On polarized (pure) Hodge structures

Question

Some simple questions, for which I know no precise reference (and would be deeply grateful for a nice one!):

1. Is it true that the category of (pure) polarized Hodge structures is abelian semi-simple, whereas the whole category of pure Hodge structures is not?

2. Should one only consider those morphisms of polarized Hodge structures that respect polarizations in order to obtain an abelian category?

3. Is it true that all pure Hodge structures 'that come from geometry' (for example, the graded pieces of the weight filtration of the singular cohomology of varieties and motives) are polarized?

Answer 1

Fortunately, these questions are easy to answer. First of all, it helps to distinguish between polarizable Hodge structures and polarized structures. For polarizable, we merely require that a polarization exists, but it is not fixed. Let Hodge structure mean pure rational Hodge structure below. Then

• The category of polarizable pure Hodge structures is abelian and semisimple (morphisms are not required to respect polarizations). This is essentially proved in Theorie de Hodge II.

• The category of arbitrary Hodge structures is abelian but not semisimple. To see the nonsemisimplicity, we can use a theorem of Oort-Zarhin [Endmorphism algebras of complex tori, Math Ann 1995], that any finite dimensional $\mathbb{Q}$-algebra is the endomorphism algebra of some complex torus, and therefore of some Hodge structure.

• All pure Hodge structures of geometric origin are polarizable. So this is a very reasonable condition to impose. For $Gr_WH^*(X)$, this is explained for example in Beilinson's Notes on absolute Hodge cohomology (although this was already implicit in Deligne's construction).