Showing papers on "Minimal model program published in 2019"

Anti-pluricanonical systems on Fano varieties

Abstract: In this paper, we study the linear systems $|-mK_X|$ on Fano varieties $X$ with klt singularities. In a given dimension $d$, we prove $|-mK_X|$ is non-empty and contains an element with "good singularities" for some natural number $m$ depending only on $d$; if in addition $X$ is $\epsilon$-lc for some $\epsilon>0$, then we show that we can choose $m$ depending only on $d$ and $\epsilon$ so that $|-mK_X|$ defines a birational map. Further, we prove Shokurov's conjecture on boundedness of complements, and show that certain classes of Fano varieties form bounded families.

K-stability of cubic threefolds

Abstract: We prove that the K-moduli space of cubic threefolds is identical to their geometric invariant theory (GIT) moduli. More precisely, the K-semistability, K-polystability, and K-stability of cubic threefolds coincide with the corresponding GIT stabilities, which could be explicitly calculated. In particular, this implies that all smooth cubic threefolds admit Kahler–Einstein (KE) metrics and provides a precise list of singular KE ones. To achieve this, our main new contribution is an estimate in dimension 3 of the volumes of Kawamata log terminal singularities introduced by Chi Li. This is obtained via a detailed study of the classification of 3-dimensional canonical and terminal singularities, which was established during the study of the explicit 3-dimensional minimal model program.

The non-archimedean SYZ fibration

Abstract: We construct non-archimedean SYZ (Strominger–Yau–Zaslow) fibrations for maximally degenerate Calabi–Yau varieties, and we show that they are affinoid torus fibrations away from a codimension-two subset of the base. This confirms a prediction by Kontsevich and Soibelman. We also give an explicit description of the induced integral affine structure on the base of the SYZ fibration. Our main technical tool is a study of the structure of minimal dlt (divisorially log terminal) models along one-dimensional strata.

Del Pezzo surfaces and Mori fiber spaces in positive characteristic

Abstract: We settle a question that originates from results and remarks by Koll\'ar on extremal ray in the minimal model program: In positive characteristics, there are no Mori fibrations on threefolds with only terminal singularities whose generic fibers are geometrically non-normal surfaces. To show this we establish some general structure results for del Pezzo surfaces over imperfect ground fields. This relies on Reid's classification of non-normal del Pezzo surfaces over algebraically closed fields, combined with a detailed analysis of geometrical non-reducedness over imperfect fields of p-degree one.

Moduli of stable maps in genus one and logarithmic geometry, I

Abstract: This is the first in a pair of papers developing a framework for the application of logarithmic structures in the study of singular curves of genus 1. We construct a smooth and proper moduli space dominating the main component of Kontsevich’s space of stable genus 1 maps to projective space. A variation on this theme furnishes a modular interpretation for Vakil and Zinger’s famous desingularization of the Kontsevich space of maps in genus 1. Our methods also lead to smooth and proper moduli spaces of pointed genus 1 quasimaps to projective space. Finally, we present an application to the log minimal model program for ℳ1,n. We construct explicit factorizations of the rational maps among Smyth’s modular compactifications of pointed elliptic curves.

The Minimal Model Program for threefolds in characteristic five.

Abstract: We show the validity of the Minimal Model Program for threefolds in characteristic five.

Kawaguchi-Silverman conjecture for surjective endomorphisms

Abstract: We prove the Kawaguchi-Silverman conjecture (KSC), about the equality of arithmetic degree and dynamical degree, for every surjective endomorphism of any (possibly singular) projective surface. In high dimensions, we show that KSC holds for every surjective endomorphism of any $\mathbb{Q}$-factorial Kawamata log terminal projective variety admitting an int-amplified endomorphism, provided that KSC holds for any surjective endomorphism with the ramification divisor being totally invariant and irreducible. In particular, we show that KSC holds for every surjective endomorphism of any rationally connected smooth projective threefold admitting an int-amplified endomorphism. The main ingredients are the equivariant minimal model program, the effectiveness of the anti-canonical divisor and a characterization of toric pairs.

Remarks on special kinds of the relative log minimal model program

Abstract: We prove $$\mathbb {R}$$ -boundary divisor versions of results proved by Birkar (Publ Math Inst Hautes Etudes Sci 115(1):325–368, 2012) and Hacon–Xu (Invent Math 192(1):161–195, 2013) on special kinds of the relative log minimal model program.

Compact moduli of elliptic K3 surfaces

Abstract: We construct various modular compactifications of the space of elliptic K3 surfaces using tools from the minimal model program, and explicitly describe the surfaces parametrized by their boundaries. The coarse spaces of our constructed compactifications admit morphisms to the Satake-Baily-Borel compactification. Finally, we show that one of our spaces is smooth with coarse space the GIT quotient of pairs of Weierstrass K3 surfaces with a chosen fiber.

On the log minimal model program for $3$-folds over imperfect fields of characteristic $p>5$

Abstract: We prove that many of the results of the LMMP hold for $3$-folds over fields of characteristic $p>5$ which are not necessarily perfect. In particular, the existence of flips, the cone theorem, the contraction theorem for birational extremal rays, and the existence of log minimal models. As well as pertaining to the geometry of fibrations of relative dimension $3$ over algebraically closed fields, they have applications to tight closure in dimension $4$.

Trisecant Flops, their associated K3 surfaces and the rationality of some Fano fourfolds.

Abstract: We provide a new construction of rationality for cubic fourfolds via Mori's theory and the minimal model program. As an application, we present the solution of the Kuznetsov's conjecture for $d=42$ (the first open case). Our methods also show an explicit connection between the rationality of cubic fourfolds belonging to the first four admissible families $\mathcal C_d$, with $d=14,26,38$, and $42$ and some birational models of minimal K3 surfaces of degree $d$ contained in well known rational Fano fourfolds.

Local and global applications of the Minimal Model Program for co-rank one foliations on threefolds

Abstract: 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.

Connected algebraic groups acting on 3-dimensional Mori fibrations

Abstract: We study the connected algebraic groups acting on Mori fibrations $X \to Y$ with $X$ a rational threefold and $\mathrm{dim}(Y) \geq 1$. More precisely, for these fibre spaces we consider the neutral component of their automorphism groups and study their equivariant birational geometry. This is done using, inter alia, minimal model program and Sarkisov program and allows us to determine the maximal connected algebraic subgroups of $\mathrm{Bir}(\mathbb{P}^3)$, recovering most of the classification results of Hiroshi Umemura in the complex case.

Golod-Shafarevich-Type Theorems and Potential Algebras

Abstract: Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Grobner basis theory and generalized Golod–Shafarevich-type theorems for potential algebras, to determine finiteness conditions in terms of the potential. We consider two-generated potential algebras. Using Grobner bases techniques and arguing in terms of associated truncated algebra we prove that they cannot have dimension smaller than 8. This answers a question of Wemyss [21], related to the geometric argument of Toda [17]. We derive from the improved version of the Golod–Shafarevich theorem, that if the potential has only terms of degree 5 or higher, then the potential algebra is infinite dimensional. We prove that potential algebra for any homogeneous potential of degree n⩾3 is infinite dimensional. The proof includes a complete classification of all potentials of degree 3. Then we introduce a certain version of Koszul complex, and prove that in the class Pn of potential algebras with homogeneous potential of degree n+1⩾4⁠, the minimal Hilbert series is Hn=11−2t+2tn−tn+1⁠, so they are all infinite dimensional. Moreover, growth could be polynomial (but nonlinear) for the potential of degree 4, and is always exponential for potential of degree starting from 5. For one particular type of potential we prove a conjecture by Wemyss, which relates the difference of dimensions of potential algebra and its abelianization with Gopakumar–Vafa invariants.

Deformations of $$\mathbb {A}^1$$ A 1 -cylindrical varieties

Abstract: An algebraic variety is called $$\mathbb {A}^{1}$$ -cylindrical if it contains an $$\mathbb {A}^{1}$$ -cylinder, i.e. a Zariski open subset of the form $$Z\times \mathbb {A}^{1}$$ for some algebraic variety Z. We show that the generic fiber of a family $$f:X\rightarrow S$$ of normal $$\mathbb {A}^{1}$$ -cylindrical varieties becomes $$\mathbb {A}^{1}$$ -cylindrical after a finite extension of the base. This generalizes the main result of Dubouloz and Kishimoto (Nagoya Math J 223:1–20, 2016) which established this property for families of smooth $$\mathbb {A}^{1}$$ -cylindrical affine surfaces. Our second result is a criterion for existence of an $$\mathbb {A}^{1}$$ -cylinder in X which we derive from a careful inspection of a relative Minimal Model Program run from a suitable smooth relative projective model of X over S.

Simple embeddings of rational homology balls and antiflips

Abstract: Let $V$ be a regular neighborhood of a negative chain of $2$-spheres (i.e. exceptional divisor of a cyclic quotient singularity), and let $B_{p,q}$ be a rational homology ball which is smoothly embedded in $V$. Assume that the embedding is simple, i.e. the corresponding rational blow-up can be obtained by just a sequence of ordinary blow-ups from $V$. Then we show that this simple embedding comes from the semi-stable minimal model program (MMP) for $3$-dimensional complex algebraic varieties under certain mild conditions. That is, one can find all simply embedded $B_{p,q}$'s in $V$ via a finite sequence of antiflips applied to a trivial family over a disk. As applications, simple embeddings are impossible for chains of $2$-spheres with self-intersections equal to $-2$. We also show that there are (infinitely many) pairs of disjoint $B_{p,q}$'s smoothly embedded in regular neighborhoods of (almost all) negative chains of $2$-spheres. Along the way, we describe how MMP gives (infinitely many) pairs of disjoint rational homology balls $B_{p,q}$ embedded in blown-up rational homology balls $B_{n,a} # \bar{\mathbb{CP}^2}$ (via certain divisorial contractions), and in the Milnor fibers of certain cyclic quotient surface singularities. This generalizes results in [Khodorovskiy-2014], [H. Park-J. Park-D. Shin-2016], [Owens-2017] by means of a uniform point of view.

Interaction between singularity theory and the minimal model program

Abstract: We survey some recent topics on singularities, with a focus on their connection to the minimal model program. This includes the construction and properties of dual complexes, the proof of the ACC conjecture for log canonical thresholds and the recent progress on the `local stability theory' of an arbitrary Kawamata log terminal singularity.

Int-amplified endomorphisms of compact Kähler spaces

Abstract: Let $X$ be a normal compact Kahler space of dimension $n$. A surjective endomorphism $f$ of such $X$ is int-amplified if $f^*\xi-\xi=\eta$ for some Kahler classes $\xi$ and $\eta$. First, we show that this definition generalizes the notation in the projective setting. Second, we prove that for the cases of $X$ being smooth, a surface or a threefold with mild singularities, if $X$ admits an int-amplified endomorphism with pseudo-effective canonical divisor, then it is a $Q$-torus. Finally, we consider a normal compact Kahler threefold $Y$ with only terminal singularities and show that, replacing $f$ by a positive power, we can run the minimal model program (MMP) $f$-equivariantly for such $Y$ and reach either a $Q$-torus or a Fano (projective) variety of Picard number one.

Manin’s b-constant in families

Abstract: We show that the b-constant (appearing in Manin’s conjecture) is constant on very general fibers of a family of algebraic varieties. If the fibers of the family are uniruled, then we show that the b-constant is constant on general fibers.

On semistable degenerations of Fano varieties

Abstract: Consider a family of Fano varieties $\pi: X \longrightarrow B i o$ over a curve germ with a smooth total space $X$. Assume that the generic fiber is smooth and the special fiber $F=\pi^{-1}(o)$ has simple normal crossings. Then $F$ is called a semistable degeneration of Fano varieties. We show that the dual complex of $F$ is a simplex of dimension $\leq \mathrm{dim}\ F$. Simplices of any admissible dimension can be realized for any dimension of the fiber. Using this result and the Minimal Model Program in dimension $3$ we reproduce the classification of the semistable degenerations of del Pezzo surfaces obtained by Fujita. We also show that the maximal degeneration is unique and has trivial monodromy in dimension $\leq3$.

Totally invariant divisors of int-amplified endomorphisms of normal projective varieties

Abstract: We consider an arbitrary int-amplified surjective endomorphism $f$ of a normal projective variety $X$ over $\mathbb{C}$ and its $f^{-1}$-stable prime divisors. We extend the early result for the case of polarized endomorphisms to the case of int-amplified endomorphisms. Assume further that $X$ has at worst Kawamata log terminal singularities. We prove that the total number of $f^{-1}$-stable prime divisors has an optimal upper bound $\dim X+\rho(X)$, where $\rho(X)$ is the Picard number. Also, we give a sufficient condition for $X$ to be rationally connected and simply connected. Finally, by running the minimal model program (MMP), we prove that, under some extra conditions, the end product of the MMP can only be an elliptic curve or a single point.

Geometric invariants and Geometric consistency of Manin’s conjecture

Abstract: Manin’s conjeture states that the asymptotic growth of the number of rational points on a Fano variety over a number field is governed by certain geometric invariants (a and b-constants). In this thesis we study the behaviour of these geometric invariants and show that Manin’s conjecture is geometrically consistent. In the first part, we study the behaviour of the b-constant in families and show that the b-constant is constant on very general fibers of a family of algebraic varieties. If the fibers of the family are uniruled, then we show that the b-constant is constant on general fibers. In the second part, we study the behaviour of the a-constant (Fujita invariant) under pull-back to generically finite covers and prove a conjecture of Lehmann-Tanimoto about finiteness of covers. In the last part, based on joint work with B. Lehmann and S. Tanimoto, we prove geometric consistency of Manin’s conjecture by showing that the rational points contributed by subvarieties or covers with larger geometric invariants are contained in a thin set.

Note on the semiampleness theorem for the log canonical divisors on klt varieties which are nef and abundant

Abstract: We give another alternative proof to the Kawamata semiampleness theorem for the log canonical divisors on klt varieties which are nef and abundant. After the first version of this article was posted to the e-print Arxiv, Prof. Fujino notified the author that the quick and essential proof ([Fujino. On Kawamata's theorem.(EMS 2011), Rem 2.7]) is already known. The author would like to thank him. More precisely, Prof. Fujino already gave the quick and essential proof ([Fujino. On Kawamata's thm.(EMS 2011), Rem 2.7], [Fujino. Finite generation of the lc ring in dim 4. (Kyoto J. Math. 50 (2010)), Rem 3.15]) from the finite generation thm (Birkar-Cascini-Hacon-McKernan [BCHM]) of the lc rings for klt pairs and from the fact (cf. Mourougane-Russo [MoRu, C.R.A.S. Math. 325 (1997)]) that a nef and abundant $\mathbf{Q}$-divisor $D$ is semiample if its graded ring is finitely generated:"For a nef and abundant lc divisor which is klt, the lc ring is finitely generated, thus it is semiample." [BCHM] first proved that the minimal model program runs for big klt lc divisors and next implied the finite generation of the lc rings for klt lc divisors which are not necessarily big from the Fujino-Mori lc bdle formula ([FM, J. Differential Geom., 56 (2000)]). Mourougane-Russo [MoRu] implies the semiampleness of a nef and abundant $\mathbf{Q}$-divisor whose graded ring is finitely generated, using the Kawamata numerically trivial fibrations ([Kawamata. Pluricanonical systems. Invent. Math. 79 (1985)]). Consequently the author withdraw the article.

Pseudolattices, del pezzo surfaces, and lefschetz fibrations

Abstract: Motivated by the relationship between numerical Grothendieck groups induced by the embedding of a smooth anticanonical elliptic curve into a del Pezzo surface, we define the notion of a quasi del Pezzo homomorphism between pseudolattices and establish its basic properties. The primary aim of the paper is then to prove a classification theorem for quasi del Pezzo homomorphisms, using a pseudolattice variant of the minimal model program. Finally, this result is applied to the classification of a certain class of genus one Lefschetz fibrations over discs.

Birational geometry of moduli spaces of perverse coherent sheaves on blow-ups

Abstract: In order to study wall crossing formula of Donaldson type invariants on the blown-up plane, Nakajima-Yoshioka constructed a sequence of blow-up/blow-down diagrams connecting the moduli space of torsion free framed sheaves on projective plane, and that on its blow-up. In this paper, we prove that Nakajima-Yoshioka's diagram realizes the minimal model program. Furthermore, we obtain a fully-faithful embedding between the derived categories of these moduli spaces.