Let $X$ be a projective normal variety over $\mathbb C$, I have several questions about semi-stable sheaves:

**Question 1.** Suppose that $E$ is a pure sheaf such that $HN_*(E)$ is the Harder-Narasimhan filtration of $E$. Let $H$ be an ample divisor and $D \in |aH|$ be a general element for $a\gg 1$. Then is $HN_{*}(E)|_D$ the HN-filtration for $E|_D$?

(I guess the statement is false because we even cannot guarantee that $HN_{*}(E)|_D$ is a filtration (i.e. the restriction of inclusion may not be inclusion). But the statement might be true if $D$ is replaced by a general complete intersection curve.)

**Question 2.** If $E, F$ are (slope) stable sheaves, then is $E \otimes F$ still stable?

**Question 3.** Suppose that $0=E^0 \subset E^1\subset \cdots \subset E^k=E$ is a filtration of $E$ such that $G^i=E^i/E^{i-1}$ is semi-stable with slopes $\mu(G^i)$ strictly decreasing. Then is above filtration the NH-filtration?

(The answer may be no. But I was wondering if there is a non-constructive way to formulate NH-filtration? If there is a such why, please indicate how it goes.)

Thank you very much!