  This article aims to investigate the curvature operator of the second kind on
K\"ahler manifolds. The first result states that an $m$-dimensional K\"ahler
manifold with $\frac{3}{2}(m^2-1)$-nonnegative (respectively,
$\frac{3}{2}(m^2-1)$-nonpositive) curvature operator of the second kind must
have constant nonnegative (respectively, nonpositive) holomorphic sectional
curvature. The second result asserts that a closed $m$-dimensional K\"ahler
manifold with $\left(\frac{3m^3-m+2}{2m}\right)$-positive curvature operator of
the second kind has positive orthogonal bisectional curvature, thus being
biholomorphic to $\mathbb{CP}^m$. We also prove that
$\left(\frac{3m^3+2m^2-3m-2}{2m}\right)$-positive curvature operator of the
second kind implies positive orthogonal Ricci curvature. Our approach is
pointwise and algebraic.
