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We prove a sharp lower bound for the Tanaka-Webster holomorphic sectionalcurvature of strictly pseudoconvex real hypersurfaces that are"semi-isometrically" immersed in a K\"ahler manifold of nonnegative holomorphicsectional curvature under an appropriate convexity condition. This gives apartial answer to a question posed by Chanillo, Chiu, and Yang regarding thepositivity of the Tanaka-Webster scalar curvature of the boundary of a strictlyconvex domain in $\mathbb{C}^2$ from 2012. In fact, the main result proves astronger positivity property, namely the $\frac12$-positivity in the sense ofCao, Chang, and Chen, for compact "convex" real hypersurfaces in a K\"ahlermanifold of nonnegative holomorphic sectional curvature. Our approach is rathersimple and uses a version of the Gauss equation for semi-isometric CRimmersions of pseudohermitian manifolds into K\"ahler manifolds.

The holomorphic sectional curvature and "convex" real hypersurfaces in K\"ahler manifolds