Let f:S⤏S be a rational self-map of a smooth complex projective surface S and η be a meromorphic two-form on S satisfying (Formula presented.) for some (Formula presented.). We show that under a mild topological assumption on f, there is a birational change of domain ψ:X⤏S such that ψ∗η has no zeros. In this context, we investigate the notion of algebraic stability for f, proving that f can be made algebraically stable if and only if it acts nicely on the poles of η. We illustrate this last result in the case η=dx∧dyxy, where we translate our stability result into a condition on whether a circle homeomorphism associated to f has rational rotation number.
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