Moduli spaces of stable pairs

Abstract

We construct a moduli space of stable pairs over a smooth projective variety, parametrizing morphisms from a fixed coherent sheaf to a varying sheaf of fixed topological type, subject to a stability condition. This generalizes the notion used by Pandharipande and Thomas, following Le Potier, where the fixed sheaf is the structure sheaf of the variety. We then describe the relevant deformation and obstruction theories. We show the existence of the virtual fundamental class in special cases. Last, we study examples of stable pairs over surfaces.iii Acknowledgements

Rank-one sheaves and stable pairs on surfaces

“…1 Compared to a quotient, we only require generic surjectivity for a stable pair, but we impose the purity condition on F . The second author [Lin18] showed that the moduli functor of equivalence classes of stable pairs is represented by a projective scheme, extending Kollár's work [Kol08]. 2 In this paper, we work in a setting similar to [BGJ16], but we focus on stable pairs rather than the Quot scheme.…”

Decorated sheaves and morphisms in tilted hearts

“…Proof. This follows from the projectivity of QHusk f (E 0 , P ) in this set-up, which is obtained via a geometric invariant theoretic construction [Lin18].…”

“…Remark 3.10. Quotient husks are also known as limit stable pairs [Lin18]. Assuming the universal of flatness for the family of t-structures, one would be able to obtain the projectivity of the Quot space over a general base S in characteristic 0, by carrying out a GIT construction [Lin18, Remark 4.6].…”

Moduli spaces of semistable pairs on projective Deligne-Mumford stacks

“…Fix a polynomial P , the moduli space of δ-(semi)stable pairs (F, ϕ) with Hilbert polynomial P was constructed in [66] by geometric invariant theory (GIT) with the assumption deg δ < dim X. In [35], the author shows the notion of δ-stable pairs defined in [66] actually generalizes the one of PT stable pairs for the case when deg δ ≥ deg P = 1 and constructs the corresponding moduli spaces. Observe that the notion of stability used in [66,35] is different from the one defined in [21,22] due to the different framing, where the latter notion has been naturally generalized to the stacky case [12] using the modified Hilbert polynomial defined in [42].…”

“…In [35], the author shows the notion of δ-stable pairs defined in [66] actually generalizes the one of PT stable pairs for the case when deg δ ≥ deg P = 1 and constructs the corresponding moduli spaces. Observe that the notion of stability used in [66,35] is different from the one defined in [21,22] due to the different framing, where the latter notion has been naturally generalized to the stacky case [12] using the modified Hilbert polynomial defined in [42]. In order to derive a stacky version of PT stable pairs, we generalize the construction of moduli spaces of stable pairs in [66,35] to the case of projective Deligne-Mumford stacks.…”