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


*

*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? 

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

*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?
 A: 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).
