Let $D, \Omega_1, ..., \Omega_m$ be irreducible bounded symmetric domains. Westudy local holomorphic maps from $D$ into $\Omega_1 \times... \Omega_m$preserving the invariant $(p, p)$-forms induced from the normalized Bergmanmetrics up to conformal constants. We show that the local holomorphic mapsextends to algebraic maps in the rank one case for any $p$ and in the rank atleast two case for certain sufficiently large $p$. The total geodesy thusfollows if $D=\mathbb{B}^n, \Omega_i = \mathbb{B}^{N_i}$ for any $p$ or if$D=\Omega_1 =...=\Omega_m$ with rank$(D)\geq 2$ and $p$ sufficiently large. Asa consequence, the algebraic correspondence between quasi-projective varieties$D / \Gamma$ preserving invariant $(p, p)$-forms is modular, where $\Gamma$ isa torsion free, discrete, finite co-volume subgroup of Aut$(D)$. This solvespartially a problem raised by Mok.
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机译：令$ D，\ Omega_1，...，\ Omega_m $为不可约的有界对称域。从$ D $到$ \ Omega_1 \ times的Westudy局部全同图，\ Omega_m $保留从归一化Bergmanmetrics到共形常数的不变$（p，p）$形式。我们表明，局部全同性映射在任何$ p $的一类情况下扩展到代数映射，而对于某些足够大的$ p $在第二种情况下至少扩展到代数映射。因此，对于任何$ p $，如果$ D = \ mathbb {B} ^ n，\ Omega_i = \ mathbb {B} ^ {N_i} $，或者如果$ D = \ Omega_1 = ... = \ Omega_m $，则总测地线如下rank $（D）\ geq 2 $和$ p $足够大。结果，准投影变体$ D / \ Gamma $保持不变$（p，p）$-形式之间的代数对应关系是模块化的，其中$ \ Gamma $是Aut $的无扭转，离散，有限同体积子群（D）$。这部分地解决了由莫提出的问题。
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