By Takuro Mochizuki

During this monograph, we de?ne and examine an algebro-geometric analogue of Donaldson invariants through the use of moduli areas of semistable sheaves with arbitrary ranks on a polarized projective floor. We may well anticipate the life of fascinating “universal kin between invariants”, which might be a common generalization of the “wall-crossing formulation” and the “Witten conjecture” for classical Donaldson invariants. Our objective is to procure a weaker model of such kin, in different short phrases, to explain a relation because the sum of integrals over the goods of m- uli areas of items with decrease ranks. thankfully, in line with a up to date first-class paintings of L. Gottsche, ¨ H. Nakajima and ok. Yoshioka, [53], a wall-crossing formulation for Donaldson invariants of projective surfaces could be deduced from this sort of weaker lead to the rank case. we are hoping that our paintings during this monograph could, at the very least tentatively, offers part of starting place for the extra research on such common relatives. within the remainder of this preface, we want to give an explanation for our motivation and a few of vital materials of this examine. See advent for our genuine difficulties and effects. Donaldson Invariants allow us to brie?y remember Donaldson invariants. We confer with [22] for extra information and detailed. We additionally confer with [37], [39], [51] and [53]. LetX be a compact easily con- ? nected orientated genuine four-dimensional C -manifold with a Riemannian metric g. enable P be a principalSO(3)-bundle on X.