Deformations of the tangent bundle and their restriction to standard rational curves (Q977139)

Let \(X\) be a complex projective manifold that is uniruled, i.e. \(X\) is covered by rational curves. A rational curve \(f: \mathbb P^1 \rightarrow X\) is said to be standard if \(f^* T_X\) decomposes as \(\mathcal O(2) \oplus \mathcal O(1)^p \oplus \mathcal O^q\). It is well-known that a general element of a covering family of minimal degree is a standard rational curve, the study of these curves and their associated varieties of minimal rational tangents are an important tool in the theory of uniruled manifolds [cf. \textit{J.-M. Hwang}, Trieste ICTP Lect. Notes 6, 335--393 (2001; Zbl 1086.14506)] for an introduction as well as the numerous articles of Hwang and Mok on this subject). In the paper under review the author addresses the following question: let \(X\) be a uniruled projective manifold, and \(f: \mathbb P^1 \rightarrow X\) a standard rational curve on it. Let \(\{V_t, t \in \Delta \}\) be a small deformation of the tangent bundle \(T_X\), i.e. a holomorphic family of vector bundles on \(X\) such that \(V_0 \simeq T_X\). Do we have \(f^* V_t \simeq f^* V_0\) for sufficiently small \(t\)? Since it is fairly easy to construct examples where the answer to this question is negative, one should ask more precisely for sufficient conditions. The main theorem is that if \(X\) satisfies the Hodge-theoretic condition \(H^{2i}(X, \Omega_X)=0\) for all \(i \geq 0\), we have \(f^* V_t \simeq f^* V_0\) for sufficiently small \(t\). The proof of the main theorem is quite interesting: since the total space of the cotangent bundle \(\Omega_X \simeq V_0^*\) is a holomorphic symplectic manifold, one can modify a construction of \textit{D. Kaledin} and \textit{M. Verbitsky} [Geom. Funct. Anal. 12, No. 6, 1265--1295 (2002; Zbl 1032.58009)] to apply an argument of \textit{J. Wierzba} [J. Algebr. Geom. 12, No. 3, 507--534 (2003; Zbl 1094.14010)] on the deformations of rational curves in holomorphic symplectic manifolds. As an application of the main theorem the author studies the question of the rigidity of the tangent bundle under small deformations.