Ample Divisor Research Articles

On the finite generation of valuation semigroups on toric surfaces

We provide a combinatorial criterion for the finite generation of a valuation semigroup associated with an ample divisor on a smooth toric surface and a non-toric valuation of maximal rank. As an application, we construct a lattice polytope such that none of the valuation semigroups of the associated polarized toric variety coming from one-parameter subgroups and centered at a non-toric point are finitely generated.

On homological mirror symmetry for the complement of a smooth ample divisor in a K3 surface

On homological mirror symmetry for the complement of a smooth ample divisor in a K3 surface

Remarks on the existence of minimal models of log canonical generalized pairs

Given an NQC log canonical generalized pair (X,B+M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(X,B+M)$$\\end{document} whose underlying variety X is not necessarily Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}$$\\end{document}-factorial, we show that one may run a (KX+B+M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(K_X+B+M)$$\\end{document}-MMP with scaling of an ample divisor which terminates, provided that (X,B+M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(X,B+M)$$\\end{document} has a minimal model in a weaker sense or that KX+B+M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_X+B+M$$\\end{document} is not pseudo-effective. We also prove the existence of minimal models of pseudo-effective NQC log canonical generalized pairs under various additional assumptions, for instance when the boundary contains an ample divisor.

Lefschetz-type theorems for the effective cone on Hyperkähler varieties

In this paper we show some Lefschetz-type theorems for the effective cone of Hyperkähler varieties. In particular we are able to show that the inclusion of any smooth ample divisor induces an isomorphism of effective cones. Moreover we deduce a similar statement for some effective exceptional divisors, which yields the computation of the effective cone of e.g. projectivized cotangent bundles or some projectivized Lazarsfeld–Mukai bundles.

Resolution and alteration with ample exceptional divisor

Resolution and alteration with ample exceptional divisor

Minimal compactifications of the affine plane with nef canonical divisors

We shall consider minimal analytic compactifications of the affine plane with singularities. In previous work, Kojima and Takahashi proved that any minimal analytic compactification of the affine plane, which has at worse log canonical singularities, is a numerical del Pezzo surface (i.e., a normal complete algebraic surface with the numerically ample anti-canonical divisor) and has only rational singularities. In this article, we show that any minimal analytic compactification of the affine plane with the nef canonical divisor has an irrational singularity.

Minimizing CM Degree and Specially K-stable Varieties

Abstract We prove that the degree of the CM line bundle for a normal family over a curve with fixed general fibers is strictly minimized if the special fiber is either a smooth projective manifold with a unique cscK metric or “specially K-stable”, which is a new class we introduce in this paper. This phenomenon, as conjectured by Odaka (cf., [ 46]), is a quantitative strengthening of the separatedness conjecture of moduli spaces of polarized K-stable varieties. The above-mentioned special K-stability implies the original K-stability and a lot of cases satisfy it, for example, K-stable log Fano, klt Calabi-Yau (i.e., $K_{X}\equiv 0$), lc varieties with the ample canonical divisor and uniformly adiabatically K-stable klt-trivial fibrations over curves (cf., [ 27]).

The F-Signature Function on the Ample Cone

Abstract For any fixed globally $F$-regular projective variety $X$ over an algebraically closed field of positive characteristic, we study the $F$-signature of section rings of $X$ with respect to the ample Cartier divisors on $X$. In particular, we define an $F$-signature function on the ample cone of $X$ and show that it is locally Lipschitz continuous. We further prove that the $F$-signature function extends to the boundary of the ample cone. We also establish an effective comparison between the $F$-signature function and the volume function on the ample cone. As a consequence, we show that for divisors that are nef but not big, the extension of the $F$-signature is zero.

Integral Points on Varieties With Infinite Étale Fundamental Group

Abstract We study integral points on varieties with infinite étale fundamental groups. More precisely, for a number field $F$ and $X/F$ a smooth projective variety, we prove that for any geometrically Galois cover $\varphi \colon Y \to X$ of degree at least $2\dim (X)^{2}$, there exists an ample line bundle $\mathscr{L}$ on $Y$ such that for a general member $D$ of the complete linear system $|\mathscr{L}|$, $D$ is geometrically irreducible and any set of $\varphi (D)$-integral points on $X$ is finite. We apply this result to varieties with infinite étale fundamental group to give new examples of irreducible, ample divisors on varieties for which finiteness of integral points is provable.

Geometric description of 〈2〉-polarised Hilbert squares of generic K3 surfaces

A generic K3 surface of degree 2t is a general complex projective K3 surface S2t whose Picard group is generated by the class of an ample divisor H∈Div(S2t) such that H2=2t with respect to the intersection form. We show that if X is the Hilbert square of a generic K3 surface of degree 2t with t≠2 which admits an ample divisor D∈Div(X) with qX(D)=2, where qX is the Beauville–Bogomolov–Fujiki form, then X is a double EPW sextic.

Kähler-Ricci flow on rational homogeneous varieties

In this work, we study the Kähler-Ricci flow on rational homogeneous varieties exploring the interplay between projective algebraic geometry and representation theory which underlies the classical Borel-Weil theorem. By using elements of representation theory of semisimple Lie groups and Lie algebras, we give an explicit description for all solutions of the Kähler-Ricci flow with homogeneous initial condition. This description enables us to compute explicitly the maximal existence time for any solution starting at a homogeneous Kähler metric and obtain explicit upper and lower bounds for several geometric quantities along the flow, including curvatures, volume, diameter, and the first non-zero eigenvalue of the Laplacian. As an application of our main result, we investigate the relationship between certain numerical invariants associated with ample divisors and numerical invariants arising from solutions of the Kähler-Ricci flow in the homogeneous setting.

Deformations of rational curves on primitive symplectic varieties and applications

We study the deformation theory of rational curves on primitive symplectic varieties and show that if the rational curves cover a divisor, then, as in the smooth case, they deform along their Hodge locus in the universal locally trivial deformation. As applications, we extend Markman's deformation invariance of prime exceptional divisors along their Hodge locus to this singular framework and provide existence results for uniruled ample divisors on primitive symplectic varieties which are locally trivial deformations of any moduli space of semistable objects on a projective $K3$ or fibers of the Albanese map of those on an abelian surface. We also present an application to the existence of prime exceptional divisors.

Log Calabi–Yau Structure of Projective Threefolds Admitting Polarized Endomorphisms

Abstract Let $X$ be a normal projective variety admitting a polarized endomorphism $f$, that is, $f^*H\sim qH$ for some ample divisor $H$ and integer $q>1$. It was conjectured by Broustet and Gongyo that $X$ is of Calabi–Yau type, that is, $(X,\Delta )$ is lc for some effective ${\mathbb {Q}}$-divisor such that $K_X+\Delta \sim _{{\mathbb {Q}}} 0$. In this paper, we establish a general guideline based on the equivariant minimal model program and the canonical bundle formula. In this way, we prove the conjecture when $X$ is a smooth projective threefold.

Remarks on nef and movable cones of hypersurfaces in Mori dream spaces

We investigate nef and movable cones of hypersurfaces in Mori dream spaces. The first result is: Let Z be a smooth Mori dream space of dimension at least four whose extremal contractions are of fiber type of relative dimension at least two and let X be a smooth ample divisor in Z, then X is a Mori dream space as well. The second result is: Let Z be a Fano manifold of dimension at least four whose extremal contractions are of fiber type and let X be a smooth anti-canonical hypersurface in Z, which is a smooth Calabi–Yau variety, then the unique minimal model of X up to isomorphism is X itself, and moreover, the movable cone conjecture holds for X, namely, there exists a rational polyhedral cone which is a fundamental domain for the action of birational automorphisms on the effective movable cone of X. The third result is: Let P:=Pn×⋯×Pn be the N-fold self-product of the n-dimensional projective space. Let X be a general complete intersection of n+1 hypersurfaces of multidegree (1,…,1) in P with dim⁡X≥3. Then X has only finitely many minimal models up to isomorphism, and moreover, the movable cone conjecture holds for X.

Pushforwards of Chow groups of smooth ample divisors, with an emphasis on Jacobian varieties

AbstractWith a homological Lefschetz conjecture in mind, we prove the injectivity of the pushforward morphism on low‐dimensional rational Chow groups, induced by the closed embedding of an ample divisor, namely, the Theta divisor inside the Jacobian variety . Here, C is a smooth irreducible complex projective curve.

A RIGIDITY PROPERTY OF PLURIHARMONIC MAPS FROM PROJECTIVE MANIFOLDS

AbstractSuppose M is a complex projective manifold of dimension $\geq 2$ , V is the support of an ample divisor in M and U is an open set in M that intersects each irreducible component of V. We show that a pluriharmonic map $f:M\to N$ into a Kähler manifold N is holomorphic whenever $f\vert _{V\,\cap \, U}$ is holomorphic.

Special MMP for log canonical generalised pairs (with an appendix joint with Xiaowei Jiang)

We show that minimal models of mathbb {Q}-factorial NQC log canonical generalised pairs exist, assuming the existence of minimal models of smooth varieties. More generally, we prove that on a mathbb {Q}-factorial NQC log canonical generalised pair (X,B+M) we can run an MMP with scaling of an ample divisor which terminates, assuming that it admits an NQC weak Zariski decomposition or that K_X+B+M is not pseudoeffective. As a consequence, we establish several existence results for minimal models and Mori fibre spaces.

Birational automorphism groups and the movable cone theorem for Calabi–Yau complete intersections of products of projective spaces

For a Calabi-Yau manifold X, the Kawamata – Morrison movable cone conjecture connects the convex geometry of the movable cone Mov‾(X) to the birational automorphism group. Using the theory of Coxeter groups, Cantat and Oguiso proved that the conjecture is true for general varieties of Wehler type, and they described explicitly Bir(X). We generalize their argument to prove the conjecture and describe Bir(X) for general complete intersections of ample divisors in arbitrary products of projective spaces. Then, under a certain condition, we give a description of the boundary of Mov‾(X) and an application connected to the numerical dimension of divisors.

On the number of vertices of Newton–Okounkov polygons

The Newton–Okounkov body of a big divisor D on a smooth surface is a numerical invariant in the form of a convex polygon. We study the geometric significance of the shape of Newton–Okounkov polygons of ample divisors, showing that they share several important properties of Newton polygons on toric surfaces. In concrete terms, sides of the polygon are associated to some particular irreducible curves, and their lengths are determined by the intersection numbers of these curves with D . As a consequence of our description, we determine the numbers k such that D admits some k -gon as a Newton–Okounkov body, elucidating the relationship of these numbers with the Picard number of the surface, which was first hinted at by work of Küronya, Lozovanu and Maclean.

Divisor topologies of CICY 3-folds and their applications to phenomenology

In this article, we present a classification for the divisor topologies of the projective complete intersection Calabi-Yau (pCICY) 3-folds realized as hypersurfaces in the product of complex projective spaces. There are 7890 such pCICYs of which 7820 are favorable, and can be subsequently useful for phenomenological purposes. To our surprise we find that the whole pCICY database results in only 11 (so-called coordinate) divisors (D) of distinct topology and we classify those surfaces with their possible deformations inside the pCICY 3-fold, which turn out to be satisfying 1 ≤ h2,0(D) ≤ 7. We also present a classification of the so-called ample divisors for all the favorable pCICYs which can be useful for fixing all the (saxionic) Kähler moduli through a single non-perturbative term in the superpotential. We argue that this relatively unexplored pCICY dataset equipped with the necessary model building ingredients, can be used for a systematic search of physical vacua. To illustrate this for model building in the context of type IIB CY orientifold compactifications, we present moduli stabilization with some preliminary analysis of searching possible vacua in simple models, as a template to be adopted for analyzing models with a larger number of Kähler moduli.