Hyperkähler Manifolds of Curves in Twistor Spaces

We discuss hypercomplex and hyperkähler structures obtained from higher degree curves in complex spaces fibring over P¹.

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Дата:	2014
Автор:	Bielawski, R.
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Опубліковано:	Інститут математики НАН України 2014
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	Hyperkähler Manifolds of Curves in Twistor Spaces / R. Bielawski // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 22 назв. — англ.

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spelling	irk-123456789-1468192019-02-12T01:24:58Z Hyperkähler Manifolds of Curves in Twistor Spaces Bielawski, R. We discuss hypercomplex and hyperkähler structures obtained from higher degree curves in complex spaces fibring over P¹. 2014 Article Hyperkähler Manifolds of Curves in Twistor Spaces / R. Bielawski // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 22 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53C26; 53C28; 32G10; 14H15 DOI:10.3842/SIGMA.2014.033 http://dspace.nbuv.gov.ua/handle/123456789/146819 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We discuss hypercomplex and hyperkähler structures obtained from higher degree curves in complex spaces fibring over P¹.
format	Article
author	Bielawski, R.
spellingShingle	Bielawski, R. Hyperkähler Manifolds of Curves in Twistor Spaces Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Bielawski, R.
author_sort	Bielawski, R.
title	Hyperkähler Manifolds of Curves in Twistor Spaces
title_short	Hyperkähler Manifolds of Curves in Twistor Spaces
title_full	Hyperkähler Manifolds of Curves in Twistor Spaces
title_fullStr	Hyperkähler Manifolds of Curves in Twistor Spaces
title_full_unstemmed	Hyperkähler Manifolds of Curves in Twistor Spaces
title_sort	hyperkähler manifolds of curves in twistor spaces
publisher	Інститут математики НАН України
publishDate	2014
url	http://dspace.nbuv.gov.ua/handle/123456789/146819
citation_txt	Hyperkähler Manifolds of Curves in Twistor Spaces / R. Bielawski // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 22 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT bielawskir hyperkahlermanifoldsofcurvesintwistorspaces
first_indexed	2025-07-11T00:41:07Z
last_indexed	2025-07-11T00:41:07Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 033, 13 pages Hyperkähler Manifolds of Curves in Twistor Spaces? Roger BIELAWSKI Institut für Differentialgeometrie, Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany E-mail: bielawski@math.uni-hannover.de URL: http://www.ifam.uni-hannover.de/~bielawski/ Received November 06, 2013, in final form March 19, 2014; Published online March 28, 2014 http://dx.doi.org/10.3842/SIGMA.2014.033 Abstract. We discuss hypercomplex and hyperkähler structures obtained from higher deg- ree curves in complex spaces fibring over P1. Key words: hyperkähler metrics; hypercomplex structures; twistor methods; projective curves 2010 Mathematics Subject Classification: 53C26; 53C28; 32G10; 14H15 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 Le- gendre 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 P3 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 P1, π : Z → P1, and equipped with an antiholomorphic involution σ, which covers the antipodal map on P1. The manifold M is recovered as a connected component of the space of σ-equivariant sections s : P1 → 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 ⊗OC π∗OP1(−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. The hypercomplex manifolds obtained this way have interesting properties. For example, they are biholomorphic, with respect to any complex structure Iζ , ζ ∈ P1, to (unramified covering of) an open subset of the smooth locus of the Hilbert scheme Z [d] ζ of d points in the fibre Zζ = 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 S1 × R3. If we consider instead the twistor space P3 − P1 of the flat R4, we shall obtain hyperkähler structures on manifolds parameterising curves in P3 not intersecting a fixed line. ?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 mailto:bielawski@math.uni-hannover.de http://www.ifam.uni-hannover.de/~bielawski/ http://dx.doi.org/10.3842/SIGMA.2014.033 http://www.emis.de/journals/SIGMA/twistors.html 2 R. Bielawski 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. 2 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 C- ringed subspaces of domains U of Cn defined by finitely many holomorphic functions in U . In particular, we allow nilpotents in the structure sheaf. For a complex space (X,OX), the topological (Urysohn–Menger), analytic (Chevalley) and algebraic (Krull) notions of local dimension dimxX coincide. A complex space is called reduced if its structure sheaf has no nilpotents. It is pure dimensional (or equidimensional) if dimxX is the same at all points of X. For a 0-dimensional space X, its length is the sum of dimensions of stalks OX,x. The cotangent sheaf Ω1 X of a complex space X is defined by glueing together the cotan- gent sheaves IV /I2 V of local model spaces (V,OCn/IV ). The tangent sheaf TX is its dual Hom(Ω1 X ,OX). The tangent space at x ∈ X is TxX = TX,x/mxTX,x, where mx is the maxi- mal ideal of the stalk OX,x. If A is a local ring with maximal ideal m and M is a module over A, then a sequence (f1, . . . , fr) ∈ m is said to be regular for M if fi+1 does not divide 0 in M/(f1M + · · ·+fiM) 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 depthOX,x = dimxX 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 codimen- sion r if every of its local defining ideals IX,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 IX ⊂ OZ , the normal sheaf of X in Z is NX/Z = HomOX (IX/I2 X ,OX). If X ⊂ Z is regularly embedded, then both NX/Z and IX/I2 X are locally free. If, moreover, Z is smooth and X is reduced, then the Hyperkähler Manifolds of Curves in Twistor Spaces 3 conormal sheaf IX/I2 X fits into an exact sequence 0→ IX/I2 X → Ω1 Z \|X → Ω1 X → 0, and consequently we have an exact sequence 0→ TX → TZ \|X → NX/Z → Ext1 ( Ω1 X ,OX ) . 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 D(Z) ν←− Y µ−→ Z (2.1) 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 : S → D(Z) such that T ' S ×D(Z) Y . 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 NY of Y in D(Z)× Z to this fibre is NX/Z . At any X ∈ D(Z), there is a canonical isomorphism TXD(Z) ' H0(X,NX/Z). The Douady spaceD(Z) is smooth atX if Ext1 OZ (IX ,OX) = 0. For a regularly embeddedX, this is equivalent to H1(X,NX/Z) = 0. For a projective Z, D(Z) is the Hilbert scheme parameterising compact subschemes of Z. 3 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 → P1 a holomorphic surjection. For a closed subspace X of Z we continue to write π for its restriction to X. Each fibre Xζ , i.e. the complex subspace (π−1(ζ),OX/π∗mζ), is a closed subspace of X. If F is a sheaf on X, we write F(k) for F ⊗ π∗OP1(k). Furthermore, we write HX = π∗H0(P1,O(1)) ⊂ H0(X,OX(1)). We consider the set X of curves C in Z such that: • π : C → P1 is finite-to-one; • the Douady space D(Z) is smooth at C; • the sheaf NC/Z(−2) is acyclic, i.e. hi(NC/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 P1 is pro- vided 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: 4 R. Bielawski (i) X is pure dimensional and Cohen–Macaulay; (ii) π is flat; (iii) π∗OX is locally free; (iv) all fibres of π have the same length. Proof. Follows from [7, Corollary 18.17] and the equality dimxX = dimπ(x) Y + dimxXπ(x) for flat maps [13, Proposition II.2.11]. � Thus, for a C ∈ X , each fibre Cζ 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 ∈ P1, consider the map NC/Z(−2) ·s−→ NC/Z(−1), where s ∈ HC 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 NC/Z(−2). We have, therefore, the exact sequence 0→ NC/Z(−2) ·s−→ NC/Z(−1) −→ NC/Z(−1)\|Cζ → 0. (3.1) Taking cohomology and considering a generic ζ shows that h0(NC/Z(−1)) = dn and h1(NC/Z(−1)) = 0. Similarly, taking a section t of OP1(2) with zeros at ζ and ζ̃ gives the exact sequence 0→ NC/Z(−2) ·t−→ NC/Z −→ NC/Z \|Cζ ⊕NC/Z \|Cζ̃ → 0, (3.2) from which we derive h0(NC/Z) = 2dn, h1(NC/Z) = 0. (3.3) The definitions and (3.3) imply Proposition 3.2. The subset Xd of X , consisting of curves of degree d, is open in D(Z) and smooth of dimension 2dn. 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 → P1 is finite-to-one and H∗(NC/Z(−2)) = 0. Then C ∈ X . Proof. The normal sheaf of a regularly embedded subspace is locally free. Hence, (3.1) implies that each fibre Cζ has length h0(NC/Z(−1))/n. Owing to Lemma 3.1, C is equidimensional and Cohen–Macaulay. Moreover, for a regularly embedded C, (3.3) implies that D(Z) is smooth at C. � Since the linear system HC is base-free, (3.2) implies that the following sequence is also exact: 0→ NC/Z(−2)→ NC/Z(−1)⊗HC → NC/Z → 0, and, consequently, there is a canonical isomorphism H0(NC/Z) ' H0(NC/Z(−1))⊗HC . (3.4) We denote by E a vector bundle over Xd, the fibre of which at C is H0(NC/Z(−1)). In the notation of (2.1), E = ν∗ ( NY ⊗µ∗OZ(−1) ) . We also write H for the trivial bundle of rank 2, with Hyperkähler Manifolds of Curves in Twistor Spaces 5 fibre HC = π∗H0(P1,O(1)). The decomposition (3.4) induces a decomposition (of holomorphic vector bundles) TXd ' E ⊗H. For every ζ ∈ P1, we have a subbundle Qζ of TXd of rank dn defined as Qζ = E ⊗ s, where s is a section of OP1(1) vanishing at ζ. In other words, the fibre of Qζ at C consists of sections of NC/Z vanishing on Cζ . Proposition 3.4. The distribution Qζ is integrable. Proof. Let X and Y be vector fields with values in Qζ , and t 7→ γt(m), t 7→ δt(m) their integral curves beginning at m. The bracket of [X,Y ] at any p ∈ Xd can be computed as lim t→0 ν̇(t)/t2, where ν(t) = δ−t(γ−t(δt(γt(m)))). Consider the corresponding deformations of curves in Z. Since X and Y take values in Qζ , the deformation ν(t) fixes the fibre Cζ , and hence ν̇(t) ∈ Qζ . Thus [X,Y ]m ∈ Qζ \|m. � Now suppose that Z is equipped with an antiholomorphic involution σ which covers the antipodal map on P1. The submanifold (Xd)σ of σ-invariant curves in Xd is either empty or of real dimension 2dn. In the latter case, (Xd)σ is canonically a hypercomplex manifold. For each complex structure Iζ , ζ ∈ P1, the bundle of (0, 1)-vectors is Qζ . We state the result as follows. Theorem 3.5. Let Z be an equidimensional complex space equipped with a holomorphic sur- jection π : Z → P1 and an antiholomorphic involution σ covering the antipodal map on P1. Then the subset Md of the smooth locus of D(Z), consisting of σ-invariant curves C of de- gree d such that H∗(NC/Z(−2)) = 0, is, if nonempty, a hypercomplex manifold of real dimension 2d(dimZ − 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 ⊂ H0(Z,O(1)) be a base-free linear system. Let X be a regularly embedded compact subspace X of Z such that h1(NX/Z) = 0, H\|X is base-free, and the kernel of the natural map NX/Z(−1)⊗H → NX/Z has no cohomology in dimensions 0, 1 and 2. We obtain again a canonical isomorphism H0(X,NX/Z) ' H0(NX/Z(−1))⊗H, so that the open subset X ⊂ D(Z), consisting of such X, is a manifold, the tangent bundle of which decomposes as TX ' 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 Qs of TX (of rank dimX/ dimH), and the proof of Proposition 3.4 can be repeated to show that Qs 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 Md, recall that, for a C ∈ Md, each fibre Cζ = C ∩π−1(ζ) is a 0-dimensional complex space of length d, and so, for each ζ ∈ P1, we have a map Ψζ : Md → Z [d] ζ , C 7→ Cζ , to the Douady space of 0-dimensional complex subspaces of Zζ of length d. This map describes the complex structures of Md: 6 R. Bielawski Proposition 3.7. The map Ψζ : (Md, Iζ) → Z [d] ζ is holomorphic and, provided that dimZζ = dimZ − 1, an unramified covering of an open subset of the smooth locus of Z [d] ζ . Proof. We show first that the map Ψζ : Xd → Z [d] ζ , Ψζ(C) = Cζ , is holomorphic. Consider the subspace T of Xd × Zζ defined as T = {(C, z) ∈ Xd × Zζ ; z ∈ C}. Since the projection ν : T → Xd 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 Zζ implies that there is a unique holomorphic map f : Xd → D(Zζ) such that T = Xd×D(Zζ) Yζ , where Yζ is the universal family on D(Zζ). It follows that f = Ψζ , and, hence, Ψζ is holomorphic on Xd. In addition, this map factors locally through Xd/Qζ ' (Md, Iζ), and hence Ψζ is holomorphic on (Md, Iζ). For ζ1 6= ζ2 ∈ P1 with dimZζ1 = dimZζ2 = dimZ − 1, consider now the map Φ : Xd 3 C 7→ ( Cζ1 , Cζ2 ) ∈ Z [d] ζ1 × Z [d] ζ2 . An argument analogous to the one given above shows that Φ is holomorphic. Since H∗(C,NC/Z(−2)) = 0, the differential of Φ is injective at any point of Xd. Thus Φ is an immersion and, since Xd and Z [d] ζ1 × Z [d] ζ2 have the same dimension, Φ is a local diffeomorphism. In particular Φ maps to the smooth locus of Z [d] ζ1 × Z [d] ζ2 and is a covering of its image. Taking ζ2 to be antipodal to ζ1, we conclude that Md is a covering of an open subset of the smooth locus of Z [d] ζ1 . � Remark 3.8. We can now describe the usual twistor space of Md as long as π : Z → P1 is flat (e.g., submersion of smooth manifolds). Consider the relative Douady space D[d] π (Z) of finite subspaces of length d in each fibre Zζ [20], and let Zd be its open subset consisting of the smooth locus in each fibre. Since π is flat, Zd is a manifold. It has an induced real structure and a canonical projection π̃ : Zd → P1. Each C in Md corresponds to a section sC of π̃ : Zd → P1, sC(ζ) = Cζ , and the normal bundle N of sC(P1) in Zd is isomorphic to π∗NC/Z – a locally free sheaf of rank 2dn, where n = dimZ − 1. It follows that N ⊗OP1(−2) = π∗ ( NC/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 mani- fold M . Its twistor space is a smooth manifold Z equipped with a projection to P1 and an antiholomorphic involution which covers the antipodal map on P1. We obtain, for each d, a hypercomplex manifoldMd, which is biholomorphic, with respect to each complex structure Iζ , to a discrete covering of an open subset of the smooth locus of the Douady space M [d] of d points in (M, Iζ). Of course, this open subset (and the manifold Md itself) could be empty. We shall now show that Md is hyperkähler if M was. First of all, let us define symplectic forms in the context we shall need them. Definition 3.10. Let X be a complex space, F a coherent sheaf, and L a line a bundle on X. An L-valued symplectic form on F is a homomorphism Λ2F → L such that the associated homomorphism F → F∗ ⊗OX L is an isomorphism. Now recall that kernel of the map dπ : TZ → TP1 is called the vertical tangent sheaf and is denoted by TZ/P1 . To obtain a hyperkähler metric we need an OZ(2)-valued symplectic form ω on TZ/P1 . This form needs to be compatible with the real structure σ in the following sense [15]. The line bundle OZ(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 TZ/P1 , we obtain a quaternionic structure on TZ/P1(−1). The form ω induces a usual (i.e. O-valued) symplectic form on TZ/P1(−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 TZ/P1(−1). We have Hyperkähler Manifolds of Curves in Twistor Spaces 7 Theorem 3.11. Let Z be a complex manifold equipped with a holomorphic submersion π : Z→P1 and an antiholomorphic involution σ covering the antipodal map on P1. In addition, suppose that we are given a σ-compatible OZ(2)-valued symplectic form on the vertical tangent bundle TZ/P1, which induces a symplectic structure in the usual sense on each fibre Zζ . Then the hypercomplex manifold Md 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 Md. Proof. The arguments of Beauville [2] show that if M is a complex manifold with a holomorphic symplectic form, then the smooth locus of 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 Zd of Md 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 Md. � Remark 3.13. Nash [19] gives a different construction of hyperkähler metrics on moduli spaces of SU(2)-monopoles, which works as long as dimZ = 3 andMd 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). Remark 3.14. Similarly, if Z has an O(n)-valued volume form on TZ/P1 , then so does Zd, and the holonomy group of Md reduces to SL(dn,H). 4 Examples 4.1 Monopoles and the generalised Legendre transform In the case of SU(2)-monopoles, one starts with the twistor space Z of S1 × R3, i.e. the total space of certain line bundle L2 over TP1 without the zero section. Write p : Z → TP1 for the projection. A curve C in Z corresponds to a curve p(C) in TP1 such that L2\|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 L2 satisfies additionally H∗(NC/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 Md 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 P1, or the projectivisation of a vector bundle over a complex manifold fibring over P1. The last situation is relevant, for example, for ALF gravitational instantons of type Dk [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→ P1, one can describe these open subsets more precisely, as in [1, Chapter 6]. Let p : Z → Σ be the projection. Then (Md, Iζ) is biholomorphic to the open subset of (Zζ) [d] p , where the subscript p denotes 0-dimensional subspaces D such that p∗(OD) is a cyclic OΣζ -sheaf. 4.2 Projective curves The twistor space Z of the flat R4 is the total space of O(1) ⊕ O(1) on P1. We can view it as P3−P1, so that curves in Z are curves in the projective space not intersecting a fixed projective 8 R. Bielawski line. In addition, the real structure of Z extends to the real structure σ of P3 (which is the twistor space of S4), so we look for σ-invariant curves in P3. Consider first a rational curve, i.e. an embedding f : P1 → P3 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 OP1(2) ⊕ OP1(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 deg- ree d splits as OP1(2d − 1) ⊕ OP1(2d − 1) [8, 12], while the restriction of OZ(1) = OP3(1)\|Z is isomorphic to OP1(d). Thus a generic rational curve of degree d ≥ 3 satisfies the condition H∗(NC/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 P1 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 Hd,g of space curves with degree d and genus g contains smooth curves with H∗(NC/P3(−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 Hd,g contains also a σ-invariant smooth curve with H∗(NC/P3(−2)) = 0, we obtain a natural pseudo-hyperkähler structure on a submanifold of Hd,g. We shall now show that the resulting metric on H3,0 (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. 4.3 ACM curves with a linear resolution We consider curves C, the structure sheaf of which admits a free resolution of the form 0→ OP3(−r − 1)r φ2−→ OP3(−r)r+1 φ1−→ OP3 −→ OC → 0. (4.1) This means that IC is defined by simultaneous vanishing of the r × r minors of the linear matrix φ2. If C is smooth, then its degree is d = 1 2r(r + 1) and its genus is equal to g = 1 6(r − 1)(r − 2)(2r + 3). A complex subspace C with a resolution (4.1) is automatically equidimensional and Cohen– Macaulay [7, Theorem 18.18], and the Douady space (i.e. Hilbert scheme) is smooth at C [14, Co- rollary 8.10]. Furthermore, Ellia [9] has shown that every such C satisfies H∗(C,NC/Z(−2)) = 0. For completeness (and to remove the unnecessary assumption of projective normality) let us reproduce his argument. Lemma 4.1 ([9]). Let C be a subscheme of P3 with a resolution (4.1). Then H∗(NC/Z(−2)) = 0. Proof. We can rewrite (4.1) as a resolution of the ideal sheaf of C: 0→ OP3(−r − 1)r φ2−→ OP3(−r)r+1 φ1−→ IC → 0. (4.2) We have H i(NC/Z(−2)) = Exti+1(IC , IC(−2)), i = 0, 1. Applying Hom(−, IC(−2)) to (4.2) and using the isomorphism Exti(OP3(k), IC(−2)) ' H i(IC(−2−k)), k ∈ Z, we obtain the exact sequence H0(IC(r − 1))r → Ext1(IC , IC(−2))→ H1(IC(r − 2))r+1 → H1(IC(r − 1))r → Ext2(IC , IC(−2))→ 0. Hyperkähler Manifolds of Curves in Twistor Spaces 9 On the other hand, tensoring the resolution (4.2) with OP3(r − 1) and with OP3(r − 2) shows that IC(r − 1) and IC(r − 2) are acyclic. � Let us now choose a σ-invariant 2-dimensional linear system H in OP3 with base B, and let Z = P3 − B be the twistor space of R4. The projection π : Z → P1 is defined by H, and it is automatically finite-to-one on any projective subscheme which is contained in P3 − 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 P1, is a (pseudo)-hyperkähler manifold Xr of dimension 2r(r + 1). We shall now show that Xr is the flat Hr(r+1)/2. Moreover, the natural biholomorphism of Proposition 3.7 identifies Xr with an open subset of the Hilbert scheme (C2)[r(r+1)/2] consisting of 0-dimensional subspaces of length r which are not subschemes of any plane curve of degree <r. Let x1, . . . , x4 be homogeneous coordinates on P3 and π : [x1, . . . , x4] 7→ [x3, x4] the chosen projection onto P1. Thus B = {[x1, x2, 0, 0]} is the base of the linear system and P3 − B is our twistor space Z. The real structure σ is given by σ : [x1, x2, x3, x4] 7→ [−x2, x1,−x4, x3]. Let C be a curve defined by (4.1), and write, with respect to some bases, φ2(x1, x2, x3, x4) = 4∑ i=1 Aixi, Ai ∈ Matr+1,r(C). Such a C does not intersect B if φ2 restricted to x3 = x4 = 0 has rank r for all non-zero (x1, x2). 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 φ2 is σ-invariant; • for r odd, the set of rows of φ2 is σ-invariant. We shall now describe the intersection Cζ of C with a fibre π−1(ζ). The map φ2 restricted to the projective plane Zζ is still injective, otherwise Cζ is 2-dimensional and intersects B. Thus Cζ also has a free resolution of the form (4.1): 0→ OP2(−r − 1)r ψ2−→ OP2(−r)r+1 ψ1−→ OP2 −→ OCζ → 0. (4.3) Let us now write ζ = [1, t], so that ψ2(x1, x2, x3) = A1x1 + A2x2 + (A3 + tA4)x3. Recall that the condition that C does not intersect B is equivalent to A1x1 + A2x2 having rank r for any [x1, x2] ∈ P1. Lemma 4.2. If A1, A2 are two (r+ 1)× r complex matrices such that A1x1 +A2x2 is injective for every (x1, x2) ∈ C2\{0}, then A1x1+A2x2 belongs to the (open) GL(r+1,C)×GL(r,C)-orbit of Sx1 + Tx2 where Sij = { 1 if i = j, 0 if i 6= j, Tij = { 1 if i = j + 1, 0 if i 6= j + 1. Moreover, the stabiliser of (A1, A2) in GL(r + 1,C)×GL(r,C) is the central subgroup ∆ = { (zId, z−1Id); z ∈ C∗ } . 10 R. Bielawski Proof. We can appeal to Kronecker’s theory of minimal indices [11, Chapter XII]. The as- sumption implies that the pencil A1 + λA2 has no minimal indices for columns and no ele- mentary 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 Sx1 + Tx2 σ- invariant (and fixed). It is then easy to see that, given an (r + 1) × r matrix Ã3, we can find a unique A3, A4 so that Ã3 = A3 + tA4 and A3x3 +A4x4 is σ-invariant. It follows that the map which sends Cζ represented by (S, T, Ã3) to Ã3 is an isomorphism between the twistor space Zr of Xr (cf. Remark 3.8) and the total space of Cr(r+1) ⊗ O(1), i.e. the twistor space of the flat Hr(r+1)/2 (possibly, although unlikely, with a non-Euclidean signature). It remains to identify the complex structure of Xr as an open subset of the Hilbert scheme (C2)[r(r+1)/2]. Recall that the Hilbert scheme (C2)[d] of d points in C2 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 H(k) =  (k + 1)(k + 2) 2 if k < r, r(r + 1) 2 if k ≥ r. This means that Cζ does not lie on any plane curve of degree <r. Conversely, if a D ∈ (P2)[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 Xr with the open stratum of (C2)[r(r+1)/2], consisting of 0-dimensional subspaces which do not lie on any plane curve of degree smaller than r. 5 Induced vector bundles and their tangent spaces We aim to define canonical connections on vector bundles over a hypercomplex manifold, ob- tained 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 D(Z) ν←− Y µ−→ Z 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 h1(NX/Z) = 0. In particular, M is a manifold and NX/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 Ê = ν∗µ ∗E is locally free on M if the function X 7→ h0(E\|X) is constant on M . In this case Ê 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 0→ V → TE → π∗(TM)→ 0 (5.1) Hyperkähler Manifolds of Curves in Twistor Spaces 11 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 Ê (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 h0(E\|X) is constant on M and h1(E\|X) = 0 for every X in M . A point of the total space of E = Ê 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 space1 \|E\| of E . The normal sheaf Ns(X) of s(X) in \|E\| fits into the exact sequence 0→ π∗E\|X → Ns(X) → π∗NX/Z → 0, (5.2) of sheaves on s(X). Since h1(E\|X) = 0, it follows that h1(Ns(X)) = 0 and the Douady space D ( \|E\| ) is smooth at s(X) of dimension h0(Ns(X)) = dimM + rankE. Since pairs (X ′, s′) define a submanifold of D ( \|E\| ) also of dimension dimM + rankE, it follows that E is an open subset of the smooth locus of D ( \|E\| ) . Therefore the tangent space TeE at e = (X, s) is cano- nically identified with H0(Ns(X)) and the sequence (5.2) induces the canonical sequence (5.1). We can phrase these considerations more precisely, if less transparently, as follows. Theorem 5.1. Let Z be a complex space and M an open connected subset of the smooth locus of D(Z), such that every X ∈M is regularly embedded. Let E be a vector bundle on Z such that h0(E\|X) is constant on M and h1(E\|X) = 0 for every X in M , and let E = Ê be the induced vector bundle on M . Then E is identified with an open subset of the smooth locus of D ( \|E\| ) , again consisting of regularly embedded subspaces. Furthermore, if Ỹ is the universal family on D ( \|E\| ) and p : Ỹ → Y , ν̃ : Ỹ → E are canonical projections, then the normal sheaf NỸ of Ỹ in E × \|E\| fits into the commutative diagram: 0 −−−−→ ν̃∗p ∗µ∗E −−−−→ ν̃∗NỸ −−−−→ ν̃∗p ∗NY −−−−→ 0yo yo yo 0 −−−−→ V −−−−→ TE −−−−→ π∗(TM) −−−−→ 0 (5.3) 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 Md 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 h0(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 Ê 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]. 6 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 con- struction is more transparent. Thus Z is equipped with a holomorphic surjection onto P1 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,NC/Z(−2)) = 0 (cf. Lemma 3.3). Recall that M is then a complex version of a hypercomplex manifold, i.e. TM ' E ⊗ C2, and for every ζ ∈ P1 the distribution Qζ = 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 . 1The reason for the inconsistence in writing \|E\| for the total space of E is that D(E) has a different meaning: it parameterises coherent quotients of E with compact support. 12 R. Bielawski Proposition 6.1. Let E be a vector bundle on Z such that C → h0(E\|C) is constant on M and E(−1)\|C is acyclic for every C ∈ M . Then the induced bundle Ê on M is equipped with a canonical linear connection ∇, which has the following property: for any ζ ∈ P1 and any m ∈ M , if u is a local section of Ê which satisfies du(X) = 0 for every X ∈ Qζ \|m, then ∇Xu = 0 for every such X. In particular, if Z is also equipped with an antiholomorphic involution σ covering the antipodal map, so that Mσ is a hypercomplex manifold, then ∇ on Ê \|Mσ is hyperholomorphic, i.e. ∇0,1 = ∂̄ for every complex structure Iζ . 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 Ê (in the course of the proof we are not going to use the bundle E from the decomposition TM ' 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 : H0(NC/Z(−1))→ H0(Ns(C)(−1)). We obtain a canonical map (where H = π∗H0(P1,O(1))) H0(NC/Z) ' H0(NC/Z(−1))⊗H φ⊗id−→ H0(Ns(C)(−1))⊗H → H0(Ns(C)), which splits (5.1). Since, for any s ∈ H with s(ζ) = 0, the image in Ns(C) of H0(Ns(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 P1), we have a description of the horizontal subspace at (C, s) as the subspace of H0(Ns(C)) linearly generated by sections vanishing on some s(C)ζ , ζ ∈ P1. 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 Qζ . The curves Ct in Z corresponding to γ(t) have a fixed intersection with the fibre Zζ . A horizontal lift γ̃(t) of γ is given by (Ct, st) such that st(Ct) ∩ Eζ is a fixed complex subspace of the fibre Eζ . Thus, for a section u of E, the parallel transport τ t(u(t)) = (C̃t, s̃t) of u(γ(t)) along γ to γ(0) satisfies s̃t(C̃t) ∩ Eζ = st(Ct) ∩ Eζ . (6.1) Suppose now that u is a section of E such that du(X) = 0 for every X ∈ Qζ \|m. This implies that if γ(t) is a curve in M with γ̇(0) = X ∈ Qζ \|m and (Γt, ψt) is the pair curve + section corresponding to u(γ(t)), then up to order 1: ψt(Γt) ∩ Eζ = ψ0(Γ0) ∩ Eζ . Comparing this and (6.1), we conclude that ∇Xu = 0 for any X ∈ Qζ \|m. The remaining statement of the Proposition (concerning Mσ) is automatic, since Qζ consists of vectors of type (0, 1) for Iζ . � Remark 6.3. A similar result remains valid for the Grassmann structures defined in Re- mark 3.6. The above argument produces a canonical linear connection on a vector bundle induced from a vector bundle E on Z with constant h0(E\|C) and vanishing H∗(E(−1)\|C). Once again, this connection has the property described in the statement of the above theorem. 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[18] Manin Yu.I., Gauge field theory and complex geometry, Grundlehren der Mathematischen Wissenschaften, Vol. 289, 2nd ed., Springer-Verlag, Berlin, 1997. [19] Nash O., A new approach to monopole moduli spaces, Nonlinearity 20 (2007), 1645–1675. [20] Pourcin G., Théorème de Douady au-dessus de S, Ann. Scuola Norm. Sup. Pisa 23 (1969), 451–459. [21] Salamon S.M., Differential geometry of quaternionic manifolds, Ann. Sci. École Norm. Sup. (4) 19 (1986), 31–55. [22] Ward R.S., On self-dual gauge fields, Phys. Lett. A 61 (1977), 81–82. http://dx.doi.org/10.1016/j.geomphys.2008.11.010 http://arxiv.org/abs/0806.0510 http://dx.doi.org/10.1007/s10455-013-9364-2 http://arxiv.org/abs/1201.0781 http://dx.doi.org/10.1007/s00220-005-1404-8 http://arxiv.org/abs/hep-th/0310084 http://dx.doi.org/10.1007/978-1-4612-5350-1 http://dx.doi.org/10.1007/BF01450541 http://dx.doi.org/10.1007/BF01459132 http://dx.doi.org/10.1007/BF01316971 http://dx.doi.org/10.1007/978-1-4419-1596-2 http://dx.doi.org/10.1007/BF01214418 http://dx.doi.org/10.1007/BF01214418 http://dx.doi.org/10.1007/BF01238851 http://dx.doi.org/10.1088/0951-7715/20/7/007 http://dx.doi.org/10.1016/0375-9601(77)90842-8 1 Introduction 2 Some background material 3 Hypercomplex and hyperkähler structures from higher degree curves 4 Examples 4.1 Monopoles and the generalised Legendre transform 4.2 Projective curves 4.3 ACM curves with a linear resolution 5 Induced vector bundles and their tangent spaces 6 Hyperholomorphic connections References

Hyperkähler Manifolds of Curves in Twistor Spaces

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