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We introduce a one-dimensional system combining the $\mathcal{PT}$-symmetriccomplex periodic potential and the $\chi ^{(2)}$ (second-harmonic-generating)nonlinearity. The imaginary part of the potential, which represents spatiallyseparated and mutually balanced gain and loss, affects only thefundamental-frequency (FF) wave, while the real potential acts on thesecond-harmonic (SH) component too. Soliton modes are constructed, and theirstability is investigated (by means of the linearization and directsimulations) in semi-infinite and finite gaps in the corresponding spectrum,starting from the bifurcation which generates the solitons from edges of thegaps' edges. Families of solitons embedded into the conttinuous spectrum of theSH component are found too, and it is demonstrated that a part of the familiesof these \textit{embedded solitons} (ESs) is stable. The analysis is focused oneffects produced by the variation of the strength of the imaginary part of thepotential, which is a specific characteristic of the $\mathcal{PT}$ system. Theconsideration is performed chiefly for the most relevant case of matched realpotentials acting on the FF\ and SH components. The case of the real potentialacting solely on the FF component is briefly considered too.

Solitons in $\mathcal{PT}$-symmetric periodic systems with the quadratic nonlinearity