# Hyperkähler geometry lecture 3

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1 Hyperkähler geometry lecture 3 Misha Verbitsky Cohomology in Mathematics and Physics Euler Institute, September 25, 2013, St. Petersburg 1

2 Broom Bridge Here as he walked by on the 16th of October 1843 Sir William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplication and cut it on a stone of this bridge. I 2 = J 2 = K 2 = IJK = 1 2

3 Geometric structures DEFINITION: A geometric structure (Elie Cartan, Charles Ehresmann) is an atlas on a manifold, with the differentials of all transition functions in a given subgroup G GL(n, R). EXAMPLE: GL(n, C) GL(2n, R) ( the complex structure ). EXAMPLE: Sp(n, R) GL(2n, R) ( the symplectic structure ). Quaternionic structures in the sense of Elie Cartan don t exist. THEOREM: Let f : H n H m be a function, defined locally in some open subset of n-dimensional quaternion space H n. Suppose that the differential Df is H-linear. Then f is a linear map. Proof (a modern one): The graph of f is a hyperkähler ( trianalytic ) submanifold in H n H m, hence geodesically complete, hence linear. 3

4 Complex manifolds (reminder) DEFINITION: Let M be a smooth manifold. An almost complex structure is an operator I : T M T M which satisfies I 2 = Id T M. The eigenvalues of this operator are ± 1. The corresponding eigenvalue decomposition is denoted T M = T 0,1 M T 1,0 (M). DEFINITION: An almost complex structure is integrable if X, Y T 1,0 M, one has [X, Y ] T 1,0 M. In this case I is called a complex structure operator. A manifold with an integrable almost complex structure is called a complex manifold. REMARK: The usual definition : complex structure is a geometric structure: an atlas on a manifold with differentials of all transition functions in GL(n, C). THEOREM: (Newlander-Nirenberg) These two definitions are equivalent. 4

5 Kähler manifolds (reminder) DEFINITION: An Riemannian metric g on an almost complex manifiold M is called Hermitian if g(ix, Iy) = g(x, y). In this case, g(x, Iy) = g(ix, I 2 y) = g(y, Ix), hence ω(x, y) := g(x, Iy) is skew-symmetric. DEFINITION: The differential form ω Λ 1,1 (M) is called the Hermitian form of (M, I, g). DEFINITION: A complex Hermitian manifold (M, I, ω) is called Kähler if dω = 0. The cohomology class [ω] H 2 (M) of a form ω is called the Kähler class of M, and ω the Kähler form. DEFINITION: Let (M, g) be a Riemannian manifold. A connection is called orthogonal if (g) = 0. It is called Levi-Civita if it is orthogonal and torsion-free. THEOREM: Let (M, I, g) be an almost complex Hermitian manifold. Then the following conditions are equivalent. (i) (M, I, g) is Kähler (ii) One has (I) = 0, where is the Levi-Civita connection. 5

6 Algebraic geometry over H. Over C, we have 3 distinct notions of algebraic geometry : 1. Scheme over C: locally ringed space with Zariski topology, each ring a quotient of a polynomical ring. 2. Complex manifold: manifold with holomorphic transition functions. 3. Kähler manifold: complex structure, metric, Levi-Civita connection, I = 0. The first notion does not work for H, because polynomial functions on H n generate all real polynomials on R 4. The second version does not work, because any quaternionic-differentiable function is linear. The third one works! Hyperkähler manifolds. 6

7 Hyperkähler manifolds Eugenio Calabi DEFINITION: (E. Calabi, 1978) Let (M, g) be a Riemannian manifold equipped with three complex structure operators I, J, K : T M T M, satisfying the quaternionic relation I 2 = J 2 = K 2 = IJK = Id. Suppose that I, J, K are Kähler. Then (M, I, J, K, g) is called hyperkähler. 7

8 Holomorphic symplectic geometry CLAIM: A hyperkähler manifold (M, I, J, K) is holomorphically symplectic (equipped with a holomorphic, non-degenerate 2-form). Recall that M is equipped with 3 symplectic forms ω I, ω J, ω K. LEMMA: The form Ω := ω J + 1ω K is a holomorphic symplectic 2-form on (M, I). Converse is also true, as follows from the famous conjecture, made by Calabi in THEOREM: (S.-T. Yau, 1978) Let M be a compact, holomorphically symplectic Kähler manifold. Then M admits a hyperkähler metric, which is uniquely determined by the cohomology class of its Kähler form ω I. Hyperkähler geometry is essentially the same as holomorphic symplectic geometry 8

9 Hyperkähler algebraic geometry is almost as good as the usual one. Define trianalytic subvarieties as closed subsets which are complex analytic with respect to I, J, K. 0. Trianalytic subvarieties are singular hyperkähler. 1. Let L be a generic quaternion satisfying L 2 = 1. Then all complex subvarieties of (M, L) are trianalytic. 2. A normalization of a hyperkähler variety is smooth and hyperkähler. This gives a desingularization ( hyperkähler Hironaka ). 3. A complex deformation of a trianalytic subvariety is again trianalytic, the corresponding moduli space is (singularly) hyperkähler. 4. Similar results are true for vector bundles which are holomorphic under I, J, K ( hyperholomorphic bundles ) 9

10 Examples of hyperkähler manifolds (reminder) EXAMPLE: An even-dimensional complex vector space. EXAMPLE: An even-dimensional complex torus. EXAMPLE: A non-compact example: T CP n (Calabi). REMARK: T CP 1 is a resolution of a singularity C 2 /±1. EXAMPLE: Take a 2-dimensional complex torus T, then the singular locus of T/±1 is of form (C 2 /±1) T. Its resolution T/±1 is called a Kummer surface. It is holomorphically symplectic. REMARK: Take a symmetric square Sym 2 T, with a natural action of T, and let T [2] be a blow-up of a singular divisor. Then T [2] is naturally isomorphic to the Kummer surface T/±1. DEFINITION: A complex surface is called K3 surface if it a deformation of the Kummer surface. THEOREM: (a special case of Enriques-Kodaira classification) Let M be a compact complex surface which is hyperkähler. Then M is either a torus or a K3 surface. 10

11 Hilbert schemes (reminder) DEFINITION: A Hilbert scheme M [n] of a complex surface M is a classifying space of all ideal sheaves I O M for which the quotient O M /I has dimension n over C. REMARK: A Hilbert scheme is obtained as a resolution of singularities of the symmetric power Sym n M. THEOREM: (Fujiki, Beauville) A Hilbert scheme of a hyperkähler surface is hyperkähler. EXAMPLE: A Hilbert scheme of K3 is hyperkähler. EXAMPLE: Let T be a torus. Then it acts on its Hilbert scheme freely and properly by translations. For n = 2, the quotient T [n] /T is a Kummer K3-surface. For n > 2, it is called a generalized Kummer variety. REMARK: There are 2 more sporadic examples of compact hyperkähler manifolds, constructed by K. O Grady. All known compact hyperkaehler manifolds are these 2 and the three series: tori, Hilbert schemes of K3, and generalized Kummer. 11

12 Induced complex structures LEMMA: Let be a torsion-free connection on a manifold, I End T M an almost complex structure, I = 0. Then I is integrable. Proof: Let X, Y T 1,0 M, then [X, Y ] = X Y Y X T 1,0 M. DEFINITION: Induced complex structures on a hyperkähler manifold are complex structures of form S 2 = {L := ai + bj + ck, a 2 + b 2 + c 2 = 1.} They are usually non-algebraic. Indeed, if M is compact, for generic a, b, c, (M, L) has no divisors. REMARK: Because of the Lemma above, induced complex structure operators are integrable. 12

14 Hodge theory: supercommutator DEFINITION: A supercommutator of pure operators on a graded vector space is defined by a formula {a, b} = ab ( 1)ã b ba. DEFINITION: A graded associative algebra is called graded commutative (or supercommutative ) if its supercommutator vanishes. EXAMPLE: The Grassmann algebra is supercommutative. DEFINITION: A graded Lie algebra (Lie superalgebra) is a graded vector space g equipped with a bilinear graded map {, } : g g g which is graded anticommutative: {a, b} = ( 1)ã b {b, a} and satisfies the super Jacobi identity {c, {a, b}} = {{c, a}, b} + ( 1)ã c {a, {c, b}} EXAMPLE: Consider the algebra End(A ) of operators on a graded vector space, with supercommutator as above. Then End(A ), {, } is a graded Lie algebra. Lemma 1: Let d be an odd element of a Lie superalgebra, satisfying {d, d} = 0, and L an even or odd element. Then {{L, d}, d} = 0. Proof: 0 = {L, {d, d}} = {{L, d}, d} + ( 1) L {d, {L, d}} = 2{{L, d}, d}. 14

15 Hodge theory: Laplacian DEFINITION: Let M be an oriented Riemannian manifold. The d -operator is a Hermitian adjoint to d with respect to the product α, β M (α, β) Vol M. DEFINITION: The anticommutator := {d, d } = dd + d d is called the Laplacian of M. It is self-adjoint and positive definite: ( x, x) = (dx, dx) + (d x, d x). Also, commutes with d and d (Lemma 1). THEOREM: (The main theorem of Hodge theory) There is a basis in the Hilbert space L 2 (Λ (M)) consisting of eigenvectors of. THEOREM: ( Elliptic regularity for ) Let α L 2 (Λ k (M)) be an eigenvector of. Then α is a smooth k-form. 15

16 Hodge theory: harmonic forms DEFINITION: The space H i (M) := ker d Λ i M is called the de Rham cohomology of M. d(λ i 1 M) DEFINITION: A form α is called harmonic if (α) = 0. REMARK: Let α be a harmonic form. Then ( x, x) = (dx, dx) + (d x, d x), hence α ker d ker d REMARK: The projection H i (M) H i (M) from harmonic forms to cohomology is injective. Indeed, a form α lies in the kernel of such projection if α = dβ, but then (α, α) = (α, dβ) = (d α, β) = 0. THEOREM: The natural map H i (M) H i (M) is an isomorphism (see the next page). 16

17 Hodge theory and the cohomology THEOREM: The natural map H i (M) H i (M) is an isomorphism. Proof. Step 1: Since d 2 = 0 and (d ) 2 = 0, one has {d, } = 0. This means that commutes with the de Rham differential. Step 2: Consider the eigenspace decomposition Λ (M) = α H α(m), where α runs through all eigenvalues of, and H α(m) is the corresponding eigenspace. For each α, de Rham differential defines a complex H 0 α (M) d H 1 α (M) d H 2 α (M) d... Step 3: On Hα (M), one has dd + d d = α. When α 0, and η closed, this implies dd (η) + d d(η) = dd η = αη, hence η = dξ, with ξ := α 1 d η. This implies that the complexes (Hα(M), d) don t contribute to cohomology. Step 4: We have proven that H (Λ M, d) = α H (H α(m), d) = H (H 0 (M), d) = H (M). 17

18 The Hodge decomposition in linear algebra (reminder) DEFINITION: The Hodge decomposition V R C := V 1,0 V 0,1 is defined in such a way that V 1,0 is a 1 -eigenspace of I, and V 0,1 a 1 - eigenspace. REMARK: Let V C := V R C. The Grassmann algebra of skew-symmetric forms Λ n V C := Λ n R V R C admits a decomposition Λ n V C = p+q=n Λ p V 1,0 Λ q V 0,1 We denote Λ p V 1,0 Λ q V 0,1 by Λ p,q V. The resulting decomposition Λ n V C = p+q=n Λ p,q V is called the Hodge decomposition of the Grassmann algebra. REMARK: The operator I induces U(1)-action on V by the formula ρ(t)(v) = cos t v + sin t I(v). We extend this action on the tensor spaces by muptiplicativity. 18

19 U(1)-representations and the weight decomposition REMARK: Any complex representation W of U(1) is written as a sum of 1-dimensional representations W i (p), with U(1) acting on each W i (p) as ρ(t)(v) = e 1 pt (v). The 1-dimensional representations are called weight p representations of U(1). DEFINITION: A weight decomposition of a U(1)-representation W is a decomposition W = W p, where each W p = i W i (p) is a sum of 1-dimensional representations of weight p. REMARK: The Hodge decomposition Λ n V C = p+q=n Λ p,q V is a weight decomposition, with Λ p,q V being a weight p q-component of Λ n V C. REMARK: V p,p is the space of U(1)-invariant vectors in Λ 2p V. 19

20 Hodge decomposition on cohomology THEOREM: (Bott) Let M be a Riemannian manifold, and V : T M T M an endomorphism satisfying V = 0. Then [V, ] = 0. In particular, if M is compact, V acts on cohomology of M. COROLLARY: The Weil operator W Λ p,q (M) = 1 (p q) acts on cohomology of a compact Kähler manifold, giving the Hodge decomposition: H (M) = H p,q (M). COROLLARY: For any hyperkähler manifold, the group SU(2) of unitary quaternions defines SU(2)-action on cohomology. REMARK: For each induced complex structure L, we have an embedding U(1) SU(2). Therefore, the Hodge decomposition for L = ai + bj + ck is induced by the SU(2)-action. 20

21 SU(2)-action on the cohomology and its applications DEFINITION: Trianalytic subvarieties are closed subsets which are complex analytic with respect to I, J, K. REMARK: Trianalytic subvarieties are hyperkähler submanifolds outside of their singularities. REMARK: Let [Z] be a fundamental class of a complex subvariety Z on a Kähler manifold. Then Z is U(1)-invariant. COROLLARY: A fundamental class of a trianalytic subvariety is SU(2)- invariant. THEOREM: Let M be a hyperkähler manifold. Then there exists a countable subset S CP 1, such that for any induced complex structure L CP 1 \S, all compact complex subvarieties of (M, L) are trianalytic. Its proof is based on Wirtinger s inequality. 21

22 Wirtinger s inequality PROPOSITION: (Wirtinger s inequality) Let V W be a real 2d-dimensional subspace in a complex Hermitian vector space (W, I, g), and ω its Hermitian form. Then Vol g V 1 2 d d! ωd V, and the equality is reached only if V is a complex subspace. COROLLARY: Let (M, I, ω, g) be a Kähler manifold, and Z M its real subvariety of dimension 2d. Then Z Vol Z 1 2 d d! Z ωd, and the equality is reached only if Z is a complex subvariety. REMARK: Notice that Z ωd is a (co)homology invariant of Z, and stays constant if we deform Z. Therefore, complex subvarieties minimize the Riemannian volume in its deformation class. 22

23 Wirtinger s inequality for hyperkähler manifolds DEFINITION: Let (M, I, J, K, g) be a hyperkähler manifold, and Z M a real 2d-dimensional subvariety. Given an induced complex structure L = ai + bj + ck, define the degree deg L (Z) := 1 2 d d! Z ωd L, where ω L(x, y) = g(x, Ly), which gives ω L = aω I + bω J + cω K. Proposition 1: Let Z (M, L) be a complex analytic subvariety of (M, L). (a) Then deg L (Z) has maximum at L. (b) Moreover, this maximum is absolute and strict, unless deg L (Z) is constant as a function of L. (c) In the latter case, Z is trianalytic. Proof. Step 1: By Wirtinger s inequality, Vol g Z deg L (Z), and the equality is reached if and only if Z is complex analytic in (M, L). This proves (a). Step 2: If the maximum is not strict, there are two quaternions L and L such that Z is complex analytic with respect to L and L. This means that T Z is preserved by the algebra of quaternions generated by L and L, hence Z is trianalytic, and deg L (Z) constant. This proves (b) and (c). 23

24 Trianalytic subvarieties in generic induced complex structures THEOREM: Let M be a hyperkähler manifold. Then there exists a countable subset S CP 1, such that for any induced complex structure L CP 1 \S, all compact complex subvarieties of (M, L) are trianalytic. Proof. Step 1: Let R H 2 (M, Z) be the set of all integer cohomology classes [Z], for which the function deg L ([Z]) = [Z] ωd L is not constant, and S the set of all strict maxima of the function deg L ([Z]) for all [Z] R. Then S is countable. Indeed, deg L ([Z]) is a polynomial function. Step 2: Now, let L CP 1 \S. For all complex subvarieties Z (M, L), deg L ([Z]) cannot have strict maximum in L. By Proposition 1 (c), this implies that Z is trianalytic. DEFINITION: A divisor on a complex manifold is a complex subvariety of codimension 1. COROLLARY: For M compact and hyperkähler, and L CP 1 generic, the manifold (M, L) has no complex divisors. In particular, it is non-algebraic. 24

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