Equivalent definitions of a Hodge structure

Question

Generally, a Hodge structure of weight $k$ on a finitely generated abelian group $H$ is defined as a decomposition of the complexification:

$$ H\otimes \mathbb C = \bigoplus_{p+q=k} H^{p,q}, $$

where the vector spaces $H^{p,q}$ must satisfy the condition $\overline{H^{p,q}}=H^{q,p}$.

If $H$ carries this structure, then it is possible to define the Hodge filtration as

$$ F^p(H)=\bigoplus_{k-q\geq p} H^{k-q,q} . $$

However, it seems that if a filtration $F^\bullet(H)$ exists with the property that

$$ F^p(H)\cap \overline{F^q(H)}= 0 \qquad \mbox{ whenever } p+q=k+1 \tag{$\ast$} $$

then a Hodge structure of weight $k$ exists on $H$ by putting

$$ H^{p,q}=F^p(H)\cap \overline{F^q(H)}. $$

Now, in Peters and Steenbrink, it is said that $(\ast)$ is equivalent to say that $F^p(H)\oplus F^{k-p+1}(H)=H\otimes\mathbb C$, while on Wikipedia this latter condition is also required (meaning that it might not be equivalent).

Question 1. Is it equivalent or not?

If the answer is yes, notice that the Hodge filtration satisfies much nicer properties than the required on the existing descending filtration (for the Hodge filtration $F^p(H)\cap\overline{F^q(H)} = 0$ unless $p+q=k$), which makes me think that the Hodge filtration of associated given by the Hodge structure induced by the former filtration would be different. Isn't that absurd?.

Question 2. Either if the answer to question 1 is yes or not, how can one prove that indeed

$$ H\otimes\mathbb C= \bigoplus_{p+q=k} F^p(H)\cap\overline{F^q(H)} ? $$

I can only see it if the filtration $F^\bullet(H)$ satisfies the same properties that the Hodge filtration.

Answer 1

Fortunately for you, I've had a headache because of the conflict between these exact references before.

Wikipedia is correct, Peters and Steenbrink are not.

Question 1. Is it equivalent? No.

Here's an example: Let $k=0$, let $H \cong \mathbb Z$. Let $$ 0= F_0 = F_1=\cdots ;\quad H \otimes \mathbb C= F_{-1}=F_{-2}=\cdots $$ You can check that the condition in Peters and Steenbrink is satisfied, but this is not a Hodge structure of weight $0$, as all the graded pieces will vanish.

In general, take any Hodge structure of weight $k$. According to the condition in Peters and Steenbrink, it is automatically a Hodge structure of any weight $k'\ge k$, since the Hodge filtration is decreasing, which is never true.