Symmetric differentials and variations of Hodge structures
This interesting paper aims at extending the main result of [\textit{Y. Brunebarbe} et al., Duke Math. J. 162, No. 14, 2797--2813 (2013; Zbl 1296.32003)] about the existence of symmetric differential forms on compact Kähler manifolds to the realm of non-compact complex manifolds using logarithmic compactifications (as defined right below). Let \(U\) be a smooth complex algebraic variety, and let \(X\) be a logarithmic compactification of \(U\), that is, a complex manifold compactifying \(U\) such that \(D:=X-U\) is a normal crossing divisor with smooth irreducible components. The main theorem of this paper states that if \(U\) supports a polarized complex variation of Hodge structures \(\mathbb{V}\) with infinite and integral monodromy, then there is a non-zero global section of the symmetric products \(S^k(\Omega_X^1(\text{log} D))\) for every \(k\) greater than some \(k_0\). It also gives a lower bound to the Sakai's dimension of the logarithmic cotangent bundle \(\Omega_X^1(\text{log} D)\). This result is shown by reducing to the case in which the period map is generically immersive and the local monodromy around \(D\) of the local system underlying \(\mathbb{V}\) is unipotent. Under these new assumptions, the author proves another important theorem of the paper under review, stating that the logarithmic cotangent bundle is big. This in turn is a consequence of an extension of the Fujita-Kawamata semi-positivity theorem, proved in the text as well. After the introduction and an expository section on Hodge theory, the core of the article is devoted to the proof of the three theorems mentioned above, using a construction of an interesting sort of nearby cycles on real graded polarized families of mixed Hodge structures defined along the way. The paper ends with some appendices covering several background topics.
scientific article; zbMATH DE number 6950104
scientific article