# Some question about (semi-)stable sheaves

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!

• Please confer Mehta-Ramanathan. Commented Oct 14, 2022 at 18:01

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,

• Thank you Juanyong! Commented Apr 17, 2023 at 2:20