English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

In this paper we are mainly interested in the case κǫ < 0, or equivalently, K(V, B) is not pseudo-effective. According to the classification philosophy, at least when B = 0, V should admit a Fano fibration structure in such cases. In §2, by using Log Minimal Model Program which is available in dimension ≤ 3 at present (cf. [Sho], [Ko]), we establish the existence of such a fibration in the polarized situation. Namely, under some reasonable assumptions, some birational transform (V , B, L) of (V, B, L) admits a fibration Φ : V −→W onto a normal variety W with dimW < dimV such that K(V , B) − κǫ(V, B, L)L = ΦA for some ample Q-bundle A on W . Such a fibration is unique up to some birational equivalence and every general fiber of Φ is Fano. In particular we have κǫ(V, B, L) ∈ Q, generalizing a result in [B]. In §3, we study the set of possible values of Kodaira energies for any fixed n = dimV , called the spectrum set. If we consider the case B = 0 only and V is allowed to have only terminal singularities, the spectrum set seems to have no negative limit point (cf. (3.2)). Using Kawamata’s result on the boundedness of QFano 3-folds, we prove the above spectrum conjecture for n ≤ 3 under the additional asumption that V is Q-factorial. On the other hand, if B 6= 0 or V is allowed to have log terminal singularities, the spectrum set becomes more complicated and has many negative limit points, even when n = 2. We just provide a few examples of this sort and present a conjecture. Of course, Mori-Kawamata theory is of fundamental importance in our method. In fact we need a slightly improved version than usual, so §1 is devoted to this purpose. The key is the notion of log ample Q-bundle. Our results in this paper form a philosophical background of the classification theory of polarized (log) varieties by Kodaira energy. See [BS], [F2], [F3] for precise classification results.

On Kodaira energy of polarized log varieties