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We study Pandharipande–Thomas’s stable pair theory on   $K3$   fibrations over curves with possibly nodal fibers. We describe stable pair invariants of the fiberwise irreducible curve classes in terms of Kawai–Yoshioka’s formula for the Euler characteristics of moduli spaces of stable pairs on   $K3$   surfaces and Noether–Lefschetz numbers of the fibration. Moreover, we investigate the relation of these invariants with the perverse (non-commutative) stable pair invariants of the   $K3$   fibration. In the case that the   $K3$   fibration is a projective Calabi-Yau threefold, by means of wall-crossing techniques, we write the stable pair invariants in terms of the generalized Donaldson–Thomas invariants of 2D Gieseker semistable sheaves supported on the fibers.

Stable Pairs on Nodal $K3$ Fibrations