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We show that all smooth Killing horizons with compact horizon sections of4-dimensional gauged N=2 supergravity coupled to any number of vectormultiplets preserve $2 c_1({\cal K})+4 \ell$ supersymmetries, where ${\cal K}$is a pull-back of the Hodge bundle of the special K\"ahler manifold on thehorizon spatial section. We also demonstrate that all such horizons with$c_1({\cal K})=0$ exhibit an SL(2,R) symmetry and preserve either 4 or 8supersymmetries. If the orbits of the SL(2,R) symmetry are 2-dimensional, thehorizons are warped products of AdS2 with the horizon spatial section.Otherwise, the horizon section admits an isometry which preserves all thefields. The proof of these results is centered on the use of index theorem inconjunction with an appropriate generalization of the Lichnerowicz theorem forhorizons that preserve at least one supersymmetry. In all $c_1({\cal K})=0$cases, we specify the local geometry of spatial horizon sections anddemonstrate that the solutions are determined by first order non-linearordinary differential equations on some of the fields.

Dynamical symmetry enhancement near N=2, D=4 gauged supergravity horizons