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We study the two-dimensional Hubbard model in the weak coupling limit, in the vicinity of half band filling, for a generic model which has nearest neighbour as well as next nearest neighbour overlap integrals. The model is hoped to be relevant to the new high-T~ superconducting oxides. A scaling theory, previously studied for the case of perfectly nested square Fermi surface at half band filling [1], is developed for the generic case, and allows a consistent treatment of the coupled lng(T) and In(T) singularities arising from density wave and Cooper pair fluctuations. When violation of perfect nesting is sufficient, d type singlet superconductivity, induced by antiferromagnetic spin fluctuations, overwhelms the latter even when the Fermi level is at the Van Hove singularity. We find that the superconducting temperature 7~ยข is of the order of the N6el temperature of the perfectly nested model in a sizeable part of the phase diagram. We also find that normal impurity scattering, when sufficiently strong, may be efficient in suppressing superconductivity. The maximum N6el temperature, in the antiferromagnetic part of the phase diagram may occur away from half band filling, and antiferromagnetism is easily suppressed by small changes in carrier concentration. The relevance of our results to actual high-T, superconducting oxides is discussed.
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