A note on positivity of the CM line bundle

Abstract

We show that positivity of the CM line associated to a family of polarised varieties is intimately related to the stability of its members. We prove that the CM line is nef on any curve which meets the stable locus, and that it is pseudoeffective (i.e. in the closure of the effective cone) as long as there is at least one stable fibre. We give examples showing that the CM line can be strictly negative or strictly positive on curves in the unstable locus.

Stable maps in higher dimensions

“…Let f : U → Y be a flat proper morphism between schemes of constant relative dimension d and L U be a line bundle on U that is relatively ample over Y . We recall the construction of the CM-line bundle (see [31,Section 2] for a more detailed account). The Knudson-Mumford expansion [48,Theorem 4] provides line bundles λ i for i = 0, .…”

“…where µ = µ(U y , L U | Uy ) is the slope of any fibre of U (and the reader is warned our convention for µ differs to that of [31]). Now suppose p : B → Y is a morphism from a normal projective variety B and consider the fibre product…”

“…Remark 4.5. In the above we do not assume that L CM has any positivity, and in fact there are examples for which it is negative [31,Example 5.2]. This is the only case we know where K-stability of a map p : (X, L) → (Y, T ) may be interesting without any positivity assumptions on T .…”

mpleness of the CM line bundle on the moduli space of canonically polarized varieties

“…The concept of the CM line bundle was first introduced in [Tia97]. Unlike the Chow line bundle, it is not positive on the entire Hilbert scheme (see [FR06]). However, it is expected to be positive on the locus where the fibers are K-polystable.…”

“…A straightforward calculation using the Grothendieck-Riemann-Roch formula (see e.g. [FR06]) shows that…”