A refinement of the stable pair invariants of Pandharipande and Thomas for non-compact Calabi-Yau spaces is introduced based on a virtual Bialynicki-Birula decomposition with respect to a C * action on the stable pair moduli space, or alternatively the equivariant index of Nekrasov and Okounkov. This effectively calculates the refined index for M -theory reduced on these Calabi-Yau geometries. Based on physical expectations we propose a product formula for the refined invariants extending the motivic product formula of Morrison, Mozgovoy, Nagao, and Szendroi for local P 1 . We explicitly compute refined invariants in low degree for local P 2 and local P 1 x P 1 and check that they agree with the predictions of the direct integration of the generalized holomorphic anomaly and with the product formula. The modularity of the expressions obtained in the direct integration approach allows us to relate the generating function of refined PT invariants on appropriate geometries to Nekrasov's partition function and a refinement of Chern-Simons theory on a lens space. We also relate our product formula to wallcrossing.

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Abstract. We study the wall-crossing of the moduli spaces M α (d, 1) of α-stable pairs with linear Hilbert polynomial dm+ 1 on the projective plane P 2 as we alter the parameter α. When d is 4 or 5, at each wall, the moduli spaces are related by a smooth blow-up morphism followed by a smooth blow-down morphism, where one can describe the blow-up centers geometrically. As a byproduct, we obtain the Poincaré polynomials of the moduli spaces M(d, 1) of stable sheaves. We also discuss the wall-crossing when the number of stable components in Jordan-Hölder filtrations is three.1. Introduction 1.1. Motivation and Results. In moduli theory, for a given quasi-projective moduli space M 0 , various compactifications stem from the different view points for the moduli points of M 0 . After we obtain various compactified moduli spaces of M 0 , it is quite natural to ask the geometric relationship among them. Sometimes, this question is answered by birational morphisms between them, which enables us to obtain some geometric information (for example, the cohomology groups) of one space from that of the other [28,5].In this paper, we study the moduli space of semistable sheaves of dimension one on smooth projective surfaces [27], which recently gains interests in both mathematics and physics. This is an example of compactifications of the relative Jacobian variety, where we regard its general point as a sheafon a smooth curve C with pole along points p i of general position. In general, the moduli space of semistable sheaves is hard to study due to the lack of geometry of its boundary points. However, if n is equal to the genus of C, the sheaf F has a unique section up to scalar. So, we may alternatively consider the general point as a sheaf with a section, which in turn leads to another compactification, so called the moduli space of α-semistable pairs (more generally, the coherent systems [18]). When α is large, it can be shown that the moduli spaces of α-stable pairs are nothing but the relative 2010 Mathematics Subject Classification. 14D20.

We study the wall-crossing of the moduli spaces M α (d, 1) of α-stable pairs with linear Hilbert polynomial dm+ 1 on the projective plane P 2 as we alter the parameter α. When d is 4 or 5, at each wall, the moduli spaces are related by a smooth blow-up morphism followed by a smooth blow-down morphism, where one can describe the blow-up centers geometrically. As a byproduct, we obtain the Poincaré polynomials of the moduli spaces M(d, 1) of stable sheaves. We also discuss the wall-crossing when the number of stable components in Jordan-Hölder filtrations is three.2010 Mathematics Subject Classification. 14D20. Key words and phrases. Semistable pairs, Wall-crossing formulae, Blow-up/down, and Betti numbers.