On Moishezon manifolds homeomorphic to \(\mathbb{P}^ n_ \mathbb{C}\) (Q1801844)

The paper studies the following conjecture and gives a partial solution to it. Conjecture \(MP_ n\). Any Moishezon complex manifold homeomorphic to \(\mathbb{P}_ \mathbb{C}^ n\) is isomorphic to \(\mathbb{P}_ \mathbb{C}^ n\). Conjecture \(MP_ n\) has been settled by \textit{F. Hirzebruch} and \textit{K. Kodaira} (1957) and Yau (1977) when the manifold is Kählerian, while the present article treats also non-Kählerian cases. Kollár (1991) and the author (1987) solved \((MP_ 3)\) in the affirmative, each supplementing the other. See the references of the article. -- The paper proves the following theorem. Let \(X\) be a Moishezon fourfold homeomorphic to \(\mathbb{P}_ \mathbb{C}^ 4\), and \(L\) a line bundle on \(X\) with \(L^ 4 = 1\). If \(h^ 0(X,L) \geq 3\), then \(X \simeq \mathbb{P}_ \mathbb{C}^ 4\). A related theorem is also proved in the paper. Let \(X\) be a Moishezon fourfold, and \(L\) a line bundle on \(X\). Assume that \(\text{Pic} X = \mathbb{Z} L\), \(c_ 1(X) = dc_ 1 (L)\) \((d \geq 5)\) and \(h^ 0(X,L) \geq 4\). Then \(X \simeq \mathbb{P}_ \mathbb{C}^ 4\). A theorem for a smooth quadric hypersurface \(\mathbb{Q}_ \mathbb{C}^ 4\) in \(\mathbb{P}_ \mathbb{C}^ 5\) similar to theorem 1 is also proved by the author [``Moishezon fourfolds homeomorphic to a quadric hypersurface \(Q^ 4\) in \(\mathbb{P}^ 5\)'', Osaka J. Math. (to appear)]. Any global (not only complex analytic but even if differentiable) deformation of \(\mathbb{P}_ \mathbb{C}^ 4\) is isomorphic to \(\mathbb{P}_ \mathbb{C}^ 4\). Conjecture \(DP_ n\) of the paper under review (``Any complex deformation of \(\mathbb{P}^ n_ \mathbb{C}\) is isomorphic to \(\mathbb{P}^ n_ \mathbb{C}\)'') in the complex analytic deformation case is a special case of Siu's theorem [cf. \textit{Y.-T. Siu}, Prospects in Complex geometry, Proc. 25th Int. Taniguchi Symp. Katata 1989, Conf., Kyoto 1989, Lect. Notes Math. 1468, 254-280 (1991; Zbl 0748.32013)].