A mixed ($\mathbb{Q}$)-hodge structure is defined to be a vector space $V/\mathbb{Q}$ with an increasing "weight" filtration of $\mathbb{Q}$- vector spaces $0\subset W_0\subset \dots$ and a decreasing "Hodge" Filtration of complex vector spaces $V=F^0\supset F^1\dots$ such that the following holds:
On the quotient pieces $W_{i+1}/W_i$ on gets an induced filtration $$F^p(W_{i}/W_{i-1}):=(W_{i}\cap F^p + W_{i-1}\otimes\mathbb{C})/(W_{i-1}\otimes\mathbb{C})$$ and everyone writes that this induced filtration is "pure of weight $i$".
My question is what precisely does this mean? Initially I thought it meant that (for the piece $W_{i}/W_{i-1}$ the relation $F^j\oplus\overline{F^{i+1-j}}=W_{i}/W_{i-1}$ holds, but this seems to be false for basic cases such as the limit mixed hodge structure on the nearby cycles of a family of elliptic curves degenerating to a nodal cubic.
Any references to thorough and clear definitions would also be greatly appreciated!
