Introduction to Hodge Structures

Svetlana Makarova MIT

1 Hodge structures

1.1 Definitions

Let V be a free Z-module of finite rank or a finite-dimensional Q-. By VC we will denote the vector space V ⊗ C obtained by extension of scalars (tensor product is over Z or Q, respectively). The complex vector space VC naturally comes with a real structure, i.e. we have an R-linear isomorphism VC → VC defined by complex conjugation.

Definition 1.1. A of weight n ∈ Z on V is given by a direct sum decom- position of the complex vector space VC:

M p,q VC = V p+q=n such that V p,q = V q,p.

Example 1.1 (A trivial example, aka Tate Hodge structure). The Tate Hodge structure is denoted by Z(1). It is the Hodge structure of weight −2 given by the free Z-module of −1,−1 rank 1 such that (Z(1)) is one-dimensional. Similarly, one defines the rational Hodge structure Q(1). We can also define twists of the Tate Hodge structures Z(n) and Q(n) in an obviuos way.

Example 1.2 (Another trivial example). Now take V to be a rank 2 free Z-module, and p,q let e1 and e2 be its basis. We can define the following decomposition: V = he1 + e2 ⊗ ii, q,p V = he1 − e2 ⊗ ii, where p 6= q. On vectors of the form v ⊗ α ∈ V ⊗ C, complex conjugation acts as v ⊗ α = v ⊗ α¯, so clearly V p,q = V q,p, and this gives a Hodge structure of weight p + q.

Example 1.3. For a compact K¨ahlermanifold, the torsion free part of the singular coho- n mology H (X, Z) comes with a natural Hodge structure of weight n given by the standard Hodge decomposition. This example will be discussed in more detail in the next section.

1 2 Harmonic theory and

Let X be a compact orientable n-dimensional Riemann manifold, g be a metric on it. We will denote the tangent bundle of X by T , the exterior powers of cotangent bundle by Ωk, with Ω0 = C∞, and let ν ∈ Ωn(X) be an orientation s.t. g(ν, ν) = 1. The global sections of Ωk will be denoted by Ak.

2.1 Hodge operator and laplacians k - k ∼ k n Ω Hom(Ω , R)=Hom(Ω , Ω )

∗ - ? Ωn−k We can define a global inner product on Ak by the formula

def Z (α, β)L2 = g(α, β)ν. X

Also define the operator d∗ def= (−1)k ∗−1 d∗. Exercise 2.1. The following facts about the Hodge operator are easy to prove. R 1.( α, β)L2 = X α ∧ ∗β. 2. Hodge operator preserves metric.

3. When restricted to Ωk, the following equality holds: ∗2 = (−1)k(n−k).

4. The operators d and d∗ are adjoint with respect to the global inner product.

5. If n is even, then d∗ = − ∗ d∗.

Analogously to the case with d∗, we can define ∂∗ def= (−1)k∗−1∂∗ and ∂¯∗ def= (−1)k∗−1∂¯∗. It is easy to verify that these operators are adjoint (with respect to the global inner product) to ∂ and ∂¯, respectively. Now we will introduce the operators which will help to split the cohomology spaces as a direct summand of the complex of differential forms A•, namely laplacians:

def ∗ ∗ ∆d = dd + d d,

def ∗ ∗ ∆∂ = ∂∂ + ∂ ∂, def ¯¯∗ ¯∗ ¯ ∆∂¯ = ∂∂ + ∂ ∂.

2 One can immediately observe from the definition that the operators ∆∂ and ∆∂¯ preserve the decomposition of forms into types. We define the space of harmonic forms to be the of the laplacian with respect to d:  ∆  Hk def= Ker Ak −−→d Ak . Exercise 2.2. The following facts about laplacians can be proved using (note that analogous statements hold for ∆∂ and ∆∂¯): ∗ ∗ 1.( α, ∆dα)L2 = (dα, dα)L2 + (d α, d α)L2 . ∗ 2. Ker ∆d = Ker d ∩ Ker d . This equality is a consequence of the first one. Theorem 2.3. The complex of differential forms A• can be decomposed into the direct k ∼ k k k k sum A = H ⊕ ∆d(A ). It follows from the decomposition that the projection H → H is an isomorphism. Proof. We will leave without proof the first part of the theorem as it heavily relies on some functional analytic result about elliptic differential operators. So we take the direct sum k ∼ k k decomposition A = H ⊕ ∆d(A ) for granted and deduce the second part of the theorem from it. For simplicity, we will write ∆ instead of ∆d for the rest of the proof. First we will prove surjectivity. Take a closed form α ∈ Ak. We can uniquely write it in the form α = β + ∆γ, where β is harmonic. So d∗dγ = α − β − dd∗γ, where the right hand side is closed, so d∗dγ is also closed. But Kerd∩Imd∗ = 0, as the two operators are adjoint, so d∗dγ = 0 and α = β + dd∗γ. The latter equality means that [α] = [β] in cohomology, so for any cohomology class we can find a harmonic representative. Surjectivity is proved. Now we will prove injectivity by taking α ∈ Hk and assuming that [α] = 0, that is α is an exact form. By the last exercise, d∗α = 0, so α ∈ Ker d∗ ∩ Im d = 0, hence injectivity follows.

To be able to induce the decomposition of cohomology classes into types from that of differential forms, one needs an additional assumption. That leads us to the definition of K¨ahlermanifolds.

2.2 Case of K¨ahlermanifolds First, we need to mention a linear algebra lemma in order to be able to give the definition of a K¨ahlermanifold. Let V be a complex vector space. Then we can introduce the real vector space W def= ∼ HomR (V, R) of R-linear functions and the complex vector space WC = W ⊗ C of C-valued R-linear functions. The latter admits the decomposition into the C-linear and C-antilinear parts: 1,0 0,1 WC = W ⊕ W . This induces a decomposition of wedge powers of W , and we set W 1,1 def= W 1,1 ∩ V2 W . C R

3 Lemma 2.4. There is a natural bijection between the set of hermitian forms on V and the set of elements of W 1,1 given by the formula: R h 7→ ω = −Im h. Exercise 2.5. Prove the lemma. Suggested formula for the inverse map ω 7→ h is h(u, v) = ω(u, Iv) − iω(u, v), where I stands for the complex structure on X. Definition 2.1. We define a K¨ahlermanifold as a X with a hermitian metric h such that the corresponding (1, 1)-form ω is closed. Example 2.1. Projective space is a K¨ahlermanifold with K¨ahler metric being Fubini– Studi metric. Example 2.2. As a corollary of the previous example, every projective complex manifold is K¨ahler,by the restriction of the K¨ahlerform. From now on, assume that X is an n-dimensional compact K¨ahlermanifold with K¨ahler metric h and K¨ahlerform ω. We denote by L : A• → A•+2 Lefschetz operator which takes C C α to ω ∧ α, and by Λ its adjoint with respect to (·, ·)L2 . Fact 2.6. For an n-dimensional K¨ahlermanifold X, for any point x ∈ X, there exists an open neighborhood with holomorphic coordinates z1,..., zn, such that in these coordinates the matrix of the metric is just the identity matrix up to second order, that is h = E + P 2 O i |zi| . Fact 2.7 (K¨ahleridentities). We have the following identities: [Λ, ∂¯] = −i∂∗, [Λ, ∂] = i∂¯∗. Fact 2.7 is proved locally with the use of 2.6, so the proof really boils down to some tedious linear algebra. Theorem 2.8. On a K¨ahlermanifold (not necessarily compact), we have the following equalities between the three laplacians: ∆d = 2∆∂ = 2∆∂¯. Proof. We will prove only the first equality. The second equality will be obtained by complex conjugation. Now recall the definition of ∆d and the formula for d to obtain: ∗ ∗ ¯ ∗ ¯∗ ∗ ¯∗ ¯ ∆d = dd + d d = (∂ + ∂)(∂ + ∂ ) + (∂ + ∂ )(∂ + ∂). Use K¨ahleridentities stated in Fact 2.7: ¯ ∗ ∗ ¯ ∆d = (∂ + ∂)(∂ − i[Λ, ∂]) + (∂ − i[Λ, ∂])(∂ + ∂) = = ∂∂∗ − i∂Λ∂ + i∂∂Λ + ∂∂¯ ∗ − i∂¯Λ∂ + i∂∂¯ Λ+ + ∂∗∂ + ∂∗∂¯ − iΛ∂∂ − iΛ∂∂¯ + i∂Λ∂ + i∂Λ∂¯.

4 The underlined parts naturally cancel each other, and also ∂∂ = 0, hence we have the following equality (do not mind the underlined parts yet):

∗ ¯ ∗ ¯ ¯ ∗ ∗ ¯ ¯ ¯ ∆d = ∂∂ + ∂∂ − i∂Λ∂ + i∂∂Λ + ∂ ∂ + ∂ ∂ − iΛ∂∂ + i∂Λ∂.

Recall another K¨ahler equality: i(Λ∂¯ − ∂¯Λ) = ∂∗. With the use of the relation ∂¯∂¯ = 0, we get: ∂∗∂¯ = i(Λ∂¯∂¯ − ∂¯Λ∂¯) = −i(∂¯Λ∂¯ − ∂¯∂¯Λ) = −∂∂¯ ∗. Use the obtained equality to obtain that the underlined terms in the penultimate equation cancel each other, then recall that ∂¯ and ∂ anticommute, and finally make use of a K¨ahler identity once more to conclude the proof:

∗ ¯ ¯ ∗ ¯ ¯ ∆d = ∂∂ − i∂Λ∂ + i∂∂Λ + ∂ ∂ − iΛ∂∂ + i∂Λ∂ = = (∂∂∗ + ∂∗∂) − i∂¯Λ∂ − i∂∂¯Λ + iΛ∂∂¯ + i∂Λ∂¯ = ¯ ¯ ∗ ∗ = ∆∂ + i[Λ, ∂]∂ + i∂[Λ, ∂] = ∆∂ + ∂ ∂ + ∂∂ = ∆∂ + ∆∂ = 2∆∂.

Corollary 2.9. ∆d is bihomogeneous, that is preserves the decomposition of forms into types.

Proof. This follows from the last theorem and from the analogous fact for ∆∂ Corollary 2.10. If α ∈ Ak (X) is harmonic, then so are its components αp,q. C Corollary 2.11. We have the following direct sum decomposition for harmonic forms:

k M p,q H (X, C) = H (X), p+q=k where Hp,q = Ker ∆ : Ap,q → Ap,q. d C C When the K¨ahlermanifold in question is compact, we can identify H• with H• by the results of the previous section, so the aforementioned decomposition of harmonic forms into types induces the direct sum decomposition of cohomology called Hodge decompositon:

k M p,q H (X, C) = H (X). p+q=k

Proposition 2.12. Hodge decomposition does not depend on K¨ahlermetric. Hp,q = Hq,p.

5 Proof. As above, X denotes the K¨ahlermanifold in question with the aforementioned assumptions. For simplicity, in this proof we will denote ∆d by ∆. p,q k p,q Let K ⊂ H (X, C) be the image of A in cohomology under the natural surjection. We will show that Hp,q = Kp,q. Obviously, Hp,q ⊂ Kp,q, so it remains to show that if p,q α ∈ K , then [α] can be represented by a harmonic form of type (p, q). We know that Ap,q = Hp,q ⊕ ∆(Ap,q) , so let β ∈ Hp,q and γ ∈ Ap,q be such that α = β + ∆γ. Recall p,q ∗ ∗ ∗ that H = Ker (d|Ap,q ) ∩ Ker (d |Ap,q ), so 0 = dα = 0 + d∆γ, hence ∆γ = dd γ + d dγ is closed, which in turn implies that d∗dγ ∈ Ker d. But Ker d ∩ Im d∗ = 0, so d∗dγ = 0, and α = β + dd∗γ. So, α is cohomologous to the harmonic form β of type (p, q), which concludes the proof.

Corollary 2.13. On a compact K¨ahlermanifold, Hp,q = Hq,p.

Proof. The statement follows from the same fact for Kp,q = Kq,p.

Corollary 2.14. The dimensions of odd cohomology are even.

3 Lefschetz decomposition

Lemma 3.1. On an n-dimensional K¨ahlermanifold (not necessarily compact), we have following commutator identity: [L, Λ]|Ak = (k − n)Id. The proof of this lemma again boils down to some tedious linear algebra in local coor- dinates, so we will omit it. The interested reader is referred to §6.2 in [1].

Corollary 3.2. We can define sl2-action on the cohomology of X via the operators L,Λ and C, where C|Ak = (k − n)Id. Lemma 3.3. Ln−k :Ωk → Ω2n−k is an isomorphism.

Proof. It is enough to prove that there is an isomorphism Ln−k : Ak → A2n−k by Serre– Swan argument. • By the last corollary, we have a structure of an sl2-representation on A . One can easily verify that this is a locally finite dimensional representation, so it splits into the direct sum of irreducible sl2-representations. We know how finite-dimensional irreducible sl2-representations look like, namely we know that if V is such a representation of dimension l + 1, then there exists a vector v ∈ V such that V is spanned by v, Lv,..., Llv, and all these vectors are eigenvectors with respect to C with eigenvalues l, l−2, . . . , −l. So we can now observe that if a form α lies in Ak, then it has eigenvalue k − n, so the representation that the images of α span is not less than (k − n + 1)-dimensional, and hence Ln−kα 6= 0. So we have proved that Ln−k is injective. But it is enough to prove injectivity, because the vector bundles in question have the same rank.

6 Definition 3.1. A differential form α ∈ Ωk, where k ≤ n, is called primitive if Ln−k+1α = 0.

Note that by representation theory, primitive forms are exactly the lowest weight vectors with respect to L, that is Λ sends them to zero, and their images under iterated L span irreducible representations.

k Theorem 3.4. Any differential form α ∈ ΩX,x is uniquely representable as a sum α = P r k−2r r L αr, where αr ∈ ΩX,x is primitive and r ≥ max(0, k − n). This theorem is proved by a similar representation-theoretic argument. We now want to define the analogous decomposition on the cohomology. For this, we first observe that L is well-defined on cohomology, because by the definition of K¨ahler manifold, ω is closed, hence defines a cohomological class. Then, we want to use the identification H• =∼ H• (which holds in the compact case), and in order to use it, we need to prove that ∆d commutes with L.

Lemma 3.5. On a K¨ahlermanifold, the operators ∆d and L commute.

Proof. Since ∆d = 2∆∂, it is enough to prove that ∆∂ commutes with L. Note that ω is d closed, hence ∂-closed (by considering the bidegree).

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ [∆∂,L] = ∂∂ L + ∂ ∂L − L∂∂ − L∂ ∂ = ∂∂ L + ∂ L∂ − ∂L∂ − L∂ ∂ = = ∂[∂∗,L] + [∂∗,L]∂ = −i∂∂¯ − i∂∂¯ = 0.

Note that we have used the adjoint to a K¨ahleridentity, namely −i∂¯ = [∂∗,L], and then the anticommutativity of ∂ and ∂¯.

It follows from the lemma that L is well-defined on H• and that the isomorphism H• =∼ H• respects the L-action. Since Λ is the adjoint to L, the same holds for Λ as well, and the commutator relation restricts to H•. Now, we can once again invoke the representation-theoretic machinery to prove the following two theorems.

Theorem 3.6 (Hard Lefschetz theorem). For a compact K¨ahlermanifold X, the operator n−k k 2n−k L : H (X, R) → H (X, R) is an isomorphism. Theorem 3.7 (Lefschetz decomposition). For each class of cohomology α ∈ Hk there P r k−2r exists a unique decomposition α = r L αr, where αr ∈ H for r ≥ max(0, k − n), and n−k+2r+1 L αr = 0.

7 4 Polarization

We have seen that if we have an n-dimensional compact K¨ahlermanifold with K¨ahlerform ω, then the operator L = ω ∧ − induces the Lefschetz decomposition

k M r k−2r H (X, R) = L Hprim (X, R). r

k Moreover, for k ≤ n, L induces an intersection form on H (X, R) by the following formula: Z D E Q(α, β) = ωn−k ∧ α ∧ β = Ln−kα, β . X This form is skew-symmetric when k is odd, and symmetric when k is even. The induced hermitian form

H(α, β) def= ikQ(α, β¯)

k on H (X, C) possesses the following properties: 1. Hodge decomposition is orthogonal with respect to this form.

2. Lefschetz decomposition is orthogonal with respect to this form.

p−q−k k(k−1) 3. The inequality i (−1) 2 H(α) ≥ 0 holds for any primitive (p, q)-form α, k = p + q.

2 If K¨ahlerclass [ω] lies in H (X, Z), then Lefschetz operator L is defined on k n−k+1 cohomology, and so we can define a subgroup H (X)prim ⊂ Ker L . Moreover, the Q is Z-valued. This structure satisfies the following definition: Definition 4.1. A polarized integral Hodge structure of weight k is an integral Hodge struc- L p,q ture (V,VC = V ) of weight k with a bilinear form Q on V which is skew-symmetric for even k and symmetric for odd k and which satisfies conditions 1 and 3 from above. So, to obtain a polarized Hodge structure from a K¨ahlermanifold, we need to choose 2 a class [ω] ∈ H (X, R) such that ω is an integral K¨ahlerform.

5 Examples

5.1 Projective space n We can easily calculate the cohomology of a P (using what- k n ever cohomology theory) and obtain that H (P , Z) = 0 for odd k or for k > 2n, and 2k n ∼ 2k H (P , Z) = Z for k ≤ n. Hodge structure on H cannot be anything but Tate Hodge structure, since the group is of rank one.

8 5.2 Weight one Hodge structures and abelian varieties

Consider a compact K¨ahlermanifold X. There is a weight one Hodge structure on Γ def= 1 H (X, Z) (consider Γ as already modded out by torsion): 1 1,0 0,1 H (X, C) = H (X) ⊕ H (X). We have the following inclusion of a into a complex vector space:

1 0,1 Γ ,→ H (X, R) → H (X).

So we can take the quotient T def= H0,1(X)/Γ and obtain a complex torus. This torus is called the Picard variety of X, that is Pic0 X = T , and it parametrizes isomorphism classes of holomorphic line bundles on X with zero Chern class. Note that a torus is also a K¨ahlermanifold, and in fact we have a 1 − 1 correspondence between Hodge structures of weight one (with no summands where at least one degree is negative) and complex tori. The identification works as follows:

∗ 1 Γ = H1(T, Z) and V = H (T, R).

5.3 K3 surfaces

We call a compact complex manifold X a if its canonical class KX is trivial and 1 H (OX ,X) = 0. It turns out that all K3 surfaces are automatically K¨ahler. Proposition 5.1. The Hodge diamond of a K3 surface X looks as follows:

1 0 0 1 20 1 0 0 1

Proof. Recall that for a E of rank r on X we have: 1 ch(E) = r + c (E) + c (E)2 − 2c (E) 1 2 1 2 and 1 1 td(E) = 1 + c (E) + c (E)2 + c (E) . 2 1 12 1 2 First, apply the Hirzebruch–Riemann–Roch theorem for E = O, O being the structure sheaf O = OX . Also, ct(O) = 1, so ch(O) = 1 and    1 1 2  1 2  χ(O) = deg 1. 1 + c1(T ) + c1(T ) + c2(T ) = c1(T ) + c2(T ) . 2 12 2 12

9 Recall that c1(T ) = c1(det T ) = c1(O) = 0, and use the fact that χ(O) = 2 for a K3 surface. So we get the following equality: c (T ) 2 = χ(O) = 2 , 12 hence c2(T ) = 24. Now apply the Hirzebruch–Riemann–Roch formula to T :     c2(T ) χ(T ) = deg ch(T ).td(T ) 2 = deg (2 − c2(T )) . 1 + = 4 − 24 = −20. 12 2

6 Morphisms of Hodge structures and strictness

6.1 Strictness Definition 6.1. Define a morphism of Hodge structures V , W of weights n and m = n+2r, respectively, as a Z-linear or Q-linear, respectively, homomorphism f : V → W such that p,q p+r,q+r its C-linear extension maps V to W . We will say that such a morphism has type (r, r). To define strictness, we first need to introduce the concept of Hodge filtration:

p def M r,n−r F VC = V . r≥p This filtration is descending, and for it we have a direct sum decomposition:

p k−p+1 VC = F VC ⊕ F VC. The subspaces defining the Hodge structure can be expressed in terms of Hodge filtration:

p,q p q V = F VC ∩ F VC. Lemma 6.1 (Strictness). If f is a morphism of Hodge structures f : V → W as in the p+r definition, then it is strict with respect to Hodge filtration, that is Im f ∩ F WC = p f(F VC). Corollary 6.2. Filtration on Im f induced from W coincides with quotient filtration induced from V . This filtration defines Hodge structure on Im f. Lemma 6.3. Restriction of Hodge structure from V to Ker f defines Hodge structure on Ker f. To sum up the aforementioned two results, we can say that Hodge structures comprise an . Polarized Hodge structures with morphisms respecting polarization form a semisimple abelian category.

10 6.2 Functoriality Given a morphism of K¨ahler manifolds f : X → Y of dimensions n and m = n + r, respectively, one can define two natural morphisms of Hodge structures. The first one is just pullback defined in the usual way:

∗ k k f : H (Y, Z) → H (X, Z).

One can easily observe that f ∗ is a morphism of Hodge structures (by considering local coordinates, for example). The second natural morphism is called Gysin map and is defined with the use of Poincare duality as the following composition (here think of all the homology and co- homology groups as already factorized by torsion):

k ∼ f∗ ∼ k+2r f∗ : H (X, Z) = H2n−k(X, Z) −→ H2n−k(Y, Z) = H (Y, Z).

One can observe that this morphism of Hodge structures is of type (r, r).

References

[1] Voisin, C. and Complex I

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