Let $(X,B)$ be a pair, and let $f \colon X \rightarrow S$ be a contraction with $-(K_X + B)$ nef over $S$. A conjecture, known as the Shokurov-Kollar connectedness principle, predicts that $f^{-1} (s) \cap \mathrm{Nklt}(X,B)$ has at most two connected components, where $s \in S$ is an arbitrary schematic point and $\mathrm{Nklt}(X,B)$ denotes the non-klt locus of $(X,B)$. In this work, we prove this conjecture, characterizing those cases in which $\mathrm{Nklt}(X,B)$ fails to be connected, and we extend these same results also to the category of generalized pairs. Finally, we apply these results and the techniques to the study of the dual complex for generalized log Calabi-Yau pairs, generalizing results of Kollar-Xu and Nakamura.

On the connectedness principle and dual complexes for generalized pairs

In this article, we describe the structure of codimension 1 foliations with canonical singularities and numerically trivial canonical class on varieties with terminal singularities, extending a result of Loray, Pereira, and Touzet to this context.

https://hal.archives-ouvertes.fr/hal-03374138/document

Codimension 1 foliations with numerically trivial canonical class on singular spaces

We prove existence of flips, special termination, the base point free theorem and, in the case of log general type, the existence of minimal models for F-dlt foliated pairs of co-rank one on a $${\mathbb {Q}}$$
 -factorial projective threefold. As applications, we show the existence of F-dlt modifications and F-terminalisations for foliated pairs and we show that foliations with canonical or F-dlt singularities admit non-dicritical singularities. Finally, we show abundance in the case of numerically trivial foliated pairs.

/pdf/mmp-for-co-rank-one-foliations-on-threefolds-1frgn9fewy.pdf

MMP for co-rank one foliations on threefolds

In this article, we describe the structure of codimension one foliations with canonical singularities and numerically trivial canonical class on varieties with terminal singularities, extending a result of Loray, Pereira and Touzet to this context

Codimension one foliations with numerically trivial canonical class on singular spaces

In this paper, we study a projective klt pair $(X, \Delta)$ with the nef anti-log canonical divisor $-(K_X+\Delta)$ and its maximally rationally connected fibration $\psi: X \dashrightarrow Y$. We prove that the numerical dimension of the anti-log canonical divisor $-(K_X+\Delta)$ on $X$ coincides with that of the anti-log canonical divisor $-(K_{X_y}+\Delta_{X_y})$ on a general fiber $X_y$ of $\psi: X \dashrightarrow Y$, which is an analogue of Ejiri-Gongyo's result formulated for the Kodaira dimension. As a corollary, we reveal a relation between positivity of the anti-canonical divisor and the rational connectedness, which gives a sharper estimate than the question posed by Hacon-$\mathrm{M^{c}}$Kernan. Moreover, in the case of $X$ being smooth, we show that a maximally rationally connected fibration $\psi: X \to Y$ can be chosen to be a morphism to a smooth projective variety $Y$ with numerically trivial canonical divisor, and further that it is locally trivial with respect to the pair $(X, \Delta)$, which can be seen as a generalization of Cao-Horing's structure theorem to klt pair cases. Finally, we study the structure of the slope rationally connected quotient for a pair $(X, \Delta)$ with $-(K_X +\Delta)$ nef, and obtain a structure theorem for projective orbifold surfaces.

/pdf/projective-klt-pairs-with-nef-anti-canonical-divisor-1hrzc2iehc.pdf

Projective klt pairs with nef anti-canonical divisor

We develop some foundational results in a higher dimensional foliated Mori theory, andshow how these results can be used to prove a structure theorem for the Kleiman-Mori coneof curves in terms of the numerical properties of $K_{\cal F}$ for rank 2 foliationson threefolds. We also make progresstoward realizing a minimal model program for rank 2 foliations on threefolds.

/pdf/higher-dimensional-foliated-mori-theory-217vamazf9.pdf

Higher Dimensional Foliated Mori Theory

We develop some basic results in a higher dimensional foliated Mori theory, and show how these results can be used to prove a structure theorem for the Kleiman-Mori cone of curves in terms of the numerical properties of $K_{\mathcal{F}}$ for rank 2 foliations on threefolds. We also make progress toward realizing a minimal model program for rank 2 foliations on threefolds.

/pdf/higher-dimensional-foliated-mori-theory-1k8dqjd2vg.pdf

Higher dimensional foliated Mori theory

We continue the study of constructing invariant Laplacians on Julia sets, and studying properties of their spectra. In this paper we focus on two types of examples: 1) Julia sets of cubic polynomials $z^3 + c$ with a single critical point; 2) formal matings of quadratic Julia sets. The general scheme introduced in earlier papers in this series involves realizing the Julia set as a circle with identifications, and attempting to obtain the Laplacian as a renormalized limit of graph Laplacians on graphs derived form the circle with identifications model. In the case of cubic Julia sets the details follows the pattern established for quadratic Julia sets, but for matings the details are quite challenging, and we have only been completely successful for one example. Once we have constructed the Laplacian, we are able to use numerical methods to approximate the eigenvalues and eigenfunctions. One striking observation from the data is that for the cubic Julia sets the multiplicities of all eigenspaces (except for the trivial eigenspace of constants) are even numbers. Nothing like this is valid for the quadratic julia sets studied earlier. We are able to explain this, based on the fact that three is an odd number, and more precisely because the dihedral-3 symmetry group has only two distinct one-dimensional irreducible representations.

Laplacians on Julia Sets III: Cubic Julia Sets and Formal Matings

We prove existence of flips, special termination, the base point free theorem and, in the case of log general type, the existence of minimal models for F-dlt foliated pairs of co-rank one on a $\mathbb Q$-factorial projective threefold. 
As applications, we show the existence of F-dlt modifications and F-terminalisations for foliated pairs and we show that foliations with canonical or F-dlt singularities admit non-dicritical singularities. Finally, we show abundance in the case of numerically trivial foliated pairs.

Calum Spicer

Papers

Higher Dimensional Foliated Mori Theory

MMP for co-rank one foliations on threefolds

Higher dimensional foliated Mori theory

Laplacians on Julia Sets III: Cubic Julia Sets and Formal Matings

MMP for co-rank one foliations on threefolds.