On boundedness of semistable sheaves (Q2077283)

Roughly speaking, a moduli space is a geometric space whose points correspond to isomorphism classes of objects we are interested in. It is of no doubt that the importance of moduli spaces in algebraic geometry. However, when we collect all the algebro-geometric objects (for instance, vector bundles on a given projective variety with fixed numerical invariants), we do not expect the existence of a nice moduli space. Hence, it is necessary to introduce the notion of stability, and try to construct moduli spaces of stable objects. To construct a projective moduli space as a result, the boundedness of the family of isomorphism classes of (semi-)stable objects is crucial. Therefore, the boundedness results of the family of isomorphism classes of \(H\)-semistable (\(H\) is an ample divisor on a smooth projective variety \(X\) over an algebraically closed field \(k\)) torsion-free coherent sheaves on \(X\) with fixed numerical invariants are foundational (cf. [\textit{M. Maruyama}, Moduli spaces of stable sheaves on schemes: restriction theorems, boundedness and the GIT construction. With collaboration of T. Abe and M. Inaba. Tokyo: Mathematical Society of Japan (2016; Zbl 1357.14017)] for \(\mathrm{char}(k)=0\), and [\textit{A. Langer}, Ann. Math. (2) 159, No. 1, 251--276 (2004; Zbl 1080.14014)] for \(\mathrm{char}(k)>0\)). The paper under review provides a new simple proof of [\textit{A. Langer}, Ann. Math. (2) 159, No. 1, 251--276 (2004; Zbl 1080.14014)] without using characteristic \(p\) methods. Instead, the author reduces the statement to the case of projective spaces, and proves Bogomolov's inequality on \(\mathbb{P}^n\) without using restriction theorem (of Flenner, or Mehta-Ramanathan). The latter one provides a restriction theorem which implies the boundedness of the families over projective spaces. This new method works both for the characteristic zero case and the characteristic \(p>0\) case. In Section 4, the author also provides an alternative proof of Bogomolov's inequality on smooth projective surfaces in the characteristic zero case without reducing to the curve case. Instead, the author uses a bound of the slope of the Frobenius pull-back and a bound the number of global sections.