Moduli of PT-semistable objects. II (Q2847118)

\textit{R. Pandharipande} and \textit{R. P. Thomas} [Invent. Math. 178, No. 2, 407--447 (2009; Zbl 1204.14026), J. Am. Math. Soc. 23, No. 1, 267--297 (2010; Zbl 1250.14035)] gave an approach to curve counting on Calabi--Yau 3-folds via \textit{stable pairs}. A stable pair consists of a pure 1-dimensional sheaf \(\mathcal{F}\), equipped with a section \(s\) with 0-dimensional cokernel. This data may be presented as a 2-term complex \(\{ \mathcal{O}_X \rightarrow \mathcal{F} \}\), and viewed as an object in the derived category \(D(X)\) of coherent sheaves on the 3-fold \(X\). In [Geom. Topol. 13, No. 4, 2389--2425 (2009; Zbl 1171.14011)], \textit{A. Bayer} exhibited a polynomial stability condition on \(D(X)\), called PT-stability, which recovers stable pairs as PT-stable objects lying in a certain heart \(\mathcal{A}\) of the derived category \(D(X)\), having trivial determinant, and satisfying a Chern class restriction. In the paper under review, the author studies moduli of PT-semistable objects for any fixed Chern class, and establishes their properties by viewing them as substacks of Lieblich's Artin stack of universally gluable complexes [\textit{M. Lieblich}, J. Algebr. Geom. 15, No. 1, 175--206 (2006; Zbl 1085.14015)].NEWLINENEWLINEThe main results are as follows. Given a smooth projective 3-fold, the author shows that the PT-semistable objects of any fixed Chern class form a universally closed Artin stack of finite type, with the stable objects forming a separated substack. In the absence of strictly semistable objects, he shows that the semistables moreover form a proper algebraic space, living inside Inaba's algebraic space of simple complexes.NEWLINENEWLINEThe main tool in this paper is a theory of semistable reduction for objects in the derived category. The valuative criterion for universal closedness is established by a careful analysis of semistable reduction applied to families of objects in \(\mathcal{A}\) with PT-semistable generic fibre, building on results from the author's previous paper [J. Algebra 339, No. 1, 203--222 (2011; Zbl 1236.14017)]. The author indicates that a similar approach should also suffice to handle other stability conditions, and higher-dimensional \(X\).