On rank 3 instanton bundles on \(\mathbb{P}^3\) (Q6606924)

Mathematical instanton bundles on \(\mathbb{P}^3\) arose as the algebraic counterpart to the (anti)self-dual solutions to the Yang-Mills equations on \(S^4\). Prompted by this connection, the study of instanton bundles has generated significant interest in the mathematical community. Instanton bundles were originally defined as rank two, stable vector bundles \(E\) with trivial determinant, satisfying a vanishing in cohomology, i.e., \(H^1(E(-2))=0\). Since then, several natural generalizations have been proposed: some authors have considered other projective varieties, while others have dealt with the higher rank case.\N\NIn this work, the authors consider the case of rank three instanton bundles on \(\mathbb{P}^3\) as a tool to study the irreducible components of the moduli spaces of rank three stable vector bundles on \(\mathbb{P}^3\). In Theorem 2.4, the authors establish a correspondence between instanton bundles and rational curves in \(\mathbb{P}^3\) via Serre's correspondence. In Section 4, they introduce the notion of generalized 't Hooft instantons, namely rank \(r\) instanton bundles \(F\) such that \(h^0(F(1)) \geq r-1\). They prove in Theorem 4.3 that the generalized rank three 't Hooft bundles are smooth points in their irreducible component in the moduli spaces of rank three stable vector bundles. The authors then focus on the case \(c_2 = 2\) and prove in Theorem 4.5 that there is a one-to-one correspondence between:\N\begin{itemize}\N\item Pairs \((F, \sigma)\) where \(F\) is a rank 3 stable instanton bundle of charge 2 on \(\mathbb{P}^3\) and \(\sigma\) is a global section of \(F(1)\);\N\item Pairs \((E, \xi)\) where \(E\) is a generic rank 2 stable reflexive sheaf on \(\mathbb{P}^3\) with Chern classes \((-1, 3, 3)\) and \(\xi \in H^0(\mathcal{E}xt^1(E(2), \mathcal{O}_{\mathbb{P}^3}))\) such that \(\xi\) is surjective.\N\end{itemize}\NThe authors exploit this correspondence to compute the dimension of \(\mathcal{I}(2)\), the family of rank three instanton bundles of charge 2. Moreover, they prove that \(\mathcal{I}(2)\) actually coincides with the whole moduli space \(\mathcal{B}(0, 2, 0)\) of stable rank three vector bundles with Chern classes \((0, 2, 0)\). Finally, in Theorem 5.3, they show that \(\mathcal{B}(0, 2, 0)\) is irreducible of dimension 16, and its generic point corresponds to a generalized 't Hooft instanton bundle.