Hyperk\"ahler Manifolds of Curves in Twistor Spaces

We discuss hypercomplex and hyperk\"ahler structures obtained from higher degree curves in complex spaces fibring over ${\mathbb{P}}^1$.


Introduction
This article is concerned with constructions of hypercomplex and hyperkähler structures from curves of arbitrary degree, and with their properties. It has been motivated by three sources. First and foremost, the work of Nash [19], who gave a new twistor construction of hyperkähler metrics on moduli spaces of SU(2) magnetic monopoles. Second, the so-called generalised Legendre transform construction of hyperkähler metrics, due to Lindström and Roček [17], which often leads to curves of higher genus. Third, the well-known fact that the smooth locus of the Hilbert scheme of (local complete intersection) curves of degree d and genus g in P 3 has, if nonempty, dimension 4d. Because of author's H-bias, the factor 4 seems to him to call for some sort of quaternionic structure.
It turns out that the above three situations can be put in a common framework. Let M be a connected hypercomplex or a hyperkähler manifold. The twistor space of M is a complex manifold Z fibring over P 1 , π : Z → P 1 , and equipped with an antiholomorphic involution σ, which covers the antipodal map on P 1 . The manifold M is recovered as a connected component of the space of σ-equivariant sections s : P 1 → Z with normal bundle N O(1) n . We now ask: what happens if we consider σ-invariant curves of higher degree in Z? It turns out that we still obtain a hypercomplex manifold, as long as we require that the normal bundle N of such a curve C satisfies the "stability condition" H * (N ⊗ O C π * O P 1 (−2)) = 0. This is the condition shown by Nash [19] to hold for spectral curves of monopoles, and used by him to describe the hypercomplex structure of monopole moduli spaces. His argument works in the general situation considered here. Moreover, this new hypercomplex manifold is (pseudo)-hyperkähler if M was a 4-dimensional hyperkähler manifold.
The hypercomplex manifolds obtained this way have interesting properties. For example, they are biholomorphic, with respect to any complex structure I ζ , ζ ∈ P 1 , to (unramified covering of) an open subset of the smooth locus of the Hilbert scheme Z [d] ζ of d points in the fibre Z ζ = π −1 (ζ) (here d is the degree of curves under consideration).
Hyperkähler monopole moduli spaces arise in the above manner from the twistor space of S 1 × R 3 . If we consider instead the twistor space P 3 − P 1 of the flat R 4 , we shall obtain This paper is a contribution to the Special Issue on Progress in Twistor Theory. The full collection is available at http://www.emis.de/journals/SIGMA/twistors.html hyperkähler structures on manifolds parameterising curves in P 3 not intersecting a fixed line. In the simplest case, that of twisted normal cubics, the resulting 12-dimensional metric is still flat, and the question arises what happens for other admissible values of genus and degree. Equally interesting is the question what happens for twistor spaces of compact hyperkähler or hypercomplex manifolds.
The differential geometry of hyperkähler manifolds obtained from higher degree curves is richer than just hyperkähler geometry. This has been already observed in [4] in the case of monopole moduli spaces. A first step in understanding this geometry is a description of natural objects on such a manifold M directly in terms of the complex manifold Z containing the higher degree curves, without passing to the usual (higher-dimensional) twistor space of the hyperkähler structure. Here we give such a description for hyperholomorphic connections on vector bundles on M . The novelty is that we construct canonical connections via a canonical splitting on the level of sections of vector bundles, without a corresponding splitting of vector bundles on Z.
Finally, we make a technical remark. If we want to consider higher degree curves, then the Kodaira moduli spaces (of complex submanifolds in a complex manifold) are not enough: such curves will almost certainly degenerate, while the hyperkähler metric will remain smooth. In fact, even for the usual twistor spaces, one is often enough led to consider a singular space Z, which already contains all the information needed, and it is then simpler to work directly with Z rather than resolving its singularities. For these reasons we replace throughout the Kodaira moduli spaces with the Douady space D(Z). We recall the necessary definitions and facts in the next section.

Some background material
We gather here some necessary facts and definitions from complex analysis. A good reference is [13].
We work in the category of complex spaces, i.e. C-ringed spaces locally modelled on Cringed subspaces of domains U of C n defined by finitely many holomorphic functions in U . In particular, we allow nilpotents in the structure sheaf.
For a complex space (X, O X ), the topological (Urysohn-Menger), analytic (Chevalley) and algebraic (Krull) notions of local dimension dim x X coincide. A complex space is called reduced if its structure sheaf has no nilpotents. It is pure dimensional (or equidimensional) if dim x X is the same at all points of X. For a 0-dimensional space X, its length is the sum of dimensions of stalks O X,x .
The cotangent sheaf Ω 1 X of a complex space X is defined by glueing together the cotangent sheaves If A is a local ring with maximal ideal m and M is a module over A, then a sequence (f 1 , . . . , f r ) ∈ m is said to be regular for M if f i+1 does not divide 0 in M/(f 1 M + · · · + f i M ) for i = 0, . . . , r − 1. The length of a maximal regular sequence is called the depth of M . A complex space X is said to be Cohen-Macaulay if depth O X,x = dim x X for all x ∈ X. A reduced 1-dimensional complex space is always Cohen-Macaulay.
A closed complex subspace X of a complex space Z is called regularly embedded of codimension r if every of its local defining ideals I X,x has depth r. If Z is a complex manifold, and X ⊂ Z is regularly embedded, then X is called a local complete intersection, abbreviated l.c.i. This is an intrinsic condition and it implies that X is Cohen-Macaulay.
For a closed complex subspace X of a complex space Z with ideal sheaf I X ⊂ O Z , the normal sheaf of X in Z is N X/Z = Hom O X (I X /I 2 X , O X ). If X ⊂ Z is regularly embedded, then both N X/Z and I X /I 2 X are locally free. If, moreover, Z is smooth and X is reduced, then the conormal sheaf I X /I 2 X fits into an exact sequence and consequently we have an exact sequence A celebrated theorem of Douady [6] states that for any complex space Z there exists a complex space D(Z) parameterising all pure-dimensional compact complex subspaces of Z. In addition, there exists a universal family on D(Z), i.e. a complex subspace Y of D(Z) × Z, which defines a double fibration with the following properties: (i) ν is flat and proper; (ii) if S is a complex space, T ⊂ S × Z a complex subspace with properties stated in (i), then there exists a unique holomorphic map f : The fibre of Y over any m ∈ D(Z) is the complex subspace X of Z corresponding to m and the restriction of the normal sheaf For a regularly embedded X, this is equivalent to H 1 (X, N X/Z ) = 0. For a projective Z, D(Z) is the Hilbert scheme parameterising compact subschemes of Z.

Hypercomplex and hyperkähler structures from higher degree curves
For us, a curve means a compact, pure 1-dimensional, Cohen-Macaulay complex space. If the curve is reduced, then the Cohen-Macaulay assumption is redundant. Let Z be a pure (n + 1)-dimensional complex space and π : Z → P 1 a holomorphic surjection. For a closed subspace X of Z we continue to write π for its restriction to X. Each fibre X ζ , i.e. the complex subspace (π −1 (ζ), O X /π * m ζ ), is a closed subspace of X. If F is a sheaf on X, we write F(k) for F ⊗ π * O P 1 (k). Furthermore, we write We consider the set X of curves C in Z such that: • π : C → P 1 is finite-to-one; • the Douady space D(Z) is smooth at C; • the sheaf N C/Z (−2) is acyclic, i.e. h i (N C/Z (−2)) = 0 for i = 0, 1.
These conditions are open, so X is an open submanifold of D(Z).
A useful characterisation of pure dimension + Cohen-Macaulay + finite cover of P 1 is provided by Lemma 3.1. Let π : X → Y be a finite-to-one closed surjective holomorphic map from a complex space X to a complex manifold Y . Then the following conditions are equivalent: (i) X is pure dimensional and Cohen-Macaulay; (ii) π is flat; (iii) π * O X is locally free; (iv) all fibres of π have the same length.
Thus, for a C ∈ X , each fibre C ζ is a 0-dimensional complex space of constant length d, which we call the degree of the curve C.
For any C ∈ X and ζ ∈ P 1 , consider the map N C/Z (−2) ·s −→ N C/Z (−1), where s ∈ H C with s(ζ) = 0. This map must be injective, since a nontrivial kernel sheaf would be supported on a 0-dimensional subspace, and so it would have nontrivial sections, which would then map to nontrivial sections of N C/Z (−2). We have, therefore, the exact sequence Taking cohomology and considering a generic ζ shows that Similarly, taking a section t of O P 1 (2) with zeros at ζ andζ gives the exact sequence from which we derive For regularly embedded subspaces some of the assumptions on C are automatically fulfilled Lemma 3.3. Let C be a regularly embedded compact subspace of Z such that π : C → P 1 is finite-to-one and Proof . The normal sheaf of a regularly embedded subspace is locally free. Hence, (3.1) implies that each fibre C ζ has length h 0 (N C/Z (−1))/n. Owing to Lemma 3.1, C is equidimensional and Cohen-Macaulay. Moreover, for a regularly embedded C, Since the linear system H C is base-free, (3.2) implies that the following sequence is also exact: and, consequently, there is a canonical isomorphism We denote by E a vector bundle over X d , the fibre of which at C is H 0 (N C/Z (−1)). In the notation of (2.1), . We also write H for the trivial bundle of rank 2, with fibre H C = π * H 0 (P 1 , O(1)). The decomposition (3.4) induces a decomposition (of holomorphic vector bundles) For every ζ ∈ P 1 , we have a subbundle Q ζ of T X d of rank dn defined as where s is a section of O P 1 (1) vanishing at ζ. In other words, the fibre of Q ζ at C consists of sections of N C/Z vanishing on C ζ .
Proof . Let X and Y be vector fields with values in Q ζ , and t → γ t (m), t → δ t (m) their integral curves beginning at m. The bracket of [X, Y ] at any p ∈ X d can be computed as lim where Consider the corresponding deformations of curves in Z. Since X and Y take values in Q ζ , the deformation ν(t) fixes the fibre C ζ , and Now suppose that Z is equipped with an antiholomorphic involution σ which covers the antipodal map on P 1 . The submanifold (X d ) σ of σ-invariant curves in X d is either empty or of real dimension 2dn. In the latter case, (X d ) σ is canonically a hypercomplex manifold. For each complex structure I ζ , ζ ∈ P 1 , the bundle of (0, 1)-vectors is Q ζ . We state the result as follows.
Theorem 3.5. Let Z be an equidimensional complex space equipped with a holomorphic surjection π : Z → P 1 and an antiholomorphic involution σ covering the antipodal map on P 1 . Then the subset M d of the smooth locus of D(Z), consisting of σ-invariant curves C of degree d such that H * (N C/Z (−2)) = 0, is, if nonempty, a hypercomplex manifold of real dimension 2d(dim Z − 1). Remark 3.6 (a generalisation). One can consider a more general situation. Let O(1) a globally generated line bundle on a complex space Z and let H ⊂ H 0 (Z, O(1)) be a base-free linear system. Let X be a regularly embedded compact subspace X of Z such that h 1 (N X/Z ) = 0, H| X is base-free, and the kernel of the natural map N X/Z (−1) ⊗ H → N X/Z has no cohomology in dimensions 0, 1 and 2. We obtain again a canonical isomorphism so that the open subset X ⊂ D(Z), consisting of such X, is a manifold, the tangent bundle of which decomposes as T X E ⊗ H. Such a decomposition is known under various names, in particular as an almost Grassmann structure or a conic structure [18]. Once again, for every s ∈ H, we have a subbundle Q s of T X (of rank dim X / dim H), and the proof of Proposition 3.4 can be repeated to show that Q s is integrable. We can also consider compatible real structures. In particular, we did not need to restrict ourselves to 1-dimensional subspaces of Z in order to obtain hypercomplex manifolds.
Returning to the hypercomplex manifold M d , recall that, for a C ∈ M d , each fibre C ζ = C ∩ π −1 (ζ) is a 0-dimensional complex space of length d, and so, for each ζ ∈ P 1 , we have a map to the Douady space of 0-dimensional complex subspaces of Z ζ of length d. This map describes the complex structures of M d : ζ is holomorphic and, provided that dim Z ζ = dim Z − 1, an unramified covering of an open subset of the smooth locus of Z Proof . We show first that the map Ψ ζ : Since the projection ν : T → X d is finite and every fibre has the same length, ν is flat. It is also clearly proper. Hence, the universal property of the Douady space of Z ζ implies that there is a unique holomorphic map f : It follows that f = Ψ ζ , and, hence, Ψ ζ is holomorphic on X d . In addition, this map factors locally through X d /Q ζ (M d , I ζ ), and hence Ψ ζ is holomorphic on (M d , I ζ ). For ζ 2 . An argument analogous to the one given above shows that Φ is holomorphic. Since H * (C, N C/Z (−2)) = 0, the differential of Φ is injective at any point of X d . Thus Φ is an immersion and, since X d and Z [d] ζ 2 and is a covering of its image. Taking ζ 2 to be antipodal to ζ 1 , we conclude that M d is a covering of an open subset of the smooth locus of Z Remark 3.8. We can now describe the usual twistor space of M d as long as π : Z → P 1 is flat (e.g., submersion of smooth manifolds). Consider the relative Douady space D [20], and let Z d be its open subset consisting of the smooth locus in each fibre. Since π is flat, Z d is a manifold. It has an induced real structure and a canonical projectionπ : Z d → P 1 . Each C in M d corresponds to a section s C ofπ : Z d → P 1 , s C (ζ) = C ζ , and the normal bundle N of s C (P 1 ) in Z d is isomorphic to π * N C/Z -a locally free sheaf of rank 2dn, where n = dim Z − 1. It follows that N ⊗ O P 1 (−2) = π * N C/Z (−2) has no cohomology, and, consequently, N O(1).
Remark 3.9. The above results can be viewed as follows. Start with a hypercomplex manifold M . Its twistor space is a smooth manifold Z equipped with a projection to P 1 and an antiholomorphic involution which covers the antipodal map on P 1 . We obtain, for each d, a hypercomplex manifold M d , which is biholomorphic, with respect to each complex structure I ζ , to a discrete  An L-valued symplectic form on F is a homomorphism Λ 2 F → L such that the associated homomorphism F → F * ⊗ O X L is an isomorphism. Now recall that kernel of the map dπ : T Z → T P 1 is called the vertical tangent sheaf and is denoted by T Z/P 1 . To obtain a hyperkähler metric we need an O Z (2)-valued symplectic form ω on T Z/P 1 . This form needs to be compatible with the real structure σ in the following sense [15]. The line bundle O Z (1) has a canonical quaternionic structure (i.e. an antilinear isomorphism with square −1) covering σ on Z. Since σ induces a real structure on T Z/P 1 , we obtain a quaternionic structure on T Z/P 1 (−1). The form ω induces a usual (i.e. O-valued) symplectic form on T Z/P 1 (−1), and we say that ω is compatible with σ if ω(σ * s, σ * t) = ω(s, t) for local sections s, t of T Z/P 1 (−1). We have Theorem 3.11. Let Z be a complex manifold of dimension 3 equipped with a holomorphic submersion π : Z→P 1 and an antiholomorphic involution σ covering the antipodal map on P 1 . In addition, suppose that we are given a σ-compatible O Z (2)-valued symplectic form on the vertical tangent bundle T Z/P 1 , which induces a symplectic structure in the usual sense on each fibre Z ζ . Then the hypercomplex manifold M d defined in Theorem 3.5 has a canonical pseudo-hyperkähler metric.
Remark 3.12. The signature of this metric can vary between different connected components of M d .
Proof . The arguments of Beauville [2] show that if M is a complex surface with a holomorphic symplectic form, then M [d] has an induced holomorphic symplectic form. Applying this construction fibrewise to Z, we obtain an O(2)-valued fibrewise symplectic form on the twistor space Z d of M d described in Remark 3.8. This form is still compatible with the induced real structure and hence, owing to [15,Theorem 3.3], it gives a pseudo-hyperkähler metric on M d .
Remark 3.13. Nash [19] gives a different construction of hyperkähler metrics on moduli spaces of SU(2)-monopoles, which works as long as M d is replaced by its open subset of l.c.i. curves (so that the normal sheaf is locally free and the Serre duality can be applied to it).

Monopoles and the generalised Legendre transform
In the case of SU(2)-monopoles, one starts with the twistor space Z of S 1 × R 3 , i.e. the total space of certain line bundle L 2 over T P 1 without the zero section. Write p : Z → T P 1 for the projection. A curve C in Z corresponds to a curve p(C) in T P 1 such that L 2 | p(C) is trivial. This is the condition satisfied by spectral curves of magnetic monopoles, and Nash [19] shows that if S is a spectral curve of a monopole, then its lift C to L 2 satisfies additionally H * (N C/Z (−2)) = 0. Thus, the moduli space of monopoles of charge d (i.e. those for which S is a curve of degree d) is a connected component of the manifold M d defined in the previous section. Already in this case one has to include singular curves (although not nonreduced ones).
More general hyperkähler metrics were considered in [3], as examples of the generalised Legendre construction of Lindström and Roček [17]. Many of these can be put in this framework, i.e. Z is the total space of a line bundle over a complex surface, or, more generally, the total space of a holomorphic principal bundle over a complex manifold fibring over P 1 , or the projectivisation of a vector bundle over a complex manifold fibring over P 1 . The last situation is relevant, for example, for ALF gravitational instantons of type D k [5].
According to Proposition 3.7, the complex structures of such hypercomplex manifolds are always those of (covering of) open subsets of the Douady spaces of 0-dimensional subspaces of fibres of Z. In the case when Z is the total space of a C * -bundle over a complex surface Σ → P 1 , one can describe these open subsets more precisely, as in [1, Chapter 6]. Let p : Z → Σ be the projection. Then (M d , I ζ ) is biholomorphic to the open subset of (Z ζ )

Projective curves
The twistor space Z of the flat R 4 is the total space of O(1) ⊕ O(1) on P 1 . We can view it as P 3 − P 1 , so that curves in Z are curves in the projective space not intersecting a fixed projective line. In addition, the real structure of Z extends to the real structure σ of P 3 (which is the twistor space of S 4 ), so we look for σ-invariant curves in P 3 .
Consider first a rational curve, i.e. an embedding f : P 1 → P 3 given by homogeneous polynomials of degree d. For d = 2, such a curve is an intersection of a line and a quadric, so its normal bundle is isomorphic to O P 1 (2) ⊕ O P 1 (4) and it does not satisfy the condition H * (N (−2)) = 0. For d ≥ 3, however, the normal bundle of a generic rational curve of degree d splits as O P 1 (2d − 1) ⊕ O P 1 (2d − 1) [8,12], while the restriction of O Z (1) = O P 3 (1)| Z is isomorphic to O P 1 (d). Thus a generic rational curve of degree d ≥ 3 satisfies the condition H * (N C/Z (−2)) = 0 and, consequently, the parameter space of such curves, which are σ-invariant and avoid a fixed line, is a 4d-dimensional (pseudo)-hyperkähler manifold. We shall see shortly, that for d = 3, i.e. for twisted normal cubics, the resulting metric is flat. This, however, is a rather special case and we do not know what to expect in the general case. We observe that the action of SO(3) on P 1 induces an isometric action rotating the complex structures, and so all complex structures are equivalent. In fact, we expect that these hyperkähler manifolds are cones over 3-Sasakian manifolds.
For higher genera, it is known [10] that the parameter space H d,g of space curves with degree d and genus g contains smooth curves with H * (N C/P 3 (−2)) = 0 for any d greater than or equal to some D(g) (e.g., D(0) = 3 and D(1) = 5). As soon as H d,g contains also a σ-invariant smooth curve with H * (N C/P 3 (−2)) = 0, we obtain a natural pseudo-hyperkähler structure on a submanifold of H d,g .
We shall now show that the resulting metric on H 3,0 (and more generally on moduli spaces of ACM (arithmetically Cohen-Macaulay) curves admitting a linear resolution) is flat. Although the metric itself is not interesting, it is still an instructive example, which shows, in particular, that if we want to have a shot at completeness of the metric, we cannot avoid including very singular and nonreduced complex subspaces of Z.

ACM curves with a linear resolution
We consider curves C, the structure sheaf of which admits a free resolution of the form This means that I C is defined by simultaneous vanishing of the r × r minors of the linear matrix φ 2 . If C is smooth, then its degree is d = 1 2 r(r + 1) and its genus is equal to g = 1 6 (r − 1)(r − 2)(2r + 3).
Let us now choose a σ-invariant 2-dimensional linear system H in O P 3 with base B, and let Z = P 3 − B be the twistor space of R 4 . The projection π : Z → P 1 is defined by H, and it is automatically finite-to-one on any projective subscheme which is contained in P 3 − B, since a 1-dimensional intersection with π −1 (ζ) will also intersect B.
Thus, the Douady space of those σ-invariant C, which admit a resolution of the form (4.1) and do not intersect a fixed P 1 , is a (pseudo)-hyperkähler manifold X r of dimension 2r(r + 1). We shall now show that X r is the flat H r(r+1)/2 . Moreover, the natural biholomorphism of Proposition 3.7 identifies X r with an open subset of the Hilbert scheme (C 2 ) [r(r+1)/2] consisting of 0-dimensional subspaces of length r which are not subschemes of any plane curve of degree <r.
Let x 1 , . . . , x 4 be homogeneous coordinates on P 3 and π : [x 1 , . . . , x 4 ] → [x 3 , x 4 ] the chosen projection onto P 1 . Thus B = {[x 1 , x 2 , 0, 0]} is the base of the linear system and P 3 − B is our twistor space Z. The real structure σ is given by Let C be a curve defined by (4.1), and write, with respect to some bases, Such a C does not intersect B if φ 2 restricted to x 3 = x 4 = 0 has rank r for all non-zero (x 1 , x 2 ). The involution σ induces a quaternionic structure σ on linear forms and such a curve C is σ-invariant as soon as • for r even, the set of columns of φ 2 is σ-invariant; • for r odd, the set of rows of φ 2 is σ-invariant.
We shall now describe the intersection C ζ of C with a fibre π −1 (ζ). The map φ 2 restricted to the projective plane Z ζ is still injective, otherwise C ζ is 2-dimensional and intersects B. Thus C ζ also has a free resolution of the form (4.1): Let us now write ζ = [1, t], so that Recall that the condition that C does not intersect B is equivalent to A 1 x 1 + A 2 x 2 having rank r for any [x 1 , x 2 ] ∈ P 1 .
Moreover, the stabiliser of (A 1 , A 2 ) in GL(r + 1, C) × GL(r, C) is the central subgroup Proof . We can appeal to Kronecker's theory of minimal indices [11,Chapter XII]. The assumption implies that the pencil A 1 + λA 2 has no minimal indices for columns and no elementary divisors. Thus it can have only minimal indices for rows, and therefore it lies in the GL(r + 1, C) × GL(r, C)-orbit of a block quasi-diagonal matrix built out of blocks of the form as in the statement of the lemma. Such a matrix cannot, however, have size (r + 1) × r, unless there is only one block. The statement about the stabiliser is then straightforward.
We can now use the action of GL(r + 1, C) × GL(r, C) in order to make Sx 1 + T x 2 σinvariant (and fixed). It is then easy to see that, given an (r + 1) × r matrixÃ 3 , we can find a unique A 3 , A 4 so thatÃ 3 = A 3 + tA 4 and A 3 x 3 + A 4 x 4 is σ-invariant. It follows that the map which sends C ζ represented by (S, T,Ã 3 ) toÃ 3 is an isomorphism between the twistor space Z r of X r (cf. Remark 3.8) and the total space of C r(r+1) ⊗ O(1), i.e. the twistor space of the flat H r(r+1)/2 (possibly, although unlikely, with a non-Euclidean signature).
It remains to identify the complex structure of X r as an open subset of the Hilbert scheme (C 2 ) [r(r+1) /2] . Recall that the Hilbert scheme (C 2 ) [d] of d points in C 2 has a natural stratification by the Hilbert function H : N → N, with H(k) equal to (k + 1)(k + 2)/2 minus the dimension of the vector space of plane curves of degree k containing X [7]. The Hilbert function can be computed from any free resolution, and in our case we obtain This means that C ζ does not lie on any plane curve of degree <r. Conversely, if a D ∈ (P 2 ) [d] does not lie on any curve of degree <r, then its ideal must be generated by forms of degree ≥r.
Comparing dimensions and using the Hilbert-Burch theorem, as in [7, § 20.4.1], shows that the minimal resolution of D is of the form (4.3). In other words, the natural biholomorphism of Proposition 3.7 identifies X r with the open stratum of (C 2 ) [r(r+1)/2] , consisting of 0-dimensional subspaces which do not lie on any plane curve of degree smaller than r.

Induced vector bundles and their tangent spaces
We aim to define canonical connections on vector bundles over a hypercomplex manifold, obtained from higher degree curves in Z, directly in terms of objects defined on Z, i.e. without passing to the usual twistor space of a hypercomplex manifold. In this section, we shall consider a general Douady space D(Z) (not necessarily a hypercomplex manifold) and describe total spaces of vector bundles arising on D(Z) as Douady spaces of some other, canonically defined, complex spaces. Let Z be a complex space and D(Z) its Douady space. The double fibration allows one to transfer holomorphic data from Z to D(Z) or to its subsets. Let M be an open connected subset of D(Z), such that every X ∈ M is regularly embedded (cf. Section 2) and satisfies h 1 (N X/Z ) = 0. In particular, M is a manifold and N X/Z a locally free sheaf. We are interested in vector bundles on M obtained from vector bundles on Z. Let E be a vector bundle on Z. The sheaf E = ν * µ * E is locally free on M if the function X → h 0 (E| X ) is constant on M .
In this case E is called the vector bundle induced by E [16]. Let π : E → M be a vector bundle on a manifold M . One way of defining connections on E is to split the canonical exact sequence of vector bundles on E, where V denotes the vertical bundle ker dπ. As the first step in defining canonical connections, we want to show that for bundles E (with E satisfying a cohomological condition) the above sequence is induced from objects on Z.
Let E be a vector bundle on Z such that h 0 (E| X ) is constant on M and h 1 (E| X ) = 0 for every X in M . A point of the total space of E = E corresponds to a pair, consisting of a regularly embedded compact subspace X in Z and a section s of E| X . The image space s(X) is well defined and is a regularly embedded compact subspace of the total space 1 |E| of E. The normal sheaf N s(X) of s(X) in |E| fits into the exact sequence 0 → π * E| X → N s(X) → π * N X/Z → 0, (5.2) of sheaves on s(X). Since h 1 (E| X ) = 0, it follows that h 1 (N s(X) ) = 0 and the Douady space D |E| is smooth at s(X) of dimension h 0 (N s(X) ) = dim M + rank E. Since pairs (X , s ) define a submanifold of D |E| also of dimension dim M + rank E, it follows that E is an open subset of the smooth locus of D |E| . Therefore the tangent space T e E at e = (X, s) is canonically identified with H 0 (N s(X) ) and the sequence (5.2) induces the canonical sequence (5.1). We can phrase these considerations more precisely, if less transparently, as follows.
Remark 5.2. The above description of the total space of an induced vector bundle as a Douady space has the following consequence. Let Z be as in Theorem 3.5 and let M be an open subset of M d consisting of regularly embedded curves. Let E be a vector σ-bundle on Z (i.e. there is an antilinear involution on E covering σ on Z) such that h 0 (E| C ) is constant on M and E(−2) is acyclic on any C ∈ M . Then the σ-invariant part of the total space of E has a natural hypercomplex structure. Indeed, it is enough to tensor (5.2) by O(−2) and apply the results of Section 3. This result is in the same spirit as [21, Theorem 7.2].

Hyperholomorphic connections
We return to hypercomplex manifolds and consider again the situation from Section 3. As in the last section, we restrict ourselves to the subset of regularly embedded curves, for which the construction is more transparent. Thus Z is equipped with a holomorphic surjection onto P 1 and M is an open connected subset of D(Z) consisting of regularly embedded compact subspaces C, such that π| C is finite-to-one and H * (C, N C/Z (−2)) = 0 (cf. Lemma 3.3). Recall that M is then a complex version of a hypercomplex manifold, i.e. T M E ⊗ C 2 , and for every ζ ∈ P 1 the distribution Q ζ = E ⊗ l, where [l] = ζ, is integrable. We now use the results of the previous section to define canonical connections on certain induced vector bundles on M . Proposition 6.1. Let E be a vector bundle on Z such that C → h 0 (E| C ) is constant on M and E(−1)| C is acyclic for every C ∈ M . Then the induced bundle E on M is equipped with a canonical linear connection ∇, which has the following property: for any ζ ∈ P 1 and any m ∈ M , if u is a local section of E which satisfies du(X) = 0 for every X ∈ Q ζ | m , then ∇ X u = 0 for every such X.
In particular, if Z is also equipped with an antiholomorphic involution σ covering the antipodal map, so that M σ is a hypercomplex manifold, then ∇ on E| M σ is hyperholomorphic, i.e. ∇ 0,1 =∂ for every complex structure I ζ .
Remark 6.2. In the case when Z is the usual twistor space of a hypercomplex manifold, i.e. the curves C have genus 0, then the condition that E(−1)| C is acyclic is equivalent to E| C to being trivial, and we recover the well-known results of [22] and [16]. In that case, already the sequence (5.2) splits.
Proof . Let us write E for E (in the course of the proof we are not going to use the bundle E from the decomposition T M E ⊗ H), and consider the canonical exact sequence (5.1). An Ehresmann connection is a splitting of this sequence. To define such a splitting, tensor (5.2) by O(−1). Acyclity of E(−1)| C implies that there is a canonical isomorphism φ : H 0 (N C/Z (−1)) → H 0 (N s(C) (−1)).
We obtain a canonical map (where H = π * H 0 (P 1 , O(1))) which splits (5.1). Since, for any s ∈ H with s(ζ) = 0, the image in N s(C) of H 0 (N s(C) (−1)) ⊗ s consists of sections which vanish on s(C) ζ (here, as in Section 3, the subscript ζ denotes the fibre of the projection to P 1 ), we have a description of the horizontal subspace at (C, s) as the subspace of H 0 (N s(C) ) linearly generated by sections vanishing on some s(C) ζ , ζ ∈ P 1 . It follows easily that the differential of the scalar multiplication E → E and the differential of the addition map E ⊕ E → E preserve the horizontal subbundles, and, therefore, our Ehresmann connection is linear.
Let us now prove the stated property of this connection. Let γ(t) be a curve in M tangent to the distribution Q ζ . The curves C t in Z corresponding to γ(t) have a fixed intersection with the fibre Z ζ . A horizontal liftγ(t) of γ is given by (C t , s t ) such that s t (C t ) ∩ E ζ is a fixed complex subspace of the fibre E ζ . Thus, for a section u of E, the parallel transport τ t (u(t)) = (C t ,s t ) of u(γ(t)) along γ to γ(0) satisfies Suppose now that u is a section of E such that du(X) = 0 for every X ∈ Q ζ | m . This implies that if γ(t) is a curve in M withγ(0) = X ∈ Q ζ | m and (Γ t , ψ t ) is the pair curve + section corresponding to u(γ(t)), then up to order 1: Comparing this and (6.1), we conclude that ∇ X u = 0 for any X ∈ Q ζ | m . The remaining statement of the Proposition (concerning M σ ) is automatic, since Q ζ consists of vectors of type (0, 1) for I ζ .
Remark 6.3. A similar result remains valid for the Grassmann structures defined in Remark 3.6. The above argument produces a canonical linear connection on a vector bundle induced from a vector bundle E on Z with constant h 0 (E| C ) and vanishing H * (E(−1)| C ). Once again, this connection has the property described in the statement of the above theorem.
Remark 6.4. The method of this and the last section reproduces also another result of Huggett and Merkulov [16]. Let Z be an arbitrary complex space, and M ⊂ D(Z) as in the previous section. Let E be a vector bundle on Z such that h 0 (E| X ) is constant on M and h i (E ⊗ X N * X/Z ) = 0 for i = 0, 1 and every X ∈ M . Then there is a canonical splitting of the sequence (5.2), which produces a canonical linear Ehresmann connection as above.