Stable pairs of 2-dimensional sheaves on 4-folds

Abstract

We identify Le Potier's moduli spaces of limit stable pairs $(F,s)$, where $F$ is a 2-dimensional sheaf on a nonsingular projective 4-fold $X$ and $s \in H^0(F)$, with the moduli spaces of polynomial stable 2-term complexes in derived category. These stable pairs are 2-dimensional analogs of Pandharipande-Thomas' stable pairs defined for 3-folds. We establish categorical correspondences involving these stable pairs, ideal sheaves of 2-dimensional subschemes of $X$, and 1-dimensional sheaves on $X$. Under some conditions on the Chern character, these lead to Hall algebra correspondences. The generalization of most of these results to higher ranks is also given. In case $X$ is Calabi-Yau, Oh-Thomas' construction gives a new set of invariants of $X$ counting these stable pairs. For certain Chern characters, these are related to the invariants of 2-dimensional stable sheaves. We calculate and study them in some cases and examples such as fibrations by abelian surfaces, local surfaces, and local Fano 3-folds. The last case in particular leads to new invariants of Fano 3-folds counting 2-dimensional stable pairs with reduced supports.