Frobenius integrability of certain \(p\)-forms on singular spaces. (Intégrabilité au sens de Frobenius pour certaines \(p\)-formes sur des espaces singuliers.) (English. French summary) Zbl 07863253
Let \(u\) be a holomorphic \(p\)-form on a compact Kähler manifold \(X\). Then \(u\) is closed, and therefore the subsheaf \(S_u \subset T_X\) consisting of holomorphic vector fields \(\xi\) such that the contraction \(\iota_\xi u := u(\xi, -)\) vanishes is integrable and defines a foliation on \(X\). This follows from Cartan’s magic formula and Frobenius’ theorem.
Demailly generalized this to \(p\)-forms with values in an anti-pseudo-effective line bundle \(L\). The present paper generalizes this further, to singular spaces and reflexive differential forms (those defined on the smooth locus) with values in an anti-pseudo-effective \(\mathbb Q\)-Cartier rank one reflexive sheaf \(\mathscr A\). This works if either \(X\) is klt, or \(X\) is log canonical and \(p = 1\).
The authors also give a corollary about log canonical spaces admitting a contact structure: their canonical divisor is not pseudo-effective.
Demailly generalized this to \(p\)-forms with values in an anti-pseudo-effective line bundle \(L\). The present paper generalizes this further, to singular spaces and reflexive differential forms (those defined on the smooth locus) with values in an anti-pseudo-effective \(\mathbb Q\)-Cartier rank one reflexive sheaf \(\mathscr A\). This works if either \(X\) is klt, or \(X\) is log canonical and \(p = 1\).
The authors also give a corollary about log canonical spaces admitting a contact structure: their canonical divisor is not pseudo-effective.
Reviewer: Patrick Graf (Bayreuth)
MSC:
| 14E30 | Minimal model program (Mori theory, extremal rays) |
| 32Q15 | Kähler manifolds |
| 32J25 | Transcendental methods of algebraic geometry (complex-analytic aspects) |
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