A generalization of the Lefschetz hyperplane-section theorem (Q5941878)

Let \(M\) be a complete Kähler manifold of positive sectional curvature, \(\dim_{\mathbb C} M =m,\) and let \(A\) and \(B\) be nonsingular complex submanifolds of \(M.\) Suppose that \(\dim_{\mathbb C} A =p,\) \(\dim_{\mathbb C} B =q,\) and \(p+q-m \geq 0.\) Denote the space of all piecewise smooth paths whose initial and endpoints lie in \(A\) and \(B,\) respectively, by \(I(A,B).\) Making use of the Morse theory [\textit{J. Milnor}, Morse theory, Ann. Math. Stud. 51, Princeton Univ. Press (1965: Zbl 0108.10401)] and its modification the author proves that the pair \((I(A,B), A\cap B)\) is homotopically \((p+q-m)\)-connected. This implies immediately a variant of the Lefschetz hyperplane-section theorem for the homotopy groups of the pairs \((A, A\cap B)\) and \((M, B)\) as well as for the corresponding homology and cohomology groups with arbitrary coefficients. As a direct consequence it is obtained the theorem of Lefschetz type due to Fulton and Lazarsfeld, and its generalization [\textit{A. Sommese} and \textit{A. Van de Ven}, Nagoya Math. J. 102, 79-90 (1986; Zbl 0564.14010)].