Moduli spaces of sheaves that are semistable with respect to a Kähler polarisation

Daniel Greb,Matei Toma

doi:10.5802/jep.116

Abstract

Using an existence criterion for good moduli spaces of Artin stacks by Alper-Fedorchuk-Smyth we construct a proper moduli space of rank two sheaves with fixed Chern classes on a given complex projective manifold that are Gieseker-Maruyama-semistable with respect to a fixed K\"ahler class.

Highlights

Ch(E)emω Todd(X), X see for example [GRT16a]
⊗ Lm), and so the above generalises the notion of Gieseker-Maruyama–stability from integral classes to real ample classes, and to all Kähler classes [ω]
When semistability is measured with respect to an ample line bundle, the construction of moduli spaces is based on Geometric Invariant Theory and is of global nature

Summary

Publié avec le soutien du Centre National de la Recherche Scientifique

Publication membre du Centre Mersenne pour l’édition scientifique ouverte www.centre-mersenne.org. The proof of the claim presented in Remark 3.7 and Lemma 3.6 of [AK16] continues to work even without the normality assumption made there, if we replace the application of Sumihiro’s Theorem (which uses the normality assumption) by the observation made in the paragraph preceding the proposition that in our setup right from the start R comes equipped with a G-equivariant locally closed embedding into the projective space associated with a finite-dimensional complex Grepresentation. — Using analytic stacks, an alternative proof can be given as follows: As in the above proof, one checks that the map (S, s)an → [S/Gs]an = [San/Gs] is smooth and the induced map on tangent spaces is an isomorphism These two conditions are equivalent to the conditions in the definition of a semi-universal family, cf [KS90, p. In our case since X is projective it follows that φ is even projective, but we will not need this fact

Let now

The proof is completed once we establish the following