Moishezon fourfolds homeomorphic to \(Q_{\mathbb{C}}^ 4\) (Q1185148)

This paper announces several results towards the followings conjecture: any Moishezon complex manifold \(X\) homeomorphic to the \(n\)-dimensional complex hyperquadric \(Q^ n\) should be isomorphic to it. In particular: for \(n=3\), this is true, and also if \(n=4\), provided \(X\) admits \(L\in\text{Pic}(X)\) with \(L^ 4=2\) and \(h^ 0(X,L)\geq 5\). As a consequence: the global deformations of \(Q^ 4\) are again \(Q^ 4\). Also: if \(X\) is a complex \(n\)-fold with \(c_ 1(X)=nc_ 1(L)\) and \(h^ 0(X,L)\geq n+1\), a classification of all irreducible components \(C\) of the scheme-theoretic intersection \(\ell\) of \((n-1)\) members of \(| L|\) is announced, provided \(C\) is a generically reduced curve. In particular: \(C\cong\mathbb{P}_ 1\), \(1\leq L\cdot C\leq 2\).