Let \(M\subset \mathbb{C}^N\) be a smooth submanifold. \(M\) is called generic if its tangent space \(T_{p}M\) at every point \(p\in M\) spans \(T_{p}\mathbb{C}^N\) over \(\mathbb{C}\) (i.e, \(T_{p}\mathbb{C}^N=T_{p}M+iT_{p}M\)). Denote by \(T^*\mathbb{C}^N\) the cotangent bundle of \(\mathbb{C}^N\) and for every \(p\in M\), by \(N_{p}M=T_{p}\mathbb{C}^N/T_{p}M\) the normal and by \(N^*_{p}M\subset T^*_{p}\mathbb{C}^N\) the conormal spaces to \(M\) in \(\mathbb{C}^N\). Let \(T^\mathbb{C} M\) be the complex tangent bundle to \(M\) and let \(T^{1,0}M\) and \(T^{0,1}M\) be the bundles of holomorphic and antiholomorphic tangent vector fields, respectively. The Levi form of \(M\) at \(p\in M\), \(L_{p}:T_{p}^{1,0}M \times T_{p}^{1,0}M\to N_{p}M\otimes \mathbb{C}\) is the Hermitian form such that the equality
\[ L_{p}(X_{p},Y_{p})=\frac{1}{2i}[X,\overline{Y}]_{p}\bmod T_{p}^{1,0}M\otimes T_{p}^{0,1}M \] holds for all vector fields \(X\) and \(Y\) in \(T^{1,0}M\). A submanifold \(M\subset\mathbb{C}^N\) is strongly pseudoconvex at \(p\in M\) if \(\xi(L_{p}(u,v))\) is positive for some \(\xi \in N^*_{p}M\). A domain \(W\subset \mathbb{C}^N\) is called edge with wedge \(M\) at \(p\) in the direction of an open cone \(\Gamma\subset N_{p}M\) if for any pair of open cones \(\Gamma'\), \(\Gamma'' \in N_{p}M\) with \(\overline{\Gamma'}\backslash \{0\}\) and \(\overline{\Gamma}\backslash \{0\} \in \Gamma''\) there exists a neighborhood \(U\) of \(p\) in \(\mathbb{C}^N\) such that (\(M\cap U)+ (\Gamma' \cap U)\subset W\) and \((M\cap U)+(\Gamma''\cap U)\) contains a neighborhood of \(p\) in \(W\), where \(N_{p}M\) is identified with any fixed complementary subspace to \(T_{p}M\) in \(\mathbb{C}^N\).
The main results of the paper are the following two theorems:
Theorem. Let \(M\subset \mathbb{C}^N\), \(N\geq2\), be a generic submanifold at a point \(p\in M\) and \(f\) be a germ at \(p\) of a holomorphic self-map of a strongly pseudoconvex wedge with wedge \(M\) at \(p\), not necessarily proper, such that \(f(z)=z+o(| z-p| ^3)\) as \(z\) approaches \(p\) nontangentially. Then \(f(z)\equiv{z}\).
Let \(M\subset C^N\), \(N\geq2\), be a generic submanifold through a point \(p,U\) and \(V\) be strongly pseudoconvex wedges with edge \(M\) at \(p\) and \(f\) be a germ at \(p\) of a holomorphic map between \(U\) and \(V\) with \(f(z)=z+o(| z-p| ^3)\) as \(z\) approaches \(p\) nontangentially. Then \(f(z)\equiv{z}\).