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Introduction to Hodge Structures

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Introduction to Hodge Structures Introduction to Hodge Structures Svetlana Makarova MIT Mathematics 1 Hodge structures 1.1 Definitions Let V be a free Z-module of finite rank or a finite-dimensional Q-vector space. 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 Hodge structure of weight n 2 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 2 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 cohomology 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 = C1, and let ν 2 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 kernel 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 linear algebra (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 2 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 2 Hk and assuming that [α] = 0, that is α is an exact form. By the last exercise, d∗α = 0, so α 2 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! ! = −Im h. Exercise 2.5. Prove the lemma. Suggested formula for the inverse map ! 7! h is h(u; v) = !(u; Iv) − i!(u; v), where I stands for the complex structure on X. Definition 2.1. We define a K¨ahlermanifold as a complex manifold X with a hermitian metric h such that the corresponding (1; 1)-form ! 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 !. We denote by L : A• ! A•+2 Lefschetz operator which takes C C α to ! ^ α, and by Λ its adjoint with respect to (·; ·)L2 . Fact 2.6. For an n-dimensional K¨ahlermanifold X, for any point x 2 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 jzij . 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 2 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).
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