arXiv subjects
Publications and source records attributed to Calum Spicer.
Let $S$ be a minimal compact complex surface of class VII with $b_2(S)=3$. We prove that if $S$ carries two distinct singular holomorphic foliations, then $S$ contains a global spherical shell. Moreover, it is an Inoue-Hirzebruch surface. The proof is based on Teleman's theorem on the existence of a cycle of rational curves and results of Dloussky and is inspired by work of Brunella.
We study minimal models and flops for foliations. We show that if $\mathcal F$ is a rank one foliation with canonical singularities on a normal projective $\mathbb Q$-factorial variety and $K_{\mathcal F}$ is pseudo-effective, then any two outputs of the $K_{\mathcal F}$-MMP are isomorphic. For co-rank one foliations on threefolds, we prove existence results for $D$-flops in the klt setting and, under additional hypotheses, in the F-dlt setting. By contrast, we construct examples showing that rank one foliations display pathologies absent from the classical MMP: flopping contractions need not admit $D$-flops, and nef and big canonical divisors need not give rise to canonical models, even in the category of algebraic spaces.
We develop an analytic construction of nowhere-vanishing harmonic $1$-forms on real loci of K3-fibred Calabi-Yau $3$-folds with collapsing Ricci-flat K\"ahler metrics. We apply our construction to examples whose real loci have connected components diffeomorphic to $S^1\times S^2$ and to both trivial and nontrivial mapping tori. As an application, we produce examples of compact $7$-manifold with holonomy $G_2$ via the Joyce-Karigiannis construction.
We prove that normal projective stable families of maximal variation, of fixed dimension, and with bounded adjoint volume are birationally bounded. This is a consequence of a substantially stronger statement, formulated a priori independently of stable families: algebraically integrable foliations of fixed dimension and bounded adjoint volume are log birationally bounded. In this way, the birational geometry of foliations provides a systematic framework for approaching classical boundedness problems for fibrations. A key input is our proof of M\textsuperscript{c}Kernan's ACC conjecture for interpolated log canonical thresholds of algebraically integrable foliations. This may be viewed as the foliated analogue of Shokurov's ACC conjecture for log canonical thresholds, proved in the classical setting by Hacon--M\textsuperscript{c}Kernan--Xu. As applications, we establish two boundedness criteria for Fano algebraically integrable adjoint foliated structures: Birkar's criterion for exceptional Fanos, and Jiang's criterion for Fanos for which both Tian's $\alpha$-invariant and the anti-canonical volume are bounded away from zero. We also obtain several results on the birational geometry of algebraically integrable adjoint foliated structures, including lower bounds for adjoint volumes, boundedness of automorphism groups, and ACC theorems for pseudo-effective thresholds, $\mathbb{R}$-complementary thresholds, and the Fano spectrum.
We survey recent progress on the birational geometry of foliations on complex varieties. We focus on the MMP viewpoint: singularities, adjunction and applications to the MMP for foliations on surfaces and to the existence of flips on threefolds.
Consider a log canonical pair $(X,B)$ such that there is a Cartier divisor $D$ for which $T_X(-\log B) \otimes \mathcal O(D)$ is locally free and globally generated. Let $\mathcal F$ be a log canonical foliation of rank 1 on $X$. We prove that there exists a divisor $\Gamma$ such that $(X, \Gamma)$ is log canonical and $K_X + \Gamma \sim K_{\mathcal F} + D$. We then apply this result to prove several statements on the birational geometry of rank 1 log canonical foliations on log homogeneous varieties.
We present a definition of stable family of foliations and show that the corresponding moduli functor for foliated surfaces is representable by a Deligne-Mumford stack.
We prove the base point free theorem for log canonical foliated pairs of rank one on a Q-factorial projective klt threefold. Moreover, we show abundance in the case of numerically trivial log canonical foliated pairs of rank one in any dimension.
We prove the existence of good minimal models for any klt algebraically integrable adjoint foliated structure of general type, and that Fano algebraically integrable adjoint foliated structures with total minimal log discrepancies and parameters bounded away from zero form a bounded family. These results serve as the algebraically integrable foliation analogues of the finite generation of the canonical rings proved by Birkar-Cascini-Hacon-M\textsuperscript{c}Kernan, and the Borisov-Alexeev-Borisov conjecture on the boundedness of Fano varieties proved by Birkar, respectively. As an application, we prove that the ambient variety of any lc Fano algebraically integrable foliation is of Fano type, provided the ambient variety is potentially klt.
For $\mathbb Q$-factorial klt algebraically integrable adjoint foliated structures, we prove the cone theorem, the contraction theorem, and the existence of flips. Therefore, we deduce the existence of the minimal model program for such structures. We also prove the base-point-freeness theorem for such structures of general type and establish an adjunction formula and the existence of $\mathbb Q$-factorial quasi-dlt modifications for algebraically integrable adjoint foliated structures.
We present an adjunction formula for foliations on varieties and we consider applications of the adjunction formula to the cone theorem for rank one foliations and the study of foliation singularities.
We show that termination of flips for $\mathbb Q$-factorial klt pairs in dimension $r$ implies existence of minimal models for algebraically integrable foliations of rank $r$ with log canonical singularities over a $\mathbb Q$-factorial klt projective variety.
We discuss a conjecture of Shokurov on the semi-ampleness of the moduli part of a general fibration.
We prove the Cone Theorem for algebraically integrable foliations. As a consequence, we show that termination of flips implies the b-nefness of the moduli part of a log canonical pair with respect to a contraction, generalising the case of lc trivial fibrations.
We explore the birational structure and invariants of a foliated surface $(X, \mathcal F)$ in terms of the adjoint divisor $K_{\mathcal F}+\epsilon K_X$, $0< \epsilon \ll 1$. We then establish a bound on the automorphism group of an adjoint general type foliated surface $(X, \mathcal F)$, provide a bound on the degree of a general curve invariant by an algebraically integrable foliation on a surface and prove that the set of $\epsilon$-adjoint canonical models of foliations of general type and with fixed volume form a bounded family.
We prove existence of flips for log canonical foliated pairs of rank one on a Q-factorial projective klt threefold. This, in particular, provides a proof of the existence of a minimal model for a rank one foliation on a threefold for a wider range of singularities, after McQuillan.
We provide several applications of the minimal model program to the local and global study of co-rank one foliations on threefolds. Locally, we prove a singular variant of Malgrange's theorem, a classification of terminal foliation singularities and the existence of separatrices for log canonical singularities. Globally, we prove termination of flips, a connectedness theorem on lc centres, a non-vanshing theorem and some hyperbolicity properties of foliations.
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.