Invariance of Plurigenera and boundedness for Generalized Pairs

Abstract

In this note, we survey some recent developments in birational geometry concerning the boundedness of algebraic varieties. We delineate a strategy to extend some of these results to the case of generalized pairs, first introduced by Birkar and Zhang, when the associated log canonical divisor is ample, and the volume is fixed. In this context, we show a version of deformation invariance of plurigenera for generalized pairs. We conclude by discussing an application to the boundedness of varieties of Kodaira dimension κ(X) = dim(X) − 1.

Boundedness of elliptic Calabi-Yau threefolds

“…Theorem 6.18 can be used to deduce analogs of Theorem 1.1 in higher dimension. So far, there have been several results addressing the boundedness in codimension 1 of elliptic Calabi-Yau varieties admitting a rational section in any dimension, see [BDCS20,FS20a,DCS21]. Unfortunately, for n ≥ 4, the current state of the art regarding elliptic Calabi-Yau n-folds f : X → Y can only guarantee that Y is bounded in codimension 1.…”

On the boundedness of $n$-folds with $κ(X)=n-1$

“…In particular, j admits a rational section. Thus, one can argue as in [DCS17,FS20], and conclude that the set of Jacobian fibrations is bounded in codimension one. To retrieve the original fibration, we make use of tools developed by Dolgachev and Gross [DG94,Gro94].…”

“…If the elliptic fibration admits a section, using techniques developed in [DCS17], one can induce a polarization that bounds the total space of the fibration. This direction is successfully explored in [FS20]. On the other hand, an elliptic fibration does not necessarily admit a rational section.…”

On the connectedness principle and dual complexes for generalized pairs

“…Below, we shall summarize the relevant changes to the proof of [10,Theorem 1.4] in order to drop the assumption on the projectivity of the pairs and on 𝑆 = Spec(C). We refer to [9] for a detailed proof. Sketch of proof.…”