Holomorphic mappings, the Schwarz-Pick lemma, and curvature (Q1895946)

The paper bounds the dimension of the complex space of rank \(k\) holomorphic mappings between compact complex algebraic manifolds \(X\) and \(Y\), which is denoted by \(\text{Hol}_k (X,Y)\). The curvature conditions imposed on \(Y\) are as follows. (i) \(Y\) has a Kähler metric with negative semidefinite Ricci curvature; and (ii) \(\wedge^k T(Y)\) has a Hermitian metric \(\Psi\) such that for each \(y \in Y\) and each simple vector \(v \neq 0\) of \(\wedge^k T(Y)\), the Hermitian bilinear form \(R(v, \overline v) ( , )\) is negative definite on the span of \(v_1, v_2, \dots, v_k\) where \(v_1 \wedge v_2 \wedge \cdots \wedge v_k = v\), and \(R(v, \overline v) ( , )\) has at least \(s_k\) negative eigenvalues on \(T_y (Y)\). Then \(\dim \text{Hol}_k (X,Y) \leq \dim Y - s_k\). For instance, if \(Y\) satisfies (i) and if \(T(Y)\) has a Hermitian metric with a positive number \(B\) such that the holomorphic sectional curvature \(< - B\) and if \(k \geq 2\), the holomorphic bisectional curvature \(\geq B/(k - 1)\), then \(\dim \text{Hol}_k (X,Y) \leq \dim Y - k\). When \(Y\) is a Hermitian locally symmetric space, the bounds improve in some cases those of Noguchi [\textit{J. Noguchi}, Invent. Math. 93, 15-34 (1988; Zbl 0651.32012)]. Explicit determinations of \(k\) are done in the paper for quotients of classical bounded symmetric domains.