2.1 Pure structures

In all the following, we denote by \mathbb{C} an algebraic closure of \mathbb{R}, and we do not suppose to have chosen a root i of the equation x^2+1=0. The theory will be invariant under complex conjugation (cf. (2.1.14)).

We denote by S the real algebraic group \mathbb{C}^*, given by restriction of scalars à la Weil from \mathbb{C} to \mathbb{R} of the group \mathbb{G}_\mathrm{m}: \begin{aligned} S &= \prod_{\mathbb{C}/\mathbb{R}}\mathbb{G}_\mathrm{m} \\S(\mathbb{R}) &= \mathbb{C}^*. \end{aligned}

The group S is a torus, i.e. it is connected and of multiplicative type. It is thus described by the free abelian group of finite type X(S) = \operatorname{Hom}(S_\mathbb{C},\mathbb{G}_\mathrm{m}) = \underline{\operatorname{Hom}}(S,\mathbb{G}_\mathrm{m})(\mathbb{C}) of its complex characters, endowed with the action of \operatorname{Gal}(\mathbb{C}/\mathbb{R})=\mathbb{Z}/(2).

The group X(S) has generators z and \overline{z}, which induce (respectively) the identity and complex conjugation: \mathbb{C}^* = S(\mathbb{R}) \to S(\mathbb{C}) \to \mathbb{G}_\mathrm{m}(\mathbb{C}) = \mathbb{C}^*. Complex conjugation exchanges z and \overline{z}.

We have a canonical map w\colon \mathbb{G}_\mathrm{m}\to S \tag{2.1.3.1} that, on the real points, induces the inclusion of \mathbb{R}^* into \mathbb{C}^*. We have zw = \overline{z}w = \mathrm{Id}. \tag{2.1.3.2}

We also have a map N\colon S\to\mathbb{G}_\mathrm{m} \tag{2.1.3.3} that on the real points can be identified with the norm N_{\mathbb{C}/\mathbb{R}}\colon\mathbb{C}^*\to\mathbb{R}^*. We have N = z\overline{z} \tag{2.1.3.4} N\circ w = (x\mapsto x^2). \tag{2.1.3.5}

A real Hodge structure is a real vector space V of finite dimension endowed with an action of the real algebraic group S.

By the general theory of groups of multiplicative type, giving a real Hodge structure on V is equivalent to giving a bigrading V^{p,q} of V_\mathbb{C}=\mathbb{C}\otimes_\mathbb{R}V that satisfies \overline{V^{pq}}=V^{qp}. The action of S and the bigrading are mutually determined via the condition:

On V^{pq}, S acts by multiplication by z^p\overline{z}^q.

Let (\mathbb{C},\times) be the multiplicative monoid, and let \overline{S} = \prod_{\mathbb{C}/\mathbb{R}} (\mathbb{C},\times). If V is a real vector space, then we can show that it is equivalent to either give an action of \overline{S} on V or to give a bigrading V^{pq} on V such that \overline{V^{pq}}=V^{qp} and V^{pq}=0 for p<0 or q<0.

Let V be a real Hodge structure, defined by a representation \sigma of S and a bigrading V^{pq}. The grading of V_\mathbb{C} by the V_\mathbb{C}^n=\sum_{p+q=n}V^{pq} is then defined over \mathbb{R}. We call this the weight grading. On V^n=V\cap V_\mathbb{C}^n, the representation \sigma w of \mathbb{G}_\mathrm{m} is multiplication by x^n.

We say that V is of weight n if V^{pq}=0 for p+q\neq n, i.e. if \sigma w is multiplication by x^n.

Let V be a real Hodge structure. The Hodge filtration on V_\mathbb{C} is defined by F^p(V_\mathbb{C}) = \sum_{p'\geqslant p} V^{p'q'}.

By (1.2.6), we have:

Let n be an integer. Construction (2.1.8) establishes an equivalence of categories between:

the category of real Hodge structures of weight n;
the category of pairs consisting of a real vector space V of finite dimension and of a filtration F on V_\mathbb{C}=\mathbb{C}\otimes_\mathbb{R}V that is n-opposite to its complex conjugate \overline{F}.

A Hodge structure H of weight n consists of

a \mathbb{Z}-module H_\mathbb{Z} of finite type (the “integral lattice”);
a real Hodge structure of weight n on H_\mathbb{R}=\mathbb{R}\otimes_\mathbb{Z}H_\mathbb{Z}.

A morphism f\colon H\to H' is a homomorphism f\colon H_\mathbb{Z}\to H'_\mathbb{Z} such that f_\mathbb{R}\colon H_\mathbb{R}\to H'_\mathbb{R} is compatible with the action of S (i.e. such that f_\mathbb{C} is compatible with the bigrading, or with the Hodge filtration).

Hodge structures of weight n form an abelian category. If H is of weight n, and H' is of weight n', then we define a Hodge structure H\otimes H' of weight n+n' by the equations:

(H\otimes H')_\mathbb{Z}=H_\mathbb{Z}\otimes H'_\mathbb{Z};
the action of S on (H\otimes H')_\mathbb{R}=H_\mathbb{R}\otimes H'_\mathbb{R} is the tensor product of the actions of S on H_\mathbb{R} and on H'_\mathbb{R}.

The bigrading (resp. Hodge filtration) of (H\otimes H')_\mathbb{C}=H_\mathbb{C}\otimes H'_\mathbb{C} is the tensor product of the bigradings (resp. Hodge filtrations (cf. (1.1.12))) of H_\mathbb{C} and of H'_\mathbb{C}.

We define in an analogous manner the Hodge structure \underline{\operatorname{Hom}}(H,H') (of weight n'-n), the Hodge structures \wedge^p H (of weight pn), and the dual Hodge structure H^*.

The internal hom \underline{\operatorname{Hom}} above and the homomorphism group are related by:

\operatorname{Hom}(H,H') is the subgroup of \underline{\operatorname{Hom}}(H,H')_\mathbb{Z} = \operatorname{Hom}_\mathbb{Z}(H_\mathbb{Z},H'_\mathbb{Z}) consisting of elements of degree (0,0).

The actions of S on H_\mathbb{R}, H'_\mathbb{R}, and \underline{\operatorname{Hom}}(H,H')_\mathbb{R}=\operatorname{Hom}(H_\mathbb{R},H'_\mathbb{R}) are related by s(f(x)) = s(f)(s(x)). This means that if f is of degree (0,0), i.e. invariant under S, then it commutes with the action of S.

If A is a Noetherian subring of \mathbb{R}, then we define a Hodge A-structure of weight n to consist of an A-module H_A of finite type along with a real Hodge structure of weight n on H_\mathbb{R}=H_A\otimes_A\mathbb{R}. This definition is mostly used for A=\mathbb{Q}. A Hodge A-structure consists of an A-module H_A of finite type along with a real Hodge structure on H_\mathbb{R}=H_A\otimes_A\mathbb{R} such that the weight grading is defined over the field of fractions of A.

The Tate Hodge structure \mathbb{Z}(1) is the Hodge structure of weight -2, of rank 1, of pure bidegree (-1,-1), with integral lattice 2\pi i\mathbb{Z}\subset\mathbb{C}.

The action of S is thus given by multiplication with the inverse of the norm (2.1.3.3).

For n\in\mathbb{Z}, we define \mathbb{Z}(n) as the n-th tensor power of \mathbb{Z}(1), so \mathbb{Z}(n) is the Hodge structure of weight -2n, of rank 1, of pure bidegree (-n,-n), with integral lattice (2\pi i)^n\mathbb{Z}\subset\mathbb{C}. The action of S is multiplication by N(x)^{-n}.

The choice in \mathbb{C} of a solution i of the equation x^2+1=0 determines, on each complex variety X of pure dimension n, an orientation \operatorname{or}_i(X). Replace i by -i gives \operatorname{or}_{-i}(X) = (-1)^n\operatorname{or}_i(X).

The choice of i also defines an element C of order 4 in S(\mathbb{R}), given by the image of i under the isomorphism S(\mathbb{R})\simeq\mathbb{C}^*.

Finally, it also defines an isomorphism between \mathbb{Z} and the integral lattice of \mathbb{Z}(n), given by multiplication by (2\pi i)^n.

When i, an orientation of X, C, or an identification \mathbb{Z}\sim\mathbb{Z}(n)_\mathbb{Z} appear in an equation, it is implicitly understood that they all follow from a single choice of the same i, and that by replacing i with -i we obtain an equivalent equation.

A polarisation of a Hodge structure H of weight n is a homomorphism (x,y)\colon H\otimes H \to \mathbb{Z}(-n) such that the real bilinear form (2\pi i)^n(x,Cy) on H_\mathbb{R} is symmetric and positive definite.

The real Tate Hodge structure is the real Hodge structure \mathbb{R}(1) underlying \mathbb{Z}(1). We similarly define \mathbb{R}(n) as underlying \mathbb{Z}(1). A polarisation of a real Hodge structure of weight n is a homomorphism (x,y)\colon H\otimes H\to \mathbb{R}(-n) such that the real bilinear form (2\pi i)^n(x,Cy) on H_\mathbb{R} is symmetric and positive definite. A polarisation is entirely defined by the positive definite quadratic form (2\pi i)^n(x,Cy) on H_\mathbb{R}, imposed with only the condition of being invariant under the compact sub-torus of S given by the kernel of N.

We have (x,y) = (Cx,Cy) = (y,C^2x) = (-1)^n(y,x). The form (x,y) is thus symmetric or alternating, depending on the parity of n.

The reader can generalise these definitions to Hodge A-structures of weight n ((2.1.12)).