On local holomorphic maps preserving invariant (p; P)-forms between bounded symmetric domains

Let D; Ω1;Ω m be irreducible bounded symmetric domains.We study local holomorphic maps from D into Ω1Ω m preserving the in-variant (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; i = BNi for any p or if D = 1 = m with rank(D) 2 and p sufficiently large. As a consequence, the algebraic correspondence between quasi-projective varieties D= preserving invariant (p; p)-forms is modular, where is a torsion free, discrete, finite co-volume subgroup of Aut(D).