On local holomorphic maps preserving invariant $(p, p)$-forms between bounded symmetric domains

Let D, Ω₁, ..., Ωₘ be irreducible bounded symmetric domains. We study local holomorphic maps from D into Ω₁ ×... Ωₘ preserving the invariant $(p, p)$-forms induced from the normalized Bergman metrics up to conformal constants. We show that the local holomorphic maps extends to algebraic maps in the rank one case for any p and in the rank at least two case for certain sufficiently large p. The total geodesy thus follows if D=Bⁿ, Ωᵢ = BNᵢ for any p or if D=Ω₁ =...=Ωₘ with rank(D)≥ 2 and p sufficiently large. As a consequence, the algebraic correspondence between quasi-projective varieties D / Γ preserving invariant $(p, p)$-forms is modular, where Γ is a torsion free, discrete, finite co-volume subgroup of Aut$(D)$. This solves partially a problem raised by Mok.