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

摘要

Let D,Omega(1),...,Omega(m) be irreducible bounded symmetric domains. We study local holomorphic maps from D into Omega 1x...Omega(m) 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 = B-n,Omega(i) = B-Ni for any p or if D = Omega(1)=...= Omega(m) with rank(D) = 2 and p sufficiently large. As a consequence, the algebraic correspondence between quasi-projective varieties D/Gamma preserving invariant (p,p)-forms is modular, where G is a torsion free, discrete, finite co-volume subgroup of Aut(D).