Rigidity theorems for primitive Fano 3-folds (Q1903146)

A fundamental problem in the classification theory of algebraic manifolds is how many different projective structures can exist on a given manifold \(X_0\). The aim of this paper is the study of projective structures on certain Fano 3-folds \(X_0\). Difficulties arise to exclude possible \(X\) with \(K_X\) ample, or \(K_X \text{nef} ((K_{X\cdot} C)\geq 0\) for every curve \(C)\). In the 3-fold case this can be excluded if we know that \(\chi ({\mathcal O}_X)>0\) using a result of \textit{Y. Miyaoka} [in: Algebraic geometry, Proc. Symp., Sendai 1985, Adv. Stud. Pure Math. 10, 449-476 (1987; Zbl 0648.14006)]. Of course, \(\chi ({\mathcal O}_{X_0})=1\), so we ask whether \(\chi ({\mathcal O}_X)\) is a topological invariant for projective 3-folds. Clearly \(\dim H^i (X, {\mathcal O}_X)\) are topological invariants for \(i=1,2\) if \(b_2\leq 2\) but whether \(\dim H^3(X, {\mathcal O}_X)\) is also invariant is a deep unsolved problem. We can force \(H^3(X, {\mathcal O}_X)\) to vanish by requiring \(b_3(X_0)=0\). So we deal only with Fano 3-folds with vanishing \(b_3\). In case \(b_2(X_0)=1\) those \(X_0\) are well understood and easy to deal with: \(X_0\) is \(\mathbb{P}_3\), \(Q_3\), one 3-fold of index 2 and a family of index 1; any \(X\) homeomorphic to \(X_0\) is again of the same type. So we turn to the case \(b_2\geq 2\); we will restrict ourselves here only to \(b_2 =2\), Fano 3-folds with \(b_2\geq 2\) are classified by \textit{S. Mori} and \textit{S. Mukai}, the most interesting case being \(b_2=2\) or 3. Such a \(X_0\) is called primitive if it is not the blow-up of another 3-fold along a smooth curve. We will also restrict ourselves to primitive \(X_0\); but certainly similar results can be proved also in the imprimitive case using the same methods. Our result is now: Theorem. Let \(X_0\) be a primitive Fano 3-fold with \(b_2=2\), \(b_3=0\). Let \(X\) be a projective smooth 3-fold homeomorphic to \(X\). Then either \(X\simeq X_0\), or \(X\simeq \mathbb{P}(E)\) with a rank 2-vector bundle \(E\) on \(\mathbb{P}_2\) whose Chern classes \((c_1,c_2)\) belong to the following set: \(\{(0,0), (-1,1)\), \((-1,0), (0,-1), (0,3)\}\) or \(X= \mathbb{P} ({\mathcal O}_{\mathbb{P}_1} (a)\oplus {\mathcal O}_{\mathbb{P}_1} (b)\oplus {\mathcal O}_{\mathbb{P}_1} (c))\) with \(a+b+c \equiv 0 (3)\). Using analogous methods, we are able in \S 7 to answer a question asked by \textit{F. Campana} in Math. Ann. 290, No. 1, 19-30 (1991; Zbl 0722.32014): If \(Z_0\) is a Moishezon non-projective twistor space, does there exist a projective threefold \(Z\) which is homeomorphic to \(Z_0\)? The answer is no, at least when \(b_2\) is odd. Let us recall that such a \(Z_0\) is the first known example of a manifold of class \({\mathcal C}\) (i.e.: bimeromorphic to a compact Kähler one) admitting arbitrarily small deformations which are not in the class \({\mathcal C}\).