Morphisms, polarizability and simplicity of rational Hodge structures

With the conventions of Compact Kähler and rational Hodge polarization conventions, all structures in this item are rational pure Hodge structures $(V,\{V^{p,q}\})$, $V_{\mathbb{C}}={⨁}_{p+q=k}{V}^{p,q}$, of one fixed weight $k$; write $f_{\mathbb{C}}$ for the $\mathbb{C}$-linear extension of a $\mathbb{Q}$-linear map.

A morphism of Hodge structures $f:(V,\{V^{p,q}\})\to (W,\{W^{p,q}\})$ is a $\mathbb{Q}$-linear map $f:V\to W$ whose $\mathbb{C}$-linear extension satisfies $f_{\mathbb{C}}(V^{p,q})\subseteq W^{p,q}(p+q=k)\mathrm{.}$ Kernels and images of such a morphism are again rational sub-Hodge structures: writing $x={\sum }_{p+q=k}{x}^{p,q}$ with $x^{p,q}\in V^{p,q}$, the relation $0=f_{\mathbb{C}}(x)={\sum }_{p+q=k}{f}_{\mathbb{C}}(x^{p,q})$ with $f_{\mathbb{C}}(x^{p,q})\in W^{p,q}$ forces $f_{\mathbb{C}}(x^{p,q})=0$ for every $(p,q)$, because the sum decomposition of $W_{\mathbb{C}}$ is direct, so $\ker (f_{\mathbb{C}})={⨁}_{p+q=k}(\ker (f_{\mathbb{C}})\cap V^{p,q});$ and $\mathrm{im}(f_{\mathbb{C}})={\sum }_{p+q=k}{f}_{\mathbb{C}}(V^{p,q})$ is direct, since $f_{\mathbb{C}}(V^{p,q})\subseteq W^{p,q}$ and the target bidegrees of a direct sum are distinct, so $\mathrm{im}(f_{\mathbb{C}})={⨁}_{p+q=k}(\mathrm{im}(f_{\mathbb{C}})\cap W^{p,q})\mathrm{.}$ The direct sum $V\oplus W$ is the Hodge structure of the same weight $k$ with $(V\oplus W{)}^{p,q}=V^{p,q}\oplus W^{p,q}$, the inclusions and projections being morphisms.

A short exact sequence $0\to U\stackrel{i}{\to }V\stackrel{\pi }{\to }W\to 0$ of Hodge structures of weight $k$ consists of morphisms that are exact as a sequence of $\mathbb{Q}$-vector spaces. It splits in the category of Hodge structures if $\pi$ admits a morphism $s:W\to V$ with $\pi \circ s={\mathrm{id}}_W$. A Hodge structure is polarizable if it admits a polarization, and simple if $V\mathrm{\ne }0$ and it has no nonzero proper rational sub-Hodge structure, that is, no $\mathbb{Q}$-subspace $U\subseteq V$ with $U\mathrm{\ne }0,V$ and $U_{\mathbb{C}}={⨁}_{p+q=k}(U_{\mathbb{C}}\cap V^{p,q})$.

• This item records, in the pure weight-$k$ case, the morphism, direct-sum, short-exact-sequence, polarizability and simplicity vocabulary of Deligne's $\mathbb{Q}$-Hodge structures, Théorie de Hodge II, (2.1.10)-(2.1.17) and (2.2.1); the orthogonal splitting of polarized Hodge structures is proved on this page. It is deliberately not routed through the mixed Hodge structures of another page of this build, which lie outside this page's prerequisite closure.
• The only assertions recorded beyond the definitions are the componentwise closure claims for kernels and images, checked in the Definition above by the directness of the bidegree decompositions; no proof phase applies to a definition. Choice-free: the item fixes definitions and verifies componentwise linear algebra over a fixed finite-dimensional structure, and no choice principle is used.