A defect relation for meromorphic maps on parabolic manifolds intersecting hypersurfaces (Q2567498)

The authors prove the second main theorem for meromorphic mappings from a parabolic manifold \(M\) into complex projective spaces \(\mathbb{P}_n(\mathbb{C})\) which extends \textit{M. Ru}'s previous results concerning defect relation for holomorphic curves intersecting nonlinear hypersurfaces [Am. J. Math. 126, No. 1, 215--226 (2004; Zbl 1044.32009)]. Let \(M\) be an admissible parabolic manifold with an exhaustion function \(\tau\) and let \(D_j\) \((j= 1,\dots, q)\) be hypersurfaces of degree \(d_j\) in a complex projective space \(\mathbb{P}_n(\mathbb{C})\). Suppose that \(q> n+ 1\) and \(\bigcap^{n+1}_{k=1} D_{j_k}= \emptyset\) for each subset \(\{j_1,\dots, j_{n+1}\}\subset\{1,\dots, q\}\). Let \(f: M\to \mathbb{P}_n(\mathbb{C})\) be an algebraically nondegenerate meromorphic mapping and denote by \(T_f(r)\) the characteristic function of \(f\) with respect to the hyperplane bundle \(H\) over \(\mathbb{P}_n(\mathbb{C})\). The proximity function \(m_f(r, D_j)\) for \(D_j\) is defined in the usual way. Then the authors prove the following second main theorem: For an arbitrary positive real number \(\varepsilon\), the inequality \[ \sum^q_{j=1} d^{-1}_j m_f(r,D_j)\leq (n+ 1+ \varepsilon) T_f(r)+ c_\varepsilon(\text{Ric}_\tau(r)+ \log^+ rTf(r)+ \log Y(r)) \] holds for all \(r\in\mathbb{R}^+\) outside a Borel subset with finite Lebesgue measure, where \(c_\varepsilon> 0\) is a constant, \(\text{Ric}_r(r)\) is a Ricci function of \(M\) and \(Y(r)\) is a real positive function on \(M\) depending on the structure of \(M\).