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!



Some question about (semi-)stable sheaves
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,


