  We consider settings providing the existence of stable two-dimensional (2D)
dissipative solitons with zero and nonzero vorticity in optical media with the
quadratic ($\chi^{(2)}$) nonlinearity. To compensate the spatially uniform loss
in both the fundamental-frequency (FF) and second-harmonic (SH) components of
the system, a strongly localized amplifying region ("hot spot",HS), carrying
the linear gain, is included, acting onto either the FF component or SH one. In
both cases, the Gaussian radial gain profile supports stable fundamental
dissipative solitons pinned to the HS. The structure of existence and stability
domains for the 2D solitons is rather complex. They demonstrate noteworthy
features, such as bistability and spontaneous symmetry breaking. A ring-shaped
gain profile acting onto the FF component supports stable vortex solitons, with
the winding number up to 5, and multipoles. Nontrivial transformation of
vortex-soliton profiles upon either growth of the gain value or stretching the
gain profile is observed.
