Some question about (semi-)stable sheaves

Question

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!

Answer 1

Concerning your Questions 1 & 2, since you adress to slope semistable sheaves and Metha-Ramanathan type results, I think every sheaf you mean is in fact torsion free (i.e. pure of dimension $\dim X$), right? Under this assumption, Question 1 is true if $D$ is replaced by a general complete intersection curve and Question 2 is also true.

Your Question 3 is just the definition of the Harder-Narasimhan filtration (thus is true), c.f. [HL10, Definition 1.3.2, p.16].

Let me explain how to prove your Question 1 & 2:

• For Question 1 (the version for general complete intersection curve), the point is that a general complete intersection curve $C$ is disjoint from the non-locally free loci (which is of codimension $\geqslant 2$ by torsion-freeness) of the factor sheaves of the HN-filtration, then the inculsions and factor sheaves are preserved under the restriction to $C$. Indeed, let $$E_0\subset E_1\subset\cdots \subset E_r=E$$ be the HN-filtration of $E$. Then for each $i$, since $E_i/E_{i-1}$ is locally free near $C$, the restriction to $C$ of the exact sequence $$0\to E_{i-1}\to E_i\to (E_i/E_{i-1})\to 0,$$ remains exact. The remaining thing is to use Metha-Ramanthan (c.f. [HL10, Theorem 7.2.1, p.197]) to show the slope semistability of each $(E_i/E_{i-1})|_C$. This result should also hold over algebraically closed fields of positive characteristic, since Metha-Ramanthan holds for arbitrary characteristic.
• As for Question 2, again let $C$ be a general complete intersection curve, then $E|_C$ and $F|_C$ are both locally free and slope stable (by Metha-Ramanthan, c.f. [HL10, Theorem 7.2.8, p.202]). And so is $(E\otimes F)|_C\simeq E_C\otimes F|_C$, as a consequence of Narasimhan-Seshadri theorem (Ulenbeck-Yau in the curve case); there is also a purely algebraic proof for this fact.

Reference(s):

[HL10] Daniel Huybrechts & Manfred Lehn: The Geometry of Moduli Spaces of Sheaves (2nd ed.), Cambridge: CUP, 2010,