  We show that all smooth Killing horizons with compact horizon sections of
4-dimensional gauged N=2 supergravity coupled to any number of vector
multiplets 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 the
horizon 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 8
supersymmetries. If the orbits of the SL(2,R) symmetry are 2-dimensional, the
horizons are warped products of AdS2 with the horizon spatial section.
Otherwise, the horizon section admits an isometry which preserves all the
fields. The proof of these results is centered on the use of index theorem in
conjunction with an appropriate generalization of the Lichnerowicz theorem for
horizons that preserve at least one supersymmetry. In all $c_1({\cal K})=0$
cases, we specify the local geometry of spatial horizon sections and
demonstrate that the solutions are determined by first order non-linear
ordinary differential equations on some of the fields.
