Higher rank flag sheaves on surfaces (Q6608139)

The paper under review stuides the moduli space of holomorphic triple \(E_1\stackrel{\phi} \longrightarrow E_2\), composed of torsion-free sheaves \(E_i, i=1,2\), and a holomorphic morphism between them, over a smooth complex surface \(S\). The triples are equipped with the Schmidt stability condition.\N\NIt is observed that when the Schmidt stability parameter \(q(m)\) becomes sufficiently large, the moduli space of triples benefits from having a perfect relative and absolute deformation-obstrction theory in some cases.\N\NFurthere the construction is generaized by gluing triple moduli spaces, and extend the earlier work of \textit{A. Gholampour} et al. [Adv. Math. 365, Article ID 107046, 50 p. (2020; Zbl 1437.14054)] where the obstruction theory of nested Hilbert schemes over the surface was studied. The generalization is achieved for the moduli space of chains \[E_1\stackrel{\phi_1}\longrightarrow E_2\stackrel{\phi_2}\longrightarrow\cdots \stackrel{\phi_{n-1}}\longrightarrow E_n\] where \(\phi_i\) are injective morphisms and \(rk (E_i)\geq 1\) for all \(i\).\N\NThere is a connection, by wallcrossing in the master space, between the theory of such higher rank flags, and the theory of Higgs pairs on the surface, which provides the means to relate the flag invariants to the local DT (Donaldson-Thomas) invariants of threefolds givenby a line bundle on the surface \(X=Tot(\mathcal{L}\to S)\).