Counting invariant of perverse coherent sheaves and its wall-crossing

We introduce moduli spaces of stable perverse coherent systems on small crepant resolutions of Calabi-Yau 3-folds and consider their DonaldsonThomas type counting invariants. The stability depends on the choice of a component (= a chamber) in the complement of finitely many lines (= walls) in the plane. We determine all walls and compute generating functions of invariants for all choices of chambers when the Calabi-Yau is the resolved conifold. For suitable choices of chambers, our invariants are specialized to Donaldson-Thomas, Pandharipande-Thomas and Szendroi invariants. Introduction In this paper we study variants of Donaldson-Thomas (DT in short) invariants [Tho00] for the crepant resolution f : Y → X = {xy − zw = 0} of the conifold where Y is the total space of the vector bundle OP1(−1)⊕OP1(−1). The ordinary DT invariants are defined by the virtual counting of moduli spaces of ideal sheaves of curves. A variant has been introduced by Pandharipande-Thomas (PT in short) recently [PT]. PT invariants are defined by the virtual counting of moduli spaces of stable coherent systems, i.e., pairs of 1-dimensional sheaves F and homomorphisms s : OY → F . Both DT and PT invariants are defined for arbitrary Calabi-Yau 3-folds. For the resolved conifold Y , yet another variant was introduced by Szendroi [Sze08] (see also [You]). His invariants are defined by the virtual counting of moduli spaces of representations of a certain noncommutative algebra. This noncommutative algebra has its origin in the celebrated works of Bridgeland [Bri02] and Van den Bergh [VdB04]. In particular, we can interpret that the invariants virtually count moduli spaces of perverse ideal sheaves , originally introduced in order to describe the flop f : Y + → X of Y as a moduli space. Those three classes of invariants have been computed for the resolved conifold and their generating functions are given by infinite products. In this paper we introduce more variants by using moduli spaces of stable perverse coherent systems , i.e., pairs of 1-dimensional perverse coherent sheaves F and homomorphisms s : OY → F . The stability condition is determined by a choice of parameter ζ = (ζ0, ζ1) in the complement of finitely many lines in R. (The number increases when the Hilbert polynomial of F becomes large.) The invariants depend only on the chamber containing ζ. When we choose the chamber appropriately, our new invariants recover DT, PT invariants for Y and