HYPERKÄHLER MANIFOLDS

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1 HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly constant orthogonal endomorphisms I, J and K of the tangent bundle which satisfy the quaternionic relations I 2 = J 2 = K 2 = IJK = 1. Note, that I, J and K give each tangent space the structure of a quaternionic vector space, so the real dimension of a hyperkähler manifold is divisible by 4. Since I, J and K are covariantly constant, a parallel transport commutes with the quaternionic multiplication and so the holonomy group is contained in O 4n GL n (H) = Sp n, the group of quaternionic unitary n n matrices. In particular, since Sp n SU 2n every hyperkähler manifold is Calabi-Yau. Examples: (1) A trivial example is H n. However, in contrast to the Kähler case, HP n is not hyperkähler and neither do its generic quaternionic submanifolds. (2) For n = 1 Sp 1 = SU2 and so every Calabi-Yau surface is hyperkähler. The only compact examples are T 4 and K3 surfaces. (3) A class of noncompact hyperkähler manifolds of real dimension 4 can be obtained by resolving the singularity of C 2 /Γ for Γ SU 2 a finite subgroup. (4) Many examples of noncompact hyperkähler manifolds arise as moduli spaces of solutions to gauge-theoretic equations. The hyperkähler structure is obtained by a hyperkähler reduction from H n. Proposition. A hyperkähler manifold M is a complex manifold with a holomorphic symplectic form. Conversely, any compact Kähler manifold with a holomorphic symplectic form is hyperkähler. Proof. Define the symplectic forms ω 1 (v, w) = g(iv, w), ω 2 (v, w) = g(jv, w) ω 3 (v, w) = g(kv, w) for v, w T M. They are of type (1, 1) in the respective complex structures. Furthermore, if we define ω + = ω 2 + iω 3, 1

2 2 PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP it is closed, non-degenerate and covariantly constant. It is holomorphic in the complex structure I. Indeed, for any vectors v, w T C M: ω + (v, (1 + ii)w) = g(jv, (1 + ii)w) + ig(kv, (1 + ii)w) = g(jv, w) + ig(kv, w) + ig(ijv, w) g(ikv, w) = g(jv, w) + ig(kv, w) ig(kv, w) g(jv, w) = 0. Therefore, it is a holomorphic symplectic form. Conversely, suppose ω is a holomorphic symplectic form on a Kähler manifold M of real dimension 2n. Then ω n is a nowhere vanishing section of the canonical bundle which gives a holomorphic trivialization. Yau s theorem ensures the existence of a Kähler metric g with vanishing Ricci tensor. Applying Bochner s theorem one concludes that ω is covariantly constant with respect to g. Hence, the holonomy group of M is contained in Sp 2n (C) U 2n = Spn, i.e. M is hyperkähler Twistor space. One can attempt to encode the data of hyperkähler manifold (M, g) in a simpler complex manifold called the twistor space [Hi-92]. Observe, that for any u = (a 1, a 2, a 3 ) R 3 I 2 u = (a 1 I + a 2 J + a 3 K) 2 = (a a a 2 3). So, if u = 1, I u is an almost complex structure which is covariantly constant and therefore integrable. Define the twistor space to be a C manifold Z = M S 2. S 2 = CP 1 has a natural complex structure I 0, so there is an almost complex structure I on Z given by where (p, u) Z, v T p M and w T u S 2. Lemma. I is integrable. I p,u (v, w) = (I u v, I 0 w), Proof. The proof involves a use of Newlander-Nirenberg theorem. Remark: Newlander-Nirenberg theorem fails for Hilbert manifolds, so the integrability of the complex structure on the twistor space of a Hilbert hyperkähler manifold has to be established independently. We have the following Theorem 1. The twistor space Z of a hyperkähler manifold M is a complex manifold of dimension 2n + 1, which has the following structures: (1) A holomorphic fiber bundle p : Z CP 1. (2) Z has a real structure σ : Z Z covering the antipodal map of CP 1. (3) There is a section ϖ of 2 T Z/CP 1 p O(2) which defines a holomorphic symplectic form on each fiber and which is real with respect to σ. (4) The bundle admits a family of global holomorphic sections, whose normal bundles are all holomorphically isomorphic to C 2n C p O(1) and which are real with respect to σ. Proof. (1) The projection p : Z CP 1 defined by (m, ζ) ζ is holomorphic by the definition of the complex structure I.

3 HYPERKÄHLER MANIFOLDS 3 (2) The antipodal map σ : S 2 S 2 extends to an antiholmorphic involution σ : Z Z, since (I u, I 0 ) ( I u, I 0 ). (3) Recall, that ω + = ω 2 + iω 3 is holomorphic in the complex structure I. Similarly, define ϖ(ζ) = ω 2 + iω 3 + 2ζω 1 ζ 2 (ω 2 iω 3 ). ϖ(ζ) is holomorphic in the complex structure I u and defines a holomorphic symplectic form on p 1 (ζ). Globally, it can be extended into a section of 2 T Z/CP 1 p O(2). The O(2) twist is due to the quadratic dependence of ϖ(ζ) on ζ. (4) For each point m M, {m} S 2 Z is a section of Z CP 1 which is holomorphic. Its normal bundle is isomorphic to C 2n C p O(1). These sections are clearly real with respect to σ. Conversely, starting from such data we can recover the hyperkähler manifold M. Theorem 2. Suppose a complex manifold Z of dimension 2n + 1 has the structures (1)-(3) given by the previous theorem. Then the parameter space of real sections (4) is a hyperkähler manifold whose twistor space is Z Hyperkähler quotient. Suppose that M is a hyperkähler manifold and a compact Lie group K acts preserving the symplectic forms ω 1, ω 2, ω 3. Then we have three moment maps µ 1, µ 2, µ 3, or, equivalently, a single moment map µ : M k R 3. Denote z k the space of K-invariants. Then we have the following Theorem 3. If ζ z R 3 is a regular value, then µ 1 (ζ)/k is a hyperkähler manifold. We can characterize hyperkähler quotients in a different way. Form µ + = µ 1 +iµ 2 and then the hyperkähler quotient is the symplectic quotient of a Kähler manifold µ 1 + (α), where α z C. Finally, consider the twistor point of view. Let Z CP 1 be a twistor space for M. Then the twistor space for the hyperkähler quotient is the fiberwise symplectic quotient of Z by the holomorphic action of G. 2. Twistor space for the moduli space In this section X is a smooth projective variety over C. G will denote a connected reductive complex group and K is its maximal compact subgroup Quotient construction of the hyperkähler structure. The following is due to Hitchin [Hi-87] for X being a curve and Fujiki [Fu] in higher dimensions. First, fix a C principal G-bundle P X and a reduction P K X of the structure group to K. We will assume that P has a vanishing first and second Chern classes. Fix an integer k dim X + 1. Define the group of complex gauge transformations to be G = Γ(X, Ad P ), where the sections are of class H k+1, it is naturally a Hilbert Lie group. Similarly, there is a group of real gauge transformations K = Γ(X, Ad P K ). We have a decomposition ad P = ad P K i ad P K, which

4 4 PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP gives ad P a complex structure. The space of all connections on P (P K ) of class H k will be denoted by A (A K ). Lemma. A is a Hilbert hyperkähler manifold. Proof. First consider A K. Its tangent space is identified with Ω 1 R (X; ad P K). Then we have a natural complex structure I on Ω 1 (X) coming from the complex structure on X. A compatible metric on A K is given by (α, β) A = (α, β) X ω n, X where (, ) X is a combination of the inner product on T X given by the Kähler metric on X, and an inner product on ad P. This is an H 0 metric, a metric for higher k has a similar form. The tangent space to A at each point is Ω 1 C (X; ad P ) and is naturally a complexification of Ω 1 R (X; ad P K). Denote the corresponding complex structure as J. If we extend I antilinearly, then IJ = JI and so A is a hyperkähler manifold. Denote the J-holomorphic symplectic form as ω + = α β ω n 1 X. X G acts holomorphically on (A, J) preserving the hyperkähler metric. This action extends to a fiberwise holomorphic action on the associated twistor space Z A. It preserves the holomorphic sympelctic form and the associated moment map is µ + (d) = iλf d, where Λ is the Lefschetz operator adjoint to a multiplication by ω X. Similarly, K acts on (A, I) preserving the Kähler form. Recall, that d can be decomposed as d = d + + θ, where d + is a connection along P K and θ is orthogonal to P K, i.e. θ Ω 1 (i ad P K ). We can decompose d + and θ further according to type: We will also form d + = d + d, θ = θ + θ. = d + θ, = d + θ. Then a result of Corlette says that the moment map is given by µ 3 = id +, θ. Denote T the resulting hyperkähler quotient of A by G. It is a moduli space of irreducible Einstein weakly harmonic connections (i.e. such that d +, θ = 0). In general this is an infinitedimensional space and we want to reduce it further to obtain a finite-dimensional subspace. There are two natural choices:

5 HYPERKÄHLER MANIFOLDS 5 (1) Denote T J T the subset of all connections d such that the curvature F d is of type (1, 1). By using the complex structure on ad P and the type decomposition of the forms we conclude that 0 = F 2,0 d + + [θ, θ ] = d θ 0 = F 0,2 d + + [θ, θ ] = d θ Then (P, ) and (P, ) are antiholomorphic and holomorphic bundles respectively. Summarizing, T J parametrizes principal bundles (P, d) with an irreducible, Einstein, weakly harmonic connection d, such that its curvature is of type (1, 1). (2) We can also consider a subset T I T of all connections d, such that the curvature F d + is of type (1, 1), θ is holomorphic with respect to d + θ and [θ θ] = 0. Clearly, T I is the moduli space of stable G-Higgs bundles. Recall a theorem of Simpson: Theorem 4. Let (P, d) be a connected principal bundle whose curvature is of type (1, 1) and Einstein. Then the metric P K is harmonic iff the Higgs field is integrable: [θ θ ] = 0. Any stable G-Higgs bundle admits an irreducible Hermitian-Einstein metric and, since the first and second Chern classes vanish, we have an identification of sets T I = TJ, where T J is a subset of T J consisting of harmonic bundles with a flat metric Harmonic bundles and λ-connections. Consider a flat G-bundle P X with a harmonic metric. Decompose d as before and for any λ C define λ = + λθ, λ = λ + θ. Lemma. (P, λ, λ ) is a holomorphic G-bundle with a flat λ-connection. We get a natural identification T J = MHod (X, G) λ Deligne moduli space. Fix a basepoint x X. Recall, that the Hodge moduli space R Hod (X, x, G) A 1 parametrizes principal G-bundles on X with integrable λ-connections and a frame at x. M Hod (X, G) is the GIT quotient of R Hod (X, x, G) by G acting by a change of frame. The fiber at 0 is identified with M Dol (X, G) and the fiber at 1 is identified with M DR (X, G). The action of G m on A 1 lifts to R Hod (X, xi, G) which then descends to M Hod (X, G) and allows one to identify the fibers at all λ 0 with M DR (X, G). The map τ : M B (X, G) (an) M B (X, G) (an) that sends a representation ρ to the dual of its complex conjugate is an antilinear involution. Using the Riemann-Hilbert correspondence M B (X, G) (an) = MDR (X, G) an it lifts to an antilinear involution on M DR (X, G) an. Define an antilinear involution by σ : M DR (X, G) an G m M DR (X, G) an G m σ (ρ, λ) = (τ(ρ), λ 1 ).

6 6 PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP The antilinear involution allows one to identify M Hod (X, G) with M Hod (X, G) over G m : M DR (X, G) an G m = MDR (X, G) an G m = MDR (X, G) an G m, where the first map is σ. Using this identification we may define the Deligne moduli space M Del (X, G) = M Hod (X, G) G m M Hod (X, G). The construction is due to Deligne and Simpson [Si]. Theorem 5. M Del (X, G) is a twistor space in the sense of theorem 1. Proof. (1) Define p : M Del (X, G) P 1 as M Hod (X, G) A 1, where A 1 is the patch with the local coordinate λ and M Hod (X, G) A 1, where A 1 is the patch with the local coordinate 1/λ. These maps agree on the intersection. (2) The real structure σ is given by σ over G m and the canonical antilinear isomorphism M Dol (X, G) an = MDol (X, G) an over 0 and. (3) The symplectic structure is the same as in Fujiki s construction. (4) Real sections are given by harmonic bundles as before. If we denote the space of real sections as M, we see that M Del (X, G) = M CP 1 as C manifolds, so we have M Dol (X, G) = M = M DR (X, G). References [Fu] A. Fujiki Hyperkähler structure on the moduli space of flat bundles, Prospects in Complex Geometry, L.N.M (1991), [HKLR] N. Hitchin, A. Karlhede, U. Lindstrøm and M. Ro cek Hyperkähler metrics and supersymmetry, Commun. Math. Phys. 108 (1987), [Hi-87] N. Hitchin The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987), [Hi-92] N. Hitchin Hyperkähler manifolds, Séminaire Bourbaki, exp. n 748, Astérisque, t. 206 (1992), [Si] C. Simpson The Hodge filtration on nonabelian cohomology. Algebraic geometry Santa Cruz 1995, Proc. Sympos. Pure Math., 62, Part 2, A.M.S. (1997),