In this article, we study minimal isometric immersions of K\"ahler manifolds into product of two real space forms. We analyse the obstruction conditions to the existence of pluriharmonic isometric immersions of a K\"ahler manifold into those spaces and we prove that the only ones into $\mathbb{ S}^{ m-1}\times\mathbb{ R}$ and $\mathbb{ H}^{ m-1}\times \mathbb{ R}$ are the minimal isometric immersions of Riemannian surfaces. Futhermore, we show that the existence of a minimal isometric immersion of a K\"ahler manifold $M^{ 2n}$ into $\mathbb{ S}^{ m-1}\times\mathbb{ R}$ and $\mathbb{ S}^{ m-k}\times \mathbb{ H}^k$ imposes strong restrictions on the Ricci and scalar curvatures of $M^{ 2n}$. In this direction, we characterise some cases as either isometric immersions with parallel second fundamental form or anti-pluriharmonic isometric immersions.
Categories: math.DG