundle E = (E , θ ) by TF(E ). The main theorem of this paper is the following: Inversion Theorem (3.2.1). — Let E and F be two Higgs bundles on a curve X of genus g ≥ 2. If TF(E ) ? = TF(F ), then E ? = F as Higgs bundles. We in fact prove this theorem by exhibiting a procedure for recovering a Higgs bundle from its transform. Furthermore, it follows easily from the theorem that the transform functor is in fact fully faithful. Acknowledgements. The original idea of developing a Fourier transform for Higgs bundles is due to Nigel Hitchin [14]. I am deeply grateful to him for generous comments, support and encouragement. Notation and conventions. Unless otherwise speci?ed, all rings and algebras are commutative and unital. We ?x an algebraically closed ?eld k. All schemes are assumed to be of ?nite type over k. All morphisms are k-morphisms and all products are products over Spec(k) unless stated otherwise. A curve always means a smooth irreducible complete (i.e., projective) curve over k. If F is an OX -module, F ∨ denotes its dual H omO (F , OX ). The canonical sheaf of a curve X is denoted X by ωX . D(X ) denotes the derived category of the (Abelian) category of OX -modules. D? (X ) (resp. D+ (X ), resp. Db (X )) is the full subcategory of objects cohomologically bounded above (resp. bounded below, resp. bounded). Dqcoh (X ) and Dcoh (X ) are the full subcategories of objects with quasi-coherent and coherent cohomology objects, respectively. These superscripts and subscripts can be combined in the obvious way. The category of OX -modules is denoted by Mod(X ), and QCoh(X ) is the thick subcategory of quasi-coherent sheaves. A commutative square Z ? ? ? ? → X ? ? ? ?f g Y ? ? ? ? → S

u v

is called Cartesian if the mapping (v, g)S : Z → X ×S Y is an isomorphism. We denote canonical isomorphisms often by "=". 2. F OURIER -M UKAI

TRANSFORMS

We develop here the general machinery of Fourier-Mukai transforms that will be necessary for our application to Higgs bundles.

4

J. BONSDORFF

We will be using the theory of derived categories; the main reference to derived categories in algebraic geometry remains Hartshorne’s seminar [10] on Grothendieck’s duality theory. Further references include Gelfand-Manin [7], KashiwaraShapira [18] and Weibel [30]. For a nice informal introduction, see Illusie [15] or the introduction of Verdier’s thesis [29]. 2.1. A base change result. (2.1.1) Consider the following diagram of schemes (here not necessarily of ?nite type over a ?eld): X1 ×S X2 X1

ww ww w ww {w w

p1

f

GG GG p2 GG GG G#

X2

f1 q1

Y1 ×S Y2

ww ww w w {www Y1 H HH HH HH HH HH $

S

2 GG GG q2 GG GG G# Y2 vv v v vv vv v {v v

f

with f = f1 ×S f2 . Recall the external tensor product over S of an OX -module F1 1 and an OX -module F2 :

2

F1 ?S F2 = ( p1 ? F1 ) ?O

L

X1 × X2 S

( p2 ? F2 ) .

We get the corresponding left-derived bifunctor (?) ?S (?) : D? (X1 ) × D? (X2 ) → D? (X1 ×S X2 ). The following theorem should be part of folklore; we include a proof of it for the lack of a suitable reference. It is essentially the derived-category version of a part of Grothendieck’s theory of "global hypertor functors" (EGA III [9], §6). D? qcoh (Xi ). Theorem (2.1.2) (Künneth formula). — For i = 1, 2 let Fi be an object of Assume that the schemes are Noetherian and of ?nite dimension, and that the fi are separated. Then (R f1? F1 ) ?S (R f2? F2 ) = R f? F1 ?S F2 if either F1 or F2 is quasi-isomorphic to a complex of S-?at sheaves. This is true in particular if either X1 or X2 is ?at over S. Proof. The Noetherian and dimensional hypotheses guarantee that the derived direct images are de?ned for complexes not bounded below. There are natural "adjunction" maps 1 → R f? L f ? giving (R f1? F1 ) ?S (R f2? F2 ) → R f? L f ? (R f1? F1 ) ?S (R f2? F2 ) .

L L L L

5

Notice that

? ? L f ? (R f1? F1 ) ?S (R f2? F2 ) = (L f ? Lq? 1 R f1? F1 ) ? (L f Lq2 R f2? F2 ) ? ? ? = (L p? 1 L f1 R f1? F1 ) ? (L p2 L f2 R f2? F2 ). L L L

Now the adjunctions L fi? R fi? → 1 give a natural map

? ? ? ? ? (L p? 1 L f1 R f1? F1 ) ? (L p2 L f2 R f2? F2 ) → (L p1 F1 ) ? (L p2 F2 ) L L

= F1 ?S F2 . Composing gives us a natural transformation (R f1? F1 ) ?S (R f2? F2 ) → R f? F1 ?S F2 . Whether this is an isomorphism is a local question; hence we may assume that S = Spec(A) and Yi = Spec(Bi ) for i = 1, 2. Suppose F1 is quasi-isomorphic to a complex of S-?at sheaves; replace F1 with this ?at resolution. Then F1 ?S F2 = F1 ?S F2 . For i = 1, 2 let Ui = (Ui,α )α be a ?nite af?ne open cover of Xi . Let U denote the open af?ne cover (U1,α ×S U2,β )α ,β of X1 ×S X2 . Notice that in all these covers ˇ ? (U , F ) denote the arbitrary intersections of the covering sets are af?ne. Let C i i ˇ simple complex associated to the Cech double complex of Fi with respect to Ui . ˇ ? (U, F ? F ) be the Cech ˇ Similarly, let C complex with respect to U. 1 S 2 ˇ Now RΓ(Xi , Fi ) is quasi-isomorphic to C? (Ui , Fi ), and hence R fi? Fi is quasiˇ ? (U , F )? . But the sheaves of these complexes are S-?at by conisomorphic to C i i struction, whence ˇ ? (U , F ) ? C ˇ ? (U , F ) (R f1? F1 ) ?S (R f2? F2 ) = C 1 1 A 2 2 Similarly R f? F1 ?S F2

L L ? L L L

L

.

ˇ ? (U, F ? F ) = C 1 S 2

?

.

ˇ ? (U , F ) ? C ˇ ? (U , F ) is Hence we are reduced to showing that the complex C 1 1 2 2 A ˇ ? (U, F ? F ). But this is showed in the proof of (6.7.6) of quasi-isomorphic to C 1 S 2 EGA III [9]. Remark (2.1.3). — If one wants to avoid the Noetherian hypothesis in the theorem, one could work with objects Fi of D? (QCoh(Xi )) and require the fi to be quasi-compact. This is essentially the viewpoint of EGA III. Corollary (2.1.4). — Let f : X → S and g : T → S be morphisms of ?nitedimensional Noetherian schemes. Let f ′ : X ×S T → T and g′ : X ×S T → X be the projections, and let F belong to D? qcoh (X ). (1) If F is quasi-isomorphic to a complex of S-?at sheaves (in particular, if f is ?at), then ? ′ Lg? R f? F = R f? Lg′ F .

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J. BONSDORFF

(2) If g is ?at, then

′ ′ g? R f? F = R f? g F.

?

Proof. Apply the Künneth formula with X1 = X , Y1 = S, f1 = f , X2 = Y2 = T , f2 = 1T , F1 = F and F2 = OT . 2.2. Integral transforms. De?nition (2.2.1). — Let S be a separated k-scheme and let X and Y be ?at Sschemes. If P is an object of Db coh (X ×S Y ), the relative integral transform de?ned P by P is the functor ΦX →Y /S : D+ (X ) → D+ (Y ) given by

? ΦP X →Y /S (?) = Rpr2 ? (pr1 (?) ? P ), L

where pr1 and pr2 are the canonical projections of X ×S Y . When S = Spec(k) we call the transform the absolute integral transform and denote it by ΦP X →Y . Proposition (2.2.2). — Let i : X ×S Y → X ×k Y be the morphism (pr1 , pr2 )k . Then Ri? = i? and i? P ΦP X →Y /S (?) = ΦX →Y (?). Proof. We have the commutative diagram

S F FF pr xx FF 2 x FF xx x FF x x x | " i X cGG Y ; GG xx x GG xx xx q p GGG xx pr1

X× Y

X × Y.

Notice that because both pr1 and p are ?at morphisms, we have

? ? ? ? ? ? ? pr? 1 = Lpr1 = L(i ? p ) = Li ? L p = Li ? p .

Using this and the projection formula, we have

? ΦP X →Y /S (?) = Rpr2 ? (pr1 (?) ? P ) L

= Rq? Ri? (Li? ( p? (?)) ? P ) = Rq? ( p? (?) ? Ri? P ). But i ?ts in a Cartesian square X ×S Y ? ? ? ? → X ×k Y ? ? ? ? S

? S/k i L

L

? ? ? ? → S ×k S .

As S/k is separated, ?S/k is a closed immersion, and consequently so is i. In particular, i? is an exact functor and therefore equal to Ri? . Hence

? i? P ΦP X →Y /S (?) = Rq? ( p (?) ? i? P ) = ΦX →Y (?) L

as claimed.

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Remark (2.2.3). — We cannot avoid using the derived tensor product in the above result, even if P is a locally free sheaf, because i? P is not ?at in general. However, as i is proper, i? P belongs always to Db coh (X × Y ).

1 (2.2.4) For ?at S-schemes X and Y and for x ∈ X , let Yx denote the ?bre pr? 1 (x), where pr1 : X ×S Y → X is the canonical projection. We have then a commutative diagram

κ (x) ? ? ? ? →

Yx ? ?

2 ? ? ? ? → X ×S Y ? ? ? ? → Y ? ? ? pr ? 1

j

pr

X

? ? ? ? → S,

in which all squares are Cartesian. Let i denote the composition of the top arrows. b For an object F of Db coh (X ×S Y ) (resp. Dcoh (Y )), we denote by Fx the "restriction" L j? F (resp. Li? F ) to Yx . For complexes of locally free sheaves these are just ordinary restrictions to Yx . If P is a locally free sheaf on X ×S Y , then for each x∈X ΦP X →Y /S (k(x)) = i? Px , where k(x) is the skyscraper sheaf k at x. Indeed, consider the commutative diagram above: the claim follows from ?at base change around the left-hand square and the projection formula applied to j. Notice that i? is exact. ? its dual, and let S be a Example (2.2.5). — Let X be an Abelian variety, X ? , normalised so that separated scheme. Let P be the Poincaré sheaf on X × X both P |X ×{0} and P{0}×X ? are the trivial line bundles. Denote by PS the pull? × S = (X × S) × (X ? × S). The relative Mukai back of this Poincaré sheaf to X × X S b b ? transform functor MS : Dcoh (X × S) → Dcoh (X × S) is the relative integral transform functor ΦPS . If S = Spec(k), we denote the transform by M . ?

(X ×S)→(X ×S)/S

The following theorem of Mukai plays a crucial role in the proof of our invertibility result (3.2.1). Theorem (2.2.6). — If S is a smooth projective variety, then the relative Mukai b ? transform MS is an equivalence of categories from Db coh (X × S) to Dcoh (X × S). Proof. See Mukai [23]. The proof is a generalisation of Mukai’s original proof of this result for the absolute transform M in [22]. Proposition (2.2.7). — Let X and Y be ?at S-schemes and P an object of Db coh (X ×S Y ). Let u : T → S be a morphism of schemes. Let iX : X(T ) → X , iY : Y(T ) → Y , and j : (X ×S Y )(T ) = X(T ) ×T Y(T ) → X ×S Y be the canonical projections. Then

? Lj LiY ? ΦP X →Y /S = ΦX

?P (T )

→Y(T ) /T

? Li? X.

Moreover, if u is a ?at morphism, then all derived pull-backs above can be replaced with normal pull-backs.

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J. BONSDORFF

Proof. Consider the commutative diagram X(T ) ×T Y(T )

p

HH HH H j HH$

X ×S Y

pr2

pr1

/X

/ X(T ) v v vv vv iX v zvv

q

Y(T )

:Y u iY uuu u uu uu

/ S dI II II u II II I /T

It is immediate that all squares are Cartesian. If u is ?at, then so are iX , iY and j; this proves the claim about replacing derived pull-backs. Since in any case X /S is ?at, pr2 is also ?at. So by (2.1.4) we can do a base change around the leftmost square. We get

? P ? LiY ΦX →Y /S (?) = LiY Rpr2? pr? 1 (?) ? P L

= Rq? L j? pr? 1 (?) ? P

? = Rq? L j? (pr? 1 (?)) ? L j P ? = Rq? p? Li? X (?) ? L j P L Lj = ΦX

?P

L

L

(T )

→Y(T ) /T

(Li? X (?)).

Proposition (2.2.8). — Let X and Y be ?at S-schemes and P an object of Db coh (X ×S Y ). Then RΓ(Y , ΦP X →Y /S (E )) = RΓ(X , E ? Rpr1? P ). Proof. We simply use the composition property of derived functors and the projection formula:

? RΓ(Y , ΦP X →Y /S (E )) = RΓ(Y , Rpr2? (pr1 E ? P )) L L

(by de?nition) (composition) (composition) (by projection formula).

= RΓ(X ×S Y , pr? 1E ? P ) = RΓ(X , Rpr1? (pr? 1 E ? P )) = RΓ(X , E ? Rpr1? P ) 2.3. WIT complexes.

L L

L

Notation (2.3.1). — Let X and Y be proper ?at S schemes. We ?x a locally free sheaf P on X ×S Y , and denote by FS the relative integral transform functor b b ΦP X →Y /S : Dcoh (X ) → Dcoh (Y ). We leave it to the reader to generalise the results of this subsection to a more general setting.

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De?nition (2.3.2). — We say that an object E of Db coh (X ) is a W ITP (n)1 p complex if H (FS (E )) = 0 for all p = n. If P is clear from the context, we shall omit the explicit reference to it. An object of Db coh (X ) is a WIT-complex if it is a W IT (n)-complex for some n. If E is a W IT (n)-complex on X , the (coherent) sheaf H n (FS (E )) on Y is called the integral transform of E , and is denoted by E .

2 De?nition (2.3.3). — We say that an object E of Db coh (X ) is an ITP (n)-complex if for each (closed) point y ∈ Y and each p = n we have

H p (Xy , Ey ? Py ) = 0, where we are using the notation of (2.2.4) for Ey , Py and Xy . Lemma (2.3.4). — Let f : X → Y be a proper morphism of (Noetherian) schemes and let E be an object of Db coh (X ) which has a Y -?at resolution. Let y ∈ Y . Then: (1) if the natural map ? p (y) : R p f? (E ) ? κ (y) → H p (Xy , Ey ) is surjective, then it is an isomorphism. (2) If ? p (y) is an isomorphism, then ? p?1 is also an isomorphism if and only if R p f? (E )is free in a open neighbourhood of y. Proof. This follows from EGA III [9] §7. However, that part of EGA can be somewhat hard to read; one could also follow the simpler proof of Hartshorne [11] Theorem III.12.11, making the fairly minor and obvious adjustments for hypercohomology. Proposition (2.3.5). — Let E be an IT (n) complex. Then E is a W IT (n)complex, and E is locally free on Y . Proof. Our schemes are Jacobson, and so it suf?ces to restrict our attention to closed points. Since pr2 is ?at, pr? 1 E is quasi-isomorphic to a complex of sheaves ?at over Y . Moreover, X is proper over S, and so pr2 is a proper morphism. We are then in position to use (2.3.4). Let y ∈ Y be a closed point. Now (pr? E ? P )y ? = Ey ? Py

1

on (X ×S Y )y = Xy . Hence by hypothesis the natural map

p ? ? p (y) : R p pr2 ? (pr? 1 E ? P ) ? κ (y) → H (Xy , (pr1 E ? P )y )

is trivially surjective, and by the base change theorem in fact isomorphic, for all p = n. As the hyper direct images of a complex of coherent sheaves are coherent for a proper map, we have R p pr2 ? (pr? 1E ? P ) = 0 for p = n by Nakayama’s lemma. This proves the ?rst part of the proposition. Now in particular Rn+1 pr2 ? (pr? 1 E ? P ) = 0. Thus, by the second part of the base change theorem, ? n (y) is an isomorphism. But as ? n?1 (y) is also surjective and thus isomorphic, Rn pr2 ? (pr? 1 E ? P ) is free in a neighbourhood of y, again by the second part of (2.3.4).

1Following Mukai, "WIT" stands for "weak index theorem". 2 "IT" stands for "Index theorem".

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J. BONSDORFF

Proposition (2.3.6). — Let X , Y and S be as in (2.3.1), and let u : T → S be a morphism of schemes. Suppose that E is an IT (n)-complex on X . Then, in the ? notation of (2.2.7), Li? X E is a W IT (n)-complex with respect to the pull-back j P of P to (X ×S Y )(T ) . Furthermore, if Li? X E denotes the corresponding Fourier transform, then

? E iY

= Li? XE .

Proof. By the assumptions and (2.3.5), ΦP X →Y /S (E ) is a locally free sheaf shifted n places to the right. Hence (2.2.7) gives

j P ? ΦP iY X →Y /S (E ) = ΦX →Y

(T ) ? (T )

/T

(Li? X E ).

j P But this shows that ΦX →Y

(T )

?

to the right. Both statements of the proposition are now immediate. 2.4. Fourier transform for curves. To ?x terminology and notation, we ?rst recall some basic facts about Jacobians of curves; for details, see Milne [20, 21]. Notation (2.4.1). — Let X be a smooth projective curve of genus g. We denote by J(X ) a Jacobian of X , i.e., a scheme representing the functor T → Pic? (X /T ). Let M be the corresponding universal sheaf on X × J(X ). Recall that J(X ) is an Abelian variety of dimension g; let J(X ) denote its dual Abelian variety, and let P be the Poincaré sheaf on J(X ) × J(X ), normalised as in (2.2.5). (2.4.2) Choosing a base point P ∈ X gives the Abel-Jacobi map iP : X → J(X ), taking the base point to 0. Notice that iP is a closed immersion. Furthermore, this ? → choice gives J(X ) a principal polarisation and hence an isomorphism ?P : J(X ) ? J(X ), which we use henceforth to identify J(X ) with its dual. Under this identi?cation, the pull-back (iP × 1J(X ) )? P is just the universal sheaf M on X × J(X ). (2.4.3) Let S be a separated k-scheme, XS = X × S, and let J(X )S = J(X ) × S be the relative Jacobian of the trivial family XS . We have a Cartesian square

2 X × J(X ) × S ? ? ? ? → J(X )S ? ? ? pr ? 1

(T )

/T

(Li? X E ) is also a locally free sheaf shifted n places

pr

XS

? ? ? ? →

S.

Let MS be the pull-back of M to X × J(X ) × S. The relative integral transform b : Db functor ΦMS coh (XS ) → Dcoh (J(X )S ) is given by

XS →J(X )S /S

ΦMS

XS →J(X )S /S

(?) = Rpr2 ? (pr? 1 (?) ? MS ),

where we can use the ordinary tensor product since MS is locally free. De?nition (2.4.4). — The relative integral transform ΦMS

XS →J(X )S /S

is called the

relative Fourier functor on X × S and is denoted by FS . If E is W IT with respect to FS , the integral transform E is called the Fourier transform of E .

11 b Proposition (2.4.5). — Let MS : Db coh (J(X ) × S) → Dcoh (J(X ) × S) denote the relative Mukai transform. Then

FS = MS ? (iP × 1S )? . Proof. Consider the diagram

2 XS ×S J(X )S ? ? ? ? → J(X )S ×S J(X )S ? ? ? ? → J(X )S ? ? ? ? p ? pr ? 1 1

j

p

XS

? ? ? ? →

iP ×1S

J(X )S

? ? ? ? →

S,

where the right-hand square is the ?bre-product diagram and j = (iP × 1S ) ×S 1J(X ) . S It is clear that the left-hand square is also commutative, and that the composition of the two top arrows is just the canonical projection pr2 . But this means that the big rectangle is Cartesian, and hence so is the left-hand square too. By de?nition, MS (?) = R p2 ? p? 1 (?) ? PS , where PS is the pull-back of the Poincaré sheaf onto J(X )S ×S J(X )S . Clearly MS = j? PS . Now by the projection formula R j? (? ? MS ) = R j? (?) ? PS . Because p1 is ?at as a base extension of a ?at morphism, we can do a base change (2.1.4) around the left-hand square to get

? p? 1 ? R(iP × 1S )? = R j? ? pr1 .

But iP × 1S is a closed immersion and thus R(iP × 1S )? = (iP × 1S )? . Putting these observations together, we get MS ((iP × 1S )? (?)) = R p2 ? p? 1 (iP × 1S )? (?) ? PS = R p2 ? R j? (pr? 1 (?)) ? PS = R p2 ? R j? pr? 1 (?) ? MS = Rpr2 ? pr? 1 (?) ? MS = FS (?). Proposition (2.4.6). — Let X be a curve of genus g and choose a base point P ∈ X as in (2.4.2); we suppose made the identi?cations given loc. cit. Let S be a k-scheme, and denote by j the embedding S ? = (X × S)P → X × S of the ?bre over ? P. Let E be a bounded complex of locally free sheaves on X × S. Then

g

H p (J(X ) × S, FS (E ? )) =

i=1

H p?i (SP , j? E ? )⊕( i?1 ) .

g? 1

Proof. By (2.2.8) we have natural isomorphisms H p (J(X ) × S, FS (E ? )) = H p (X × S, E ? ? Rpr1? M(S) ) for all p. Lemma (2.4.6.1). — With the notation of the proposition, Rpr1? M(S) is the zero-differential complex C ? where C i is the direct sum of 1 ≤ i ≤ g, zero otherwise.

g?1 i?1

copies of j? OS for

12

J. BONSDORFF

Consider the Cartesian square X × S × J(X ) ? ? ? ? → X × J(X ) ? ? ?q ? pr

1

p′

X ×S

? ? ? ? →

p

X.

By ?at base change around the square we get (2.4.6.1.1) Rpr1? M(S) = Rpr1? p′ M = p? Rq? M .

i ×1 ?

In order to compute Rq? M on X , we consider the Cartesian square

P X × J(X ) ? ? ? ? → J(X ) × J(X ) ? ? ?π ? q 1

X

? ? ? ? →

iP

J(X ).

Now by the general base-change (2.1.4) we have Rq? M = Rq? (iP × 1 )? P

? = LiP Rπ1? P .

But Rπ1? P = k(0)[?g], the skyscraper sheaf at 0 shifted g places to the right (see the proof of the theorem of §13 in Mumford [24]). Notice that iP is a reg?i O = ? N ular embedding; using Koszul resolutions it follows that LiP P? X X /J(X ) , the zero-differential exterior-algebra complex of the conormal sheaf of X in J(X ), ? k(0) = ? N concentrated in degrees ?g + 1 to 0. Similarly LiP X /J(X ) (P), the exterior algebra of the ?bre at P, whence the lemma follows immediately taking into account the shift by ?g. Using the projection formula we have H p (X × S, E ? ? j? OS ) = H p (SP , j? E ? ).

P

The proposition now follows from the lemma because hypercohomology commutes with direct sums. 3. T RANSFORMS

OF

H IGGS

BUNDLES

We shall now apply the Fourier-transform machinery developed in the previous section to stable Higgs bundles on curves. 3.1. De?nitions and basic properties. De?nition (3.1.1). — A Higgs bundle on a smooth projective curve is a pair E = (E , θ ), where E is a locally free sheaf on X , and θ is a morphism E → E ? ωX . → ωX is The morphism θ is often called the Higgs ?eld. The Higgs bundle OX ? called trivial. The rank and degree (i.e., the ?rst Chern class) of a Higgs bundle (E , θ ) mean the rank and degree of the underlying sheaf E . If E = (E ? → E ? ωX ) and F =

θ

0

13

(F ? → F ? ωX ) are Higgs bundles, by a morphism E → F we understand a morphism of sheaves ? : E → F making the square E ? ? ? ? → E ? ωX ? ? ?? ?1 ? ? F ? ? ? ? → F ? ωX

η θ

η

commutative. → E ? ωX ) be a Higgs bundle on X . Then we can consider it (3.1.2) Let E = (E ? as a complex of sheaves concentrated in degrees 0 and 1, and hence as an object in ? Db coh (X ). When we write E ? F or H (X , E ) etc., we consider the Higgs bundle as a sheaf complex this way. Notice that the image of E in Db coh (X ) does not uniquely determine the isomorphism class of the Higgs bundle (E ? → E ? ωX ). In fact, multiplying θ by a non-zero constant gives a quasi-isomorphic complex; however, the resulting Higgs bundle is not in general isomorphic. → E ? ωX ) is called stable if for any De?nition (3.1.3). — A Higgs bundle (E ? locally free subsheaf F of E satisfying θ (F ) ? F ? ωX , we have deg E deg F < . rk F rk E → E ? ωX ) be a non-trivial stable Higgs bundle Theorem (3.1.4). — Let E = (E ? on X with deg(E ) = 0. Then H p (X , E ) = 0 for p = 1. Proof. Hausel [12] Corollary (5.1.4.). Notice that H p (X , E ) = 0 automatically for p > 2 because dim(X ) = 1 and the length of the complex E is 2. Proposition (3.1.5). — If a Higgs bundle E is stable, then so is E ? L , where L is an element of Pic? (X ). Proof. Let F ? E ? L be a subbundle stable under θ ? 1 L . Then F ? L ?1 is a subbundle of E stable under θ . But tensoring with L affects neither the ranks nor the degrees of E and F , and hence the lemma follows from the stability of the Higgs bundle E . (3.1.6) Let E = (E ? → E ? ωX ) be a Higgs bundle and α ∈ H 0 (X , ωX ) a global 1-form. Then 1E ? α is canonically identi?ed with a morphism E → E ? ωX . We

E ? ? ? ? → E ? ωX ) by E (α ). denote the Higgs bundle (E ?

θ

θ

θ

θ

θ

θ +1 ?α

Lemma (3.1.7). — Let E be a stable Higgs bundle. Then E (α ) is also stable for any α ∈ H 0 (X , ωX ). Proof. Let F ? E be a subbundle stable under θα = θ + 1 ? α . Let t ∈ Γ(U , F ). Then θα (t ) = θ (t ) + t ? α ∈ Γ(U , F ? ωX ). But t ? α ∈ Γ(U , F ? ωX ) too, and hence θ (t ) ∈ Γ(U , F ? ωX ). Thus F is stable under θ , and the lemma follows from the stability of E .

14

J. BONSDORFF

We shall now introduce an important construction of algebraic families of Higgs bundles. For details about projective bundles see for example EGA II [8] §4. → E ? ωX ) be a Higgs bundle on a curve X of genus g, (3.1.8) Let E = (E ? and let π : X → Spec(k) be the structural morphism. Then the k-rational points of the vector bundle (or af?ne space) V((π? ωX )∨ ) are canonically identi?ed with the elements of H 0 (X , ωX ); we use the notation H 0 (X , ωX ) also for this scheme if no confusion seems likely. Let D = π ? ((π? ωX )∨ ) = (π ? π? ωX )∨ ; we have the canonical adjunction morphism

θ

? : D ∨ = π ? π? ωX → ωX .

? : D ∨ → ωX ? E nd (E ) be the morphism Let ? t → ? (t ) ? 1E . On the other hand, let ψ : OX → ωX ? E nd (E ) be the map that takes 1 to θ . Putting these together we get a morphism ? + ψ : D ∨ ⊕ OX → ωX ? E nd (E ). γ =? Because D ⊕ OX = π ? ((π? ωX )∨ ⊕ k), we have a canonical isomorphism . PX (D ⊕ OX ) = X × Pk ((π? ωX )∨ ⊕ k) = X × P(H 0(X , ωX ) ⊕ k) ? = X × Pg k Let p : P = PX (D ⊕ OX ) → X be the projection. There is the canonical surjection p? (D ⊕ OX ) → OP (1), and so by dualising a canonical OP (?1) → p? (D ∨ ⊕ OX ). Composing this morphism with p? γ we get a morphism OP (?1) → p? (ωX ? E nd (E )), or in other words a global section of p? (ωX ? E nd (E )) ? OP (1). We interpret this section as a morphism Θ : p? E → p? E ? p? ωX ? OP (1), and denote this complex of sheaves (in degrees 0 and 1) on P by H ? (E). In more pedestrian terms, let (αi )i be a basis of H 0 (X , ωX ), and let (αi? )i be the dual basis of H 0 (X , ωX )∨ . Let t : k → k be the canonical coordinate on k; ? , . . . , α ? ) forms a basis of the global sections of O (1), and H 0 (X , ω ) then (t , α1 g X Pg corresponds to the open af?ne subscheme of Pg with t = 0. Now Θ = θ ? t + ∑ αi ? 1 ? αi? .

i=1 g

Remark (3.1.9). — Notice that for α ∈ H 0 (X , ωX ) the restriction of H ? (E) to X × {α } is just E (α ) of (3.1.6). Proposition (3.1.10). — Let E be a stable Higgs bundle of degree 0 and rank ≥ 2 on a curve X of genus g ≥ 2. Then the complex H ? (E) on X × Pg is W IT (1) g g b with respect to the relative Fourier functor FPg : Db coh (X × P ) → Dcoh (J(X ) × P ). Moreover, the Fourier transform (H ? (E)) is a locally free sheaf on J(X ) × Pg . Proof. By (2.3.5) we are reduced to showing that H ? (E) is IT (1) with respect to M(Pg ) . We consider two cases. Let U denote the open subset H 0 (X , ωX ) in Pg .

15

A) Let (ξ , α ) ∈ J(X ) × U . Then (using the notation of (2.2.4)) ? (H ? (E)) = E (α ),

(ξ ,α )

and we need to show that H p (X , E (α ) ? Mξ ) = 0 for p = 1. But this follows from (3.1.5), (3.1.7) and (3.1.4). Notice that for a rank-1 Higgs bundle E one of the bundles E (α ) would be trivial, and the vanishing theorem (3.1.4) would fail. B) Let (ξ , z) ∈ J(X ) × (Pg ? U ). We consider the second hypercohomology spectral sequence:

II pq E2

= H p (X , H q ((H ? (E))(ξ ,z) ? Mξ )) ? H p+q (X , (H ? (E))(ξ ,z) ? Mξ ). (H ? (E))(ξ ,z) ? ? → E ? ωX ) = (E ?

1 ?α

But

for a 1-form α = 0, determined up to multiplication by a non-zero scalar . Now 1 ? α is clearly an injective map of sheaves; let S be its cokernel. Thus the E2 terms of the spectral sequence are

IIE pq 2

=

q

H 0 (X , S ? Mξ )

H 1 (X , S ? Mξ )

0

0

p

But S is a direct sum of skyscraper sheaves supported on the divisor of zeroes of the one-form, and since skyscraper sheaves are ?asque, we have H 1 (X , S ? Mξ ) = 0. Hence H0 (X , (H ? (E))(ξ ,z) ? Mξ ) = H2 (X , (H ? (E))(ξ ,z) ? Mξ ) = 0. De?nition (3.1.11). — Let E be a stable Higgs bundle of degree 0 and rank r ≥ 2 on a curve X of genus g ≥ 2. Then the locally free sheaf (H ? (E)) on J(X ) × Pg k is called (by abuse of language) the total Fourier transform of E and is denoted by TF(E ). Proposition (3.1.12). — Let E and X be as in (3.1.10), and let α ∈ H 0 (X , ωX ). Then TF(E )α ? = E (α ), where the left-hand side denotes the absolute Fourier transform. Proof. By the proof of (3.1.10) H ? (E) is IT (1). Now the proposition follows from Remark (3.1.9) and Proposition (2.3.6) applied to the immersion {α } → H 0 (X , ωX ) → Pg . → E ? ωX ) be a non-trivial stable Higgs Proposition (3.1.13). — Let E = (E ? bundle of degree 0 on a curve X of genus g ≥ 2. Then the rank of the total Fourier transform TF(E ) is (2g ? 2) rk(E ).

θ

16

J. BONSDORFF

Proof. It follows from (3.1.10) and (3.1.12) that rk(TF(E )) = dim H1 (X , E ). Consider the ?rst hypercohomology spectral sequence

I pq E2

= H p (H q (X , E )) ? H p+q (X , E ).

H 1 (θ )

The E1 -terms of the sequence are:

IE pq 1

=

q

H 1 (X , E )

/ H 1 (X , E ? ω ) X

H 0 (X , E )

H 0 (θ )

/ H 0 (X , E ? ω ) X

p

pq = IE pq , and hence The sequence clearly degenerates at E2 , i.e., IE∞ 2 I 0,0 E2 I 1,1 E2

? = H0 (X , E ) and ? H2 (X , E ). =

But these hypercohomologies vanish by (3.1.4), and thus H 0 (X , θ ) is injective and H 1 (X , θ ) is surjective. On the other hand,

0,1 1,0 ⊕ IE∞ = ker H 1 (X , θ ) ⊕ coker H 0 (X , θ ), H1 (X , E ) ? = IE∞

and hence dim H1 (X , E ) = dim H 1 (X , E ) ? dim H 1 (X , E ? ωX ) + dim H 0 (X , E ? ωX ) ? dim H 0 (X , E ) = χ (E ? ωX ) ? χ (E ). But as deg(E ) = 0, the Riemann-Roch theorem gives

χ (E ) = (1 ? g) rk(E ) and χ (E ? ωX ) = (g ? 1) rk(E ),

whence the result follows immediately. Proposition (3.1.14). — Let E be a stable Higgs bundle of rank r ≥ 2 and degree 0 on a curve X of genus g ≥ 2. Then dimk H p (J(X ) × Pg , TF(E )) = rg when 1 ≤ p ≤ g, and zero otherwise. Proof. Let P ∈ X be a base point giving an embedding iP : X → J(X ), and denote 1 by j the embedding Pg → X × Pg of the ?bre pr? X (P). Then by (2.4.6) H p (J(X ) × Pg , TF(E )) = H p+1 (J(X ) × Pg, FP (H ? (E))) (3.1.14.1)

g

g?1 , p?1

=

i=1

H p+1?i (Pg , j? H ? (E))⊕( i?1 ) .

g? 1

We apply the ?rst hypercohomology spectral sequence

I pq E2

= H p (H q (Pg , j? H ? (E))) ? H p+q (Pg , j? H ? (E)).

17

The E1 -terms are given by

IE pq 1

=

q

r ) H 1 (Pg , OP g

/ H 1 (Pg , O g (1)r ) P

r ) H 0 (Pg , OP g

d

/ H 0 (Pg , O g (1)r ). P

p

The standard results on the cohomology of a projective space (Hartshorne [11] 0,1 = E 1,1 = 0. Furthermore, it is clear from the de?nition III.5.1) show that the E1 1 ? (3.1.8) of H (E) that d = H 0 (Pg , j? Θ) is an injection. Thus we see that dim H p (Pg , j? H ? (E)) = rg if p=1, 0 otherwise.

Thus in the direct sum of (3.1.14.1) we have non-zero cohomology only when i = p, and the result follows immediately. → E ? ωX ) be a stable non-trivial Higgs Proposition (3.1.15). — Let E = (E ? bundle on a smooth projective curve X of genus g ≥ 2, with r = rk(E ) ≥ 2 and deg(E ) = 0. Then

? ch(TF(E )) = rk(E ) g ? 1 + (g ? 1)pr? P ch(OPg (1)) + t .(1 ? prP ch(OPg (1))) ,

θ

where t is the class of the Θ-divisor on J(X ). Proof. This is an easy application of the Grothendieck-Riemann-Roch formula. 3.2. Invertibility. Theorem (3.2.1). — Let E and F be two Higgs bundles on a curve X of genus g ≥ 2. If TF(E ) ? = TF(F ), then E ? = F as Higgs bundles. Proof. We show this by actually exhibiting a process of recovering a Higgs bundle E from its total Fourier transform TF(E ). Step 1. Choose a base point P ∈ X as in (2.4.2), and let iP : X → J(X ) be the corresponding embedding. Denote by j the immersion iP × 1J(X ) . Then by (2.4.5) FPg = MPg ? j? . By (2.2.6) MPg is a category equivalence; let G be its inverse. Now by de?nition TF(E ) = FPg (H ? (E))[1], and hence G(TF(E ))[?1] = j? (H ? (E)). Lemma (3.2.1.1). — The differential Θ of the complex H ? (E) is injective. Let U ? X × Pg be an open subset and s ∈ Γ(U , pr? E ) a non-zero section. There is a point z = (x, p) ∈ U for which s(z) = 0. Because E is locally free, it follows (using Nakayama’s lemma) that there is an open neighbourhood V ? U of z such that s(z′ ) = 0 for z′ ∈ V . If Θ(z)(s(z)) = 0, it follows from the de?nition of Θ that there is a point y ∈ V with Θ(y)(s(y)) = 0, and in particular ΘU (s) = 0. But this shows that Θ is injective as a morphism of presheaves and hence as a sheaf morphism too. Thus the lemma is proved.

18

J. BONSDORFF

By the lemma there is an exact sequence (3.2.1.2) 0 → pr? E ? → pr? (E ? ωX ) ? pr? OPg (1) → R → 0,

Θ

and consequently H ? (E ) is quasi-isomorphic to R [?1]. It follows from this that g G(TF(E )) = j? R in Db coh (X × P ). Since j? R is an honest sheaf, G(TF(E )) = j? R also in Mod(X × Pg ). This means that we can recover the cokernel R of H ? (E) on X × Pg as j? (G(TF(E ))). Step 2. Tensor (3.2.1.2) with pr? OPg (?1) and obtain the exact sequence ? → pr? (E ? ωX ) → R ? pr? OPg (?1) → 0. (3.2.1.3) 0 → pr? E ? pr? OPg (?1) ? We shall use the long exact RprX ? -sequence associated to (3.2.1.3). By the projection formula

? ? RprX ? (pr? X E ? prP OPg (?1)) = E ? RprX ? prP OP (?1), Θ?1

and

RprX ? (pr? X (E

? ωX )) = E ? ωX ? RprX ? OX ×P .

Now it follows from base change and the standard formulas for the cohomology of projective spaces that

1 ? prX ? pr? P OP (?1) = R prX ? prP OP (?1) = 0,

and

prX ? OX ×P = OX . ? It follows then from the long exact sequence that prX ? (R ? pr? P OP (?1)) = E ? ωX , and that we may consequently recover the underlying sheaf E of E from R by ∨. twisting by OP (?1), projecting down to X , and twisting by ωX Step 3. It remains to recover the Higgs ?eld θ . This will be done after discarding much of the information contained in R . We choose a non-zero α ∈ H 0 (X , ωX ), and we let U = Spec(A) be an open af?ne subscheme of X over which α does not vanish; then α gives a trivialisation of ωX on U . Clearly it is enough to recover θ over U . Let V be the subvectorspace of H 0 (X , ωX ) generated by α . We can consider V as a closed subscheme of the open subscheme H 0 (X , ωX ) of P(H 0 (X , ωX ) ⊕ k). Furthermore, we consider U × V as a subscheme of U × P(H 0 (X , ωX ) ⊕ k), and let S be the restriction of R to U × V ; it is just the cokernel of Θ restricted to U × V . Notice that U × V ? = Spec(A[T ]). On U the underlying sheaf E of E corresponds to an A-module M and θ corresponds to an endomorphism u of M . Furthermore, the pull-back of E to U × V corresponds to M [T ] = M ?A A[T ]. By the de?nition of Θ (3.1.8), Θ|U ×V corresponds to the A[T ]-linear map

ψ = 1M ? T + u ? 1A[T ] .

But ψ ?ts into the exact sequence → M [T ] → Mu → 0, M [T ] ? where Mu is the A[T ]-module with T acting on M as u (cf. Bourbaki [3], Ch. III §8 no. 10). Hence S = (Mu )? . But the structure of A[T ]-module of Mu determines u and hence θ |U .

ψ

19

Remark (3.2.2). — Lemma 6.8 in Simpson [27] gives a description of Higgs bundles on X as coherent sheaves on the total space of the cotangent bundle of X . The scheme U × V in Step 3 of the proof is the total space of the cotangent bundle of U , and the coherent sheaf S on U × V is the one that corresponds to E |U under Simpson’s correspondence. Corollary (3.2.3). — The functor TF from the category of stable non-trivial Higgs bundles on X with vanishing Chern classes to Mod(J(X ) × Pg ) is fully faithful. Proof. Let E and E′ be Higgs bundles on X and let R and R ′ be the cokernels of H ? (E ) and H ? (E′ ) respectively. Because the relative Mukai transform is an equivalence of categories, we have Hom(TF(E ), TF(E′ )) = Hom(R , R ′ ). Thus faithfulness is clear. On the other hand, let ? : R → R ′ ; using the notation of the proof of the theorem, the previous remark shows that ? |U ×V gives a morphism of Higgs bundles E |U → E′ |U . But as the genus of X is at least 2, the canonical linear system |ωX | has no base points. Hence we can cover X by open sets like U ; it is clear that the morphisms thus obtained glue to give a morphism E → E′ . R EFERENCES

[1] A. Bondal and D. Orlov: Semiorthogonal decomposition for algebraic varieties (1995), preprint alg-geom/9506012. [2] — Reconstruction of a variety from the derived category and groups of autoequivalences (1997), preprint alg-geom/9712029. [3] Nicolas Bourbaki: Algèbre Ch. 1–3, Hermann, Paris, 1970. [4] Tom Bridgeland: Fourier-Mukai transforms for surfaces and moduli spaces of stable sheaves, Ph.D. thesis, University of Edinburgh (1998). [5] — Equivalences of triangulated categories and Fourier-Mukai transforms, Bull. London Math. Soc. 31, no. 1 (1999) 25–34. [6] Simon K. Donaldson and Peter B. Kronheimer: The Geometry of Four-Manifolds, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1990. [7] Sergei Gelfand and Yuri Manin: Methods of Homological Algebra, Springer-Verlag, Berlin, 1996. [8] Alexander Grothendieck and Jean Dieudonné: ?léménts de géométrie algébrique II, Inst. Hautes ?tudes Sci. Publ. Math. 8 (1961) . [9] — ?léménts de géométrie algébrique III, Inst. Hautes ?tudes Sci. Publ. Math. 11,17 (1961,1963) . [10] Robin Hartshorne: Residues and Duality, Lecture Notes in Mathematics 20, Springer-Verlag, Berlin, 1966. [11] — Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag, New York, 1977. [12] Tamás Hausel: Vanishing of intersection numbers on the moduli space of Higgs bundles, Adv. Theor. Math. Phys. 2, no. 5 (1998) 1011–1040. [13] Nigel J. Hitchin: Self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987) 59–126. [14] — The Dirac operator, in Martin Bridson and Simon Salamon (Editors), Topics in Geometry and Topology, Oxford University Press, Oxford, to appear. [15] Luc Illusie: Catégories dérivées et dualité: travaux de J.-L. Verdier, Enseign. Math. (2) 36, no. 3-4 (1990) 369–391. [16] Marcos Jardim: Nahm Transform for Doubly Periodic Instantons, Ph.D. thesis, University of Oxford (1999), available as math.DG/9912028. [17] — Construction of doubly-periodic instantons, Comm. Math. Phys. 216, no. 1 (2001) 1–15.

20

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[18] Masaki Kashiwara and Pierre Schapira: Sheaves on Manifolds, Grundlehren 292, SpringerVerlag, Berlin, 1990. [19] Antony Maciocia: Generalized Fourier-Mukai transforms, J. Reine Angew. Math. 480 (1996) 197–211. [20] John S. Milne: Abelian varieties, in Gary Cornell and Joseph H. Silverman (Editors), Arithmetic Geometry (Storrs, 1984), Springer-Verlag, New York, 1986, (pp. 103–150). [21] — Jacobian varieties, in Gary Cornell and Joseph H. Silverman (Editors), Arithmetic Geometry (Storrs, 1984), Springer-Verlag, New York, 1986, (pp. 167–212). ? ) with its application to Picard sheaves, Nagoya [22] Shigeru Mukai: Duality between D(X ) and D(X Math. J. 84 (1984) 153–175. [23] — Fourier functor and its application to the moduli of bundles on an abelian variety, in Algebraic geometry, Sendai, 1985, North-Holland, Amsterdam, 1987, (pp. 515–550). [24] David Mumford: Abelian Varieties, Oxford University Press, Oxford, 1970. [25] Jean-Pierre Serre: Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier 6 (1956) 1–42. [26] Carlos T. Simpson: Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc. 1, no. 4 (1988) 867–918. [27] — Moduli of representations of the fundamental group of a smooth projective variety. II, Inst. Hautes ?tudes Sci. Publ. Math. 80 (1994) 5–79 (1995). [28] — The Hodge ?ltration on nonabelian cohomology, in Algebraic geometry—Santa Cruz 1995, Amer. Math. Soc., Providence, RI, 1997, (pp. 217–281). [29] Jean-Louis Verdier: Des catégories dérivées des catégories abéliennes, Astèrisque 239 (1996) [30] Charles A. Weibel: Introduction to Homological Algebra, Cambridge studies in advanced mathematics 38, Cambridge University Press, Cambridge, 1994. M ATHEMATICAL I NSTITUTE , 24?29 S T G ILES , OXFORD OX1 3LB, U NITED K INGDOM E-mail address: ?? ? ? ??× ?? ? ????????
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