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In this note we present an obstruction to the existence of solutions to theself-dual Einstein-Maxwell-Higgs equations on a compact surface, which dependson the multiplicities of the zeroes of the \emph{Higgs field} and the\emph{vortex number} $N$. In particular, we exhibit infinitely many examples ofHiggs fields for which solutions cannot exist.

Obstructions to the existence of solutions of the self-dual Einstein-Maxwell-Higgs equations on a compact surface