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We study local holomorphic maps between K\"ahler manifolds preserving$(p,p)$-forms. In this direction, we prove that any such local holomorphic map$F$ is a holomorphic isometry up to a scalar constant provided that $p$ isstrictly less than the complex dimension of the domain of $F$. We then studylocal holomorphic maps between finite dimensional complex space formspreserving invariant $(p,p)$-forms. It was proved by Calabi that there does notexist a local holomorphic isometry between complex space forms $M$ and $N$provided that $M$ and $N$ are of different types. In this article, wegeneralize this result to local holomorphic maps between complex space forms$M$ and $N$ preserving invariant $(p,p)$-forms whenever $M$ and $N$ are ofdifferent types except for the case where the universal covers of $M, N$ arebiholomorphic to $\mathbb{C}^m, \mathbb{P}^n$, respectively and $2\le p=m<n$.We also obtain some results in more general settings, including the study onindefinite K\"ahler manifolds and relatives for K\"ahler manifolds.

On local holomorphic maps between K\"ahler manifolds preserving $(p,p)$-forms