Moduli spaces of semistable pairs on projective Deligne-Mumford stacks

Abstract

We generalize the construction of a moduli space of semistable pairs parametrizing isomorphism classes of morphisms from a fixed coherent sheaf to any sheaf with fixed Hilbert polynomial under a notion of stability to the case of projective Deligne-Mumford stacks. We study the deformation and obstruction theories of stable pairs, and then prove the existence of virtual fundamental classes for some cases of dimension two and three. This leads to a definition of Pandharipande-Thomas invariants on three-dimensional smooth projective Deligne-Mumford stacks.

Relative orbifold Pandharipande-Thomas theory and the degeneration formula

“…In this section, we will give a construction of relative moduli spaces of semistable pairs, which is a relative version of the one in [33]. Let p : X → S be a family of projective Deligne-Mumford stacks with a moduli scheme π : X → X and a relative polarization (E, O X (1)).…”

“…A pair (F , ϕ) on X /S or a coherent sheaf F on X /S is said to be of dimension d if dim F s = d for any geometric point Spec k s − → S. We say a pair (F , ϕ) on X /S or a coherent sheaf F on X /S is pure if F s is pure for every geometric point s of S. A pair (F , ϕ) on X /S is called nontrivial if for each geometric point s of S, the morphism ϕ s : F 0,s → F s is not zero (also called nontrivial). By [33,Remark 2.8], for each geometric point s of S, the fiber X s is a projective Deligne-Mumford stack with a moduli scheme X s and possesses a generating sheaf E s . Let P be a polynomial of degree d, we call a pair (F , ϕ) on X /S of type P if P Es (F s ) = P for each geometric point s of S. A pair (F , ϕ) on X /S with F flat over S is of type P for some polynomial P due to the following lemma.…”

“…In [57], the author also gives the definition of the absolute and relative orbifold DT theory (see also [8,17] for the prior definition with Calabi-Yau condition) and derives the corresponding degeneration formulas. In [33], the author constructs moduli spaces of orbifold PT stable pairs and defines virtual fundamental classes which are integrated to give the absolute orbifold PT invariants for 3-dimensional smooth projective Deligne-Mumford stacks (see also [5] for the prior definition in the case of Calabi-Yau 3-orbifolds). However, it is naturally expected to have the relative orbifold PT theory and the corresponding degeneration formula.…”

“…In order to define the relative orbifold PT invariants, one needs to consider the construction of relative moduli spaces of semistable pairs, which is a relative version of the one in [33]. We first state the result of this construction as follows.…”

“…Let π M s : X × S M s → M s and πX : X × S M s → X be the projections. As in the absolute case (see [33,Theorem 1.6]), if each fiber of p is smooth, we have the relative version.…”