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Postivity of the logarithmic cotangent and Shafarevich-Viehweg conjecture. (Positivité du cotangent logarithmique et conjecture de Shafarevich-Viehweg.) (French) Zbl 1388.32019

Séminaire Bourbaki. Volume 2015/2016. Exposés 1104–1119. Avec table par noms d’auteurs de 1948/49 à 2015/16. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-855-8/pbk). Astérisque 390, 27-63, Exp. No. 1105 (2017).
This survey covers recent work on positivity of the cotangent bundle and its logarithmic (i.e., orbifold) analogue. In particular the article summarises two papers of F. Campana and M. Păun [Compos. Math. 152, No. 11, 2350–2370 (2016; Zbl 1388.32018); “Foliations with positive slopes and birational stability of orbifold cotangent bundles”, Preprint, arXiv:1508.02456], and also describes work of B. Taji [Compos. Math. 152, No. 7, 1421–1434 (2016; Zbl 1427.14031)] and of K. Jabbusch and S. Kebekus [Math. Z. 269, No. 3–4, 847–878 (2011; Zbl 1238.14024)].
One consequence of the work being described is the following result: if \(X\) is a smooth projective variety with \(K_X\) pseudo-effective, then for any integer \(m\geq 1\) and any torsion-free rank-\(r\) quotient \(Q\) of \((\Omega^1_X)^{\otimes m}\), the line bundle \(\det Q=(\bigwedge\nolimits^r Q)^{**}\) is also pseudo-effective.
This is a strengthening of an old result of Miyaoka, and can be seen as saying that positivity of \(K_X\) implies positivity of \(\Omega_X\), in a suitable sense. However, the work of Campana and Păun and of Taji is set in the wider context of orbifolds, which in this context are pairs \((X,\Delta)\) where \(X\) is a smooth projective variety and \(\Delta\) is a normal crossings \({\mathbb Q}\)-divisor with coefficients between \(0\) and \(1\). A major part of the work under discussion consists of setting out and establishing the properties of the orbifold cotangent bundle. As that is carried out across several papers, the careful summary here is a useful reference where everything is collected together. The proofs, of course, are only sketched here.
Another important part of the work of Campana and Păun [loc. cit.] is to give a criterion for a foliation \(F\) on \(X\) to be algebraically integrable, meaning that the leaves are open in their Zariski closures, which are then rationally connected. This holds if \(F\) has positive slope relative to some moving class on \(X\): the article under review includes an account of this terminology and the basic facts needed.
These results (the orbifold version is needed) yield a proof of an important conjecture of Viehweg to the effect that the base of any sufficiently general (maximal variation) family of canonically polarised varieties will always be of log general type. This result (also due to Campana and Păun) is further generalised by Taji, who incorporates it into a wider theorem which, at the opposite extreme, includes the conjecture of Campana refined by Jabbusch and Kebekus [loc. cit.]. All these developments are covered in the article under review, some in more detail than others but always with indications of where to find out more.
There are three sections. Section 1 covers algebraically integrable foliations, slopes, and positivity, all in an orbifold context: this is the common background to all the papers under discussion. Section 2 is devoted to the orbifold cotangent bundle and gives the precise orbifold version of the main theorems of Campana and Păun on the algebraic integrability of foliations and pseudo-effectivity (stated above for the case \(\Delta=0\)). Here complete proofs are given, showing why it is necessary to consider orbifolds even if only the \(\Delta=0\) case is immediately needed. Section 3 is devoted to examples, Taji’s extended result, and the refinements of Jabbusch and Kebekus (and some others).
For the entire collection see [Zbl 1370.00002].

MSC:

32Q10 Positive curvature complex manifolds
14D22 Fine and coarse moduli spaces
14J10 Families, moduli, classification: algebraic theory
14D99 Families, fibrations in algebraic geometry
14E99 Birational geometry
32J27 Compact Kähler manifolds: generalizations, classification
32S65 Singularities of holomorphic vector fields and foliations
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
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