Greb, Daniel; Toma, Matei Compact moduli spaces for slope-semistable sheaves. (English) Zbl 1423.14076 Algebr. Geom. 4, No. 1, 40-78 (2017). The authors generalize the algebro-geometric construction of the moduli spaces of slop-semistable sheaves over surfaces, which was obtained independently by J. Le Potier [Lond. Math. Soc. Lect. Note Ser. 179, 213–240 (1992; Zbl 0788.14045)] and J. Li [J. Differ. Geom. 37, No. 2, 417–466 (1993; Zbl 0809.14006)], to the case where the underlying variety is of higher dimension.They considered the slop-semistability with respect to the complete intersection of \(n-1\) very ample divisors \(H_1,\dots,H_{n-1}\), which equals to the slop-semistability in the usual sense if \(H_1=\dots=H_{n-1}\). They also called that \((H_1,\dots,H_{n-1})\)-semistability.At first the boundedness of the set of all \((H_1,\dots,H_{n-1})\)-semistable sheaves with fixed Chern classes can be easily proven, analogue to the case with the usual slop-semistability. Then one can find an open subset \(R^{\mu ss}\) containing all such sheaves in some Quot-scheme. However after that there came a problem that the flatness of a family of sheaves may fail to preserve under restriction to hyperplan sections. Therefore in order to have enough sections of some determinant line bundle only well-defined for flat families, they had to replace \(R^{\mu ss}\) by the weak normalization \((R^{\mu ss}_{\mathrm{red}})^{wn}\) of its reduction \(R^{\mu ss}_{\mathrm{red}}\). The price is that the final moduli space \(M^{\mu ss}:=\text{Proj}(\bigoplus_{k\geq 0} H^0((R^{\mu ss}_{\mathrm{red}})^{wn},\mathcal{L}^{\otimes kN})^{SL(V)})\) holds the universal property only in the category of weakly normal schemes.Let \(H_1=\dots=H_{n-1}=H\), they also showed that \(M^{\mu ss}\) gives a compactification (with algebraic structure) of the moduli space of \(\mu\)-stable reflexive sheaves.A very important advantage of this \((H_1,\dots,H_{n-1})\)-semistability conception is that the authors resolved the pathological wall-crossing phenomena for moduli spaces of sheaves on higher-dimensional complex manifolds. More precisely, they showed that if one defines the walls by equations homogeneous of degree \(n-1\) rather than linear, then one gets chamber structure and local finiteness of the walls in the ample cone, where in each chamber there is exactly one stability condition realized by some multipolarisation \((H_1,\dots,H_{n-1})\) but not always \(H_1=\dots=H_{n-1}\). Reviewer: Yao Yuan (Beijing) Cited in 1 ReviewCited in 13 Documents MSC: 14D20 Algebraic moduli problems, moduli of vector bundles 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 58D27 Moduli problems for differential geometric structures Keywords:slop-stability; moduli of sheaves; wall-crossing; Donaldson-Uhlenbeck compactification; determinant line bundle Citations:Zbl 0788.14045; Zbl 0809.14006 PDFBibTeX XMLCite \textit{D. Greb} and \textit{M. Toma}, Algebr. Geom. 4, No. 1, 40--78 (2017; Zbl 1423.14076) Full Text: DOI arXiv