Vector bundles without intermediate cohomology and the trichotomy result (Q6623828)

Let \((X,{\mathcal O}(1))\) be a projective variety \(X\) polarized by a very ample line bundle \({\mathcal O}(1)\). A vector bundle \(E\) on \(X\) is said to be arithmetically Cohen-Macaulay (ACM) (or not to have intermediate cohomology) if \(h^i(X, E(t))=0\) for any integer \(t\) and \(0 <i <\dim X\). In particular, \((X,{\mathcal O}(1))\) is called arithmetically Cohen-Macaulay when so is \({\mathcal O}(1)\). The paper under review is an expository survey on some results on ACM vector bundles. \N\NIn the first section the cases \(X\) a projective space and \(X\) a quadric are considered. For \(({\mathbb P}^n,{\mathcal O}(1))\), as a consequence of a criterion of \N\textit{G. Horrocks} [Proc. Lond. Math. Soc. (3) 14, 689--713 (1964; Zbl 0126.16801)] (see Theorem 1.1), the ACM bundles decompose as a sum of line bundles. For quadrics (see Theorem 1.6 by \textit{H. Knörrer} [Invent. Math. 88, 153--164 (1987; Zbl 0617.14033)]), the spinor bundles appear as a new indecomposable piece of ACM bundles. These two cases are called finite, in the sense that up to twist by line bundles, indecomposable ACM bundles are a finite set. Moreover, in this first section of the paper under review some other results are presented, of particular interest on Grassmannians where, besides the universal bundles, big families of indecomposable ACM bundles appear. \N\NThis is complemented in Section 2, where the derived categories of the projective space, the quadrics and the Grassmannians (see Theorem 2.8 by \N\textit{M. M. Kapranov} [Invent. Math. 92, No. 3, 479--508 (1988; Zbl 0651.18008)]) \Nare presented. ACM bundles on Grassmannians make families of arbitrarily large dimension, what is said that Grassmannians are wild. \N\NLast section is devoted to present the trichotomy result \N(see Theorem 3.1 by \textit{D. Faenzi} and \textit{J. Pons-Llopis} [Épijournal de Géom. Algébr., EPIGA 5, Article 8, 37 p. (2021; Zbl 1480.14013)]): \NAny reduced ACM variety \(X \subset {\mathbb P}^n\) of positive dimension is: either (1) finite (only finitely many indecomposable ACM sheaves over \(X\) up to twist by line bundles); or (2) tame (the moduli space of indecomposable non-isomorphic ACM sheaves of any rank is a finite or countable union of points or curves); or (3) wild (\(X\) supports families of arbitrarily large dimension of indecomposable non-isomorphic ACM sheaves). The classification of the finite and tame cases is also provided.