2.2 Hodge theory
Let X be a compact Kähler variety (for example, smooth and projective). By the holomorphic Poincaré lemma, the de Rham complex \Omega_X^\bullet is a resolution of the constant sheaf \mathbb{C}. We thus have an isomorphism (1.4.2) \operatorname{H}^\bullet(X,\mathbb{C}) \sim \operatorname{\mathbb{H}}^\bullet(X,\Omega_X^\bullet) and the stupid filtration on \Omega_X^\bullet (1.4.5) defines the hypercohomology spectral sequence E_1^{pq} = \operatorname{H}^q(X,\Omega_X^q) \Rightarrow\operatorname{H}^{p+q}(X,\mathbb{C}) \tag{2.2.1.1} which abuts to the Hodge filtration of \operatorname{H}^\bullet(X,\mathbb{C}).
By Hodge theory (Hodge 1952; Weil 1958), we have:
- The spectral sequence (2.2.1.1) degenerates: E_1=E_\infty.
- The Hodge filtration of \operatorname{H}^n(X,\mathbb{C}) is n-opposite to the complex conjugate filtration.
Let V be a local system of real vector spaces over X, i.e. a sheaf of \mathbb{R}-vector spaces that is locally isomorphic to a constant sheaf \mathbb{R}^n. Suppose that there exists on V a bilinear form Q\colon V\otimes V\to\mathbb{R} that is locally constant and defined. If X is connected, this is the case if V is defined by a representation of a finite quotient of the fundamental group of X.
Statements (a) and (b) in (2.2.1) remain true as they are, without needing to change the proofs, for cohomology with coefficients in V_\mathbb{C}=V\otimes_\mathbb{R}\mathbb{C}, since the spectral sequence E_1^{pq} = \operatorname{H}^q(X,\Omega_X^p(V)) \Rightarrow\operatorname{H}^{p+q}(X,V_\mathbb{C}) induced by the de Rham resolution of V_\mathbb{C} by \Omega_X^\bullet(V_\mathbb{C}) degenerates, and abuts to a filtration on \operatorname{H}^n(X,V_\mathbb{C}) that is n-opposite to the complex conjugate filtration.
The vector space \operatorname{H}^n(X,V) is thus endowed with a canonical real Hodge structure of weight n.
We show in (Deligne 1968) that the statements in (2.2.1) remain true when X is a non-singular complete algebraic, not necessarily Kähler, variety. The proof from loc. cit., based on a reduction to the projective case by Chow’s lemma and resolution of singularities, extends to the case of (2.2.2).
Let \mathcal{L} be an invertible sheaf. Here are two ways of defining the class c_1(\mathcal{L}).
The sheaf \mathcal{L} defines an element c in \operatorname{H}^1(X,\mathcal{O}^*). Its image under \mathrm{d}f/f\colon\mathcal{O}^*\to\Omega^1 lies in \operatorname{H}^1(X,\Omega^1). More precisely, \mathrm{d}f/f defines a morphism of complexes \mathrm{d}\log\colon \mathcal{O}^*[-1] \to (0\to\Omega_X^1\to\Omega_X^2\to\ldots) = \sigma_{\geqslant1}(\Omega_X^\bullet). This complex maps to \Omega_X^\bullet, whence \mathrm{d}\log\colon \mathcal{O}^*[-1]\to\Omega_X^\bullet and the image of c under \mathrm{d}\log is in \operatorname{\mathbb{H}}^2(\Omega_X^\bullet). This construction still makes sense for an algebraic variety over an arbitrary field k. For k=\mathbb{C}, we further have \operatorname{H}^2(X,\mathbb{C}) \xrightarrow{\sim}\operatorname{\mathbb{H}}^2(X,\Omega_X^\bullet) whence a class c'_1(\mathcal{L}) \in \operatorname{H}^2(X,\mathbb{C}).
The exponential exact sequence 0 \to \mathbb{Z}(1) \to \mathcal{O} \to \mathcal{O}^* \to 0 defines a homomorphism \partial\colon \operatorname{H}^1(X,\mathcal{O}^*)\to\operatorname{H}^2(X,\mathbb{Z}(1)) whence a class \partial c = c''_1(\mathcal{L}) \in \operatorname{H}^2(X,\mathbb{Z}(1)).
If we have a choice of i, then this class is identified with c'''_1(\mathcal{L}) \in \operatorname{H}^2(X,\mathbb{Z}) and for \mathcal{L}=\mathcal{O}(D), this c'''_1(\mathcal{L}) is exactly the integer cohomology class defined by D (with the orientations being defined by i).
We can prove:
For \alpha the natural injection of \mathbb{Z}(1) into \mathbb{C}, we have -\alpha c''_1(\mathcal{L}) = c'_1(\mathcal{L}).
For \beta the natural injection of \mathbb{Z} into \mathbb{C}, we have -\beta c'''_1(\mathcal{L}) = \frac{1}{2\pi i}c'_1(\mathcal{L}).
We prove this by considering the following diagram of complexes, in which \approx denotes a quasi-isomorphism, and whose upper triangle anti-commutes up to homotopy: \begin{CD} \mathbb{C}@>\approx>> \Omega_X^\bullet @<<< \sigma_{\geqslant1}(\Omega_X^\bullet) \\@| @. @| \\\mathbb{C}@<<< (\mathbb{C}\to\mathcal{O}) @>\approx>\mathrm{d}> \sigma_{\geqslant1}(\Omega_X^\bullet) \\@AAA @AAA @AA{\mathrm{d}\log}A \\\mathbb{Z}(1) @<<< (\mathbb{Z}(1)\to\mathcal{O}) @>\approx>\exp> \mathcal{O}^*[-1] \end{CD}
Let X be a non-singular projective variety of pure dimension n. A choice of i defines an orientation of X and an isomorphism \mathbb{Z}(1)\simeq\mathbb{Z}. The corresponding trace morphism \operatorname{H}^{2n}(X,\mathbb{Z}(n)) \to \mathbb{Z} that is induced does not depend on the choice of i.
By Hodge, for i\leqslant n, the morphism L^{n-i} = -\wedge c''_1(\mathcal{O}(1))^{n-i}\colon \operatorname{H}^i(X,\mathbb{Z}) \to \operatorname{H}^{2n-i}(X,\mathbb{Z}(n-i)) is an isomorphism, and, combined with Poincaré duality \operatorname{H}^i(X,\mathbb{Z})\otimes\operatorname{H}^{2n-i}(X,\mathbb{Z}(n-i)) \to \operatorname{H}^{2n}(X,\mathbb{Z}(n-i)) \to \mathbb{Z}(-i) gives a polarisation on the primitive part \operatorname{Ker}(L^{n-i+1}) of \operatorname{H}^i(X,\mathbb{Z}).
We thus deduce that the rational Hodge structures \operatorname{H}^i(X,\mathbb{Q}) are polarisable.