Let $k\in\mathbb Z$, let $V_{\mathbb Q}$ be a finite-dimensional $\mathbb Q$-[vector space](/page/Vector%20Space), and define

\begin{align*} V_{\mathbb C}:=V_{\mathbb Q}\otimes_{\mathbb Q}\mathbb C. \end{align*}

Equip $V_{\mathbb C}$ with the complex conjugation induced by complex conjugation on the factor $\mathbb C$. Let $F^\bullet V_{\mathbb C}$ be a decreasing $\mathbb Z$-indexed filtration by complex vector subspaces such that $F^pV_{\mathbb C}=V_{\mathbb C}$ for all sufficiently small $p$ and $F^pV_{\mathbb C}=0$ for all sufficiently large $p$.

Then $F^\bullet V_{\mathbb C}$ is the Hodge filtration of a pure $\mathbb Q$-Hodge structure of weight $k$ on $V_{\mathbb Q}$ if and only if, for every $p\in\mathbb Z$,

\begin{align*} V_{\mathbb C}=F^pV_{\mathbb C}\oplus \overline{F^{k-p+1}V_{\mathbb C}}. \end{align*}

In this case the associated [Hodge decomposition](/theorems/2745) is uniquely determined by

\begin{align*} V^{p,q}=F^pV_{\mathbb C}\cap \overline{F^qV_{\mathbb C}} \end{align*}

for all $p,q\in\mathbb Z$ with $p+q=k$, and by $V^{p,q}=0$ for all $p,q\in\mathbb Z$ with $p+q\ne k$.