Boundedness of pluricanonical maps of varieties of general type

Pluricanonical maps of varieties of Albanese fiber dimension two

In this paper we prove that for any smooth projective variety of Albanese fiber dimension two and of general type, the 6-canonical map is birational. And we also show that the 5-canonical map is birational for any such variety with some geometric restrictions.

Cohomological support loci and Pluricanonical systems on irregular varieties

For an irregular variety $X$ of general type, we show that if a general fiber $F$ of the Albanese morphism of $X$ satisfies certain Hodge theoretic condition, the $0$-th cohomological support loci of $K_X$ generates the Picard variety of $X$ . We then show that the condition that the $0$-th cohomological support loci of $K_X$ generates the Picard variety of $X$ can often be applied to prove the birationality of certain pluricanonical maps of $X$.

Pushforwards of pluricanonical bundles under morphisms to abelian varieties

Let $f \colon X \to A$ be a morphism from a smooth projective variety to an abelian variety (over the field of complex numbers). We show that the sheaves $f_* ω_X^{\otimes m}$ become globally generated after pullback by an isogeny. We use this to deduce a decomposition theorem for these sheaves when $m \ge 2$, analogous to that obtained by Chen-Jiang when $m = 1$. This is in turn applied to effective results for pluricanonical linear series on irregular varieties with canonical singularities.

Pluricanonical systems of projective varieties of general type

We prove that there exists a positive integer $ν_{n}$ depending only on $n$ such that for every smooth projective $n$-fold of general type $X$ defined over {\bf C}, $\mid mK_{X}\mid$ gives a birational rational map from $X$ into a projective space for every $m\geq ν_{n}$. This theorem gives an affirmative answer to Severi's conjecture. The key ingredients of the proof are the theory of AZD which was originated by the aurhor and the subadjunction formula for AZD's of logcanoncial divisors.

Anti-pluricanonical systems on Fano varieties

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 $ε$-lc for some $ε>0$, then we show that we can choose $m$ depending only on $d$ and $ε$ 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.

关于一般型簇的多重典范有界性

On pluricanonical boundedness of varieties of general type

本文给出 Chen–Jiang 定理的新证明，修正原证明漏洞，并利用该方法改进 Lacini 的有界性定理，证明一般型 n 维簇的 r-典范映射具有双有理有界纤维，从而肯定回答 Chen 和 Jiang 2014 年提出的问题。

On direct images of twisted pluricanonical sheaves on normal varieties

We study the depth properties of certain direct image sheaves on normal varieties. Let $f: Y\rightarrow X$ be a proper morphism of relative dimension $d$ from a smooth variety onto a normal variety such that the preimage $E$ of the singular locus of $X$ is a divisor. We show that for any integer $m>0$, the higher direct image $R^df_*ω^{\otimes m}_Y(aE)$ modulo the torsion subsheaf is $S_2$, provided that $a$ is sufficiently large. In case $f$ is birational, we give criteria on $a$ for the direct image $f_*ω_Y(aE)$ to coincide with $ω_X$. We also introduce an index measuring the singularities o…

Pluricanonical maps of varieties of maximal Albanese dimension

Let $X$ be a smooth complex projective algebraic variety of maximal Albanese dimension. We give a characterization of $κ(X)$ in terms of the set $V^0(X,ω_{X})$ $:=\{P\in {\text{\rm Pic}}^0(X)|h^0(X, ω_X \otimes P) \ne 0\}$. An immediate consequence of this is that the Kodaira dimension $κ(X)$ is invariant under smooth deformations. We then study the pluricanonical maps $ϕ_m:X -> \Bbb{P} (H^0(X,mK_X))$. We prove that if $X$ is of general type, $ϕ_m$ is generically finite for $m\geq 5$ and birational for $m\geq 5 \text{\rm dim} (X) +1$. More generally, we show that for $m\geq 6$ the image of $ϕ_…

Syzygies, Pluricanonical Maps, and the Birational Geometry of Varieties of Maximal Albanese Dimension

In this thesis we looked into three different problems which share, as a common factor, the exstensive use of the Fourier--Mukai transform as research tool. In the first Part we investigated the syzygies of Kummer varieties (i.e. quotients of abelian varieties by the $\mathbb{Z}/2\Z$-action induced by the group operation), extending to higher syzygies results on projective normality and degree of equations of Sasaki, Kempf, and Khaled. The second Part of this Thesis (partially written in collaboration with Z.Jiang and M. Lahoz) is dedicated to the study of pluricanonical linear systems on vari…

On generic vanishing for pluricanonical bundles

We study cohomology support loci and higher direct images of (log) pluricanonical bundles of smooth projective varieties or log canonical pairs. We prove that the 0-th cohomology support loci of log pluricanonical bundles are finite unions of torsion translates of subtori. We also construct an example of morphism from a smooth projective variety to an abelian variety such that a higher direct image of a pluricanonical bundle to the abelian variety is not a GV-sheaf.

Pluricanonical systems of projective varieties of general type II

This is a revised version of the second half of my paper math.AG/9909021. We prove that there exists a positive integer $ν_{n}$ depending only on $n$ such that for every smooth projective $n$-fold of general type $X$ defined over complex numbers, $\mid mK_{X}\mid$ gives a birational rational map from $X$ into a projective space for every $m\geq ν_{n}$.

Pluricanonical Maps and the Fujita Conjecture

We describe examples showing the sharpness of Fujita's conjecture on adjoint bundles also in the general type case, and use these examples to formulate related bold conjectures on pluricanonical maps of varieties of general type.

Frobenius stable pluricanonical systems on threefolds of general type in positive characteristic

This paper aims to investigate effectivity problems of pluricanonical systems on varieties of general type in positive characteristic. In practice, we will consider a sub-linear system $|S^0_{-}(X, K_X + nK_X)| \subseteq |H^0(X, K_X +nK_X)|$ generated by certain Frobenius stable sections, and prove that for a minimal terminal threefold $X$ of general type with either $q(X)>0$ or Gorenstein singularities, if $n\geq 28$ then $|S^0_{-}(X, K_X + nK_X)| \neq \emptyset$; if $n\geq 42$ then the linear system $|S^0_{-}(X, K_X + nK_X)|$ defines a birational map.

关于对数多典范丛的正像层与非消失猜想的评述

Remarks on the Direct Image Sheaf of Logarithmic Pluricanonical Bundles and the Non-Vanishing Conjecture

本文利用 Chen-Jiang 分解证明不规则簇上 lc 对的非消失猜想可由低维情形推出，并将 Catanese-Fujita-Kawamata 分解推广至 klt 情形以建立截面存在性。

双曲几何与积同源于积的变种及阿贝尔覆盖

Pluricanonical Geometry of Varieties Isogenous to a Product and Abelian Covers

研究同源于积的三次曲面的典范映射，证明当$p_g \ge 5$时4-典范映射为理性映射，并构造最大典范度的例子。

Curvature semipositivity of relative pluricanonical systems

We prove semipositivity of the curvature of the Narashimhan-Simha metric on an arbitrary flat projective family of varieties with only canonical singularities. This also implies that the direct image of pluricanonical systems have a natural continuous hermitian strucutre with semipositive curvature curvature current in the sense of Nakano. This is an improvement of the author's preprint ``Refined semipositivity and moduli of canonical models (2005)''.

Cohomological support loci of varieties of Albanese fiber dimension one

Let $X$ be a smooth projective variety of Albanese fiber dimension 1 and of general type. We prove that the translates through 0 of all components of $V^0(ω_X)$ generate $\Pic^0(X)$. We then study the pluricanonical maps of $X$. We show that $|4K_X|$ induces a birational map.

Pluricanonical maps of stable log surfaces

Stable surfaces and their log analogues are the type of varieties naturally occuring as boundary points in moduli spaces. We extend classical results of Kodaira and Bombieri to this more general setting: if $(X,Δ)$ is a stable log surface with reduced boundary (possibly empty) and $I$ is its global index, then $4I(K_X+Δ)$ is base-point-free and $8I(K_X+Δ)$ is very ample. These bounds can be improved under further assumptions on the singularities or invariants, for example, $5(K_X+Δ)$ is very ample if $(X,Δ)$ has semi-canonical singularities.

Log-plurigenera in stable families

We study the flatness of log-pluricanonical sheaves on stable families of varieties. v2: many changes in presentation, results unchanged.

On Kawamata's theorem

We give an alternate proof of the main theorem of Kawamata's paper: Pluricanonical systems on minimal algebraic varieties. Our proof also works for varieties in class $\mathcal C$. We note that our proof is completely different from Kawamata's.

Boundedness of Q-Fano varieties of Picard number one

We prove the birational boundedness of Q-Fano varieties with Picard number one and dimension $n$ for every $n$. We do not use Mori theory.

Boundedness of varieties of log general type

We survey recent results on the boundedness of the moduli functor of stable pairs

On The Projective Normality And Normal Presentation On Higher Dimensional Varieties With Nef Canonical Bundle

In this article we prove new results on projective normality and normal presentation of adjunction bundle associated to an ample and globally generated line bundle on higher dimensional smooth projective varieties with nef canonical bundle. As one of the consequences of the main theorem, we give bounds on very ampleness and projective normality of pluricanonical linear systems on varieties of general type in dimensions three, four and five. These improve known such results.

Boundedness of the base varieties of certain fibrations

It is conjectured that the base varieties of the Iitaka fibrations are bounded when the Iitaka volumes are bounded above. We confirm this conjecture for Iitaka $ε$-lc Fano type fibrations.

Boundedness of moduli of varieties of general type

We show that the family of semi log canonical pairs with ample log canonical class and with fixed volume is bounded.

极小稳定模型模栈上的几何沙法列维奇有界猜想

Geometric Shafarevich boundedness conjecture for families of polarized varieties

本文研究极小稳定模型模栈上的几何沙法列维奇有界猜想，主要贡献是证明了该猜想的成立。

On the Boundedness of Globally $F$-split varieties

This paper proposes the use of $F$-split and globally $F$-regular conditions in the pursuit of BAB type results in positive characteristic. The main technical work comes in the form of a detailed study of threefold Mori fibre spaces over positive dimensional bases. As a consequence we prove the main theorem, which reduces birational boundedness for a large class of varieties to the study of prime Fano varieties.

On birational boundedness of Fano fibrations

We investigate birational boundedness of Fano varieties and Fano fibrations. We establish an inductive step towards birational boundedness of Fano fibrations via conjectures related to boundedness of Fano varieties and Fano fibrations. As corollaries, we provide approaches towards birational boundedness and boundedness of anti-canonical volumes of varieties of $ε$-Fano type. Furthermore, we show birational boundedness of $3$-folds of $ε$-Fano type.

Minimal models and boundedness of stable varieties

We consider a class of stable smoothable n-dimensional varieties, the analogs of stable curves. Assuming the minimal model program in dimension n+1, we prove that this class is bounded. From Kollar's method of constructing projective moduli spaces we get as a corollary that minimal model program in dimension n+1 implies the existence of a projective coarse moduli space for stable smoothable n-folds.

非典范极化射影簇族的有界性结果

Boundedness results for families of non-canonically polarized projective varieties

本文证明在光滑拟射影曲线上，具有固定希尔伯特多项式且半丰富典范丛的非等平凡、光滑射影极化簇族是有界的，将Arakelov、Parshin和Kovács–Lieblich的有界性结果推广到非典范极化情形。

Logarithmic Kodaira-Akizuki-Nakano vanishing and Arakelov-Parshin boundedness for singular varieties

The article has two parts. The first part is devoted to proving a singular version of the logarithmic Kodaira-Akizuki-Nakano vanishing theorem of Esnault and Viehweg. This is then used to prove other vanishing theorems. In the second part these vanishing theorems are used to prove an Arakelov-Parshin type boundedness result for families of canonically polarized varieties with rational Gorenstein singularities.

Boundedness results for singular Fano varieties and applications to Cremona groups

This survey paper reports on work of Birkar, who confirmed a long-standing conjecture of Alexeev and Borisov-Borisov: Fano varieties with mild singularities form a bounded family once their dimension is fixed. Following Prokhorov-Shramov, we explain how this boundedness result implies that birational automorphism groups of projective spaces satisfy the Jordan property, answering a question of Serre in the positive.

Strict Arakelov inequality for a family of varieties of general type

Let $f:\, X\to Y$ be a semistable non-isotrivial family of $n$-folds over a smooth projective curve with discriminant locus $S \subseteq Y$ and with general fibre $F$ of general type. We show the strict Arakelov inequality \[\frac{\mathrm{deg}\, f_*ω_{X/Y}^ν}{\mathrm{rank}\, f_*ω_{X/Y}^ν} < {nν\over 2}\cdot\mathrm{deg}\,Ω^1_Y(\log S),\] for all $ν\in \mathbb N$ such that the $ν$-th pluricanonical linear system $|ω^ν_F|$ is birational. This answers a question asked by Möller, Viehweg and the third named author.

Boundedness of varieties of Fano type with alpha-invariants and volumes bounded below

We show that fixed dimensional klt weak Fano pairs with alpha-invariants and volumes bounded away from $0$ and the coefficients of the boundaries belong to the set of hyperstandard multiplicities $Φ(\mathscr{R})$ associated to a fixed finite set $\mathscr{R}$ form a bounded family. We also show $α(X,B)^{d-1}\mathrm{vol}(-(K_X+B))$ are bounded from above for all klt weak Fano pairs $(X,B)$ of a fixed dimension $d$.

Arakelov Type Inequalities and Deformation Boundedness of polarized varieties

We give two kinds of generalizations of Arakelov type inequalities for higher dimensional families. These results give higher dimensional generalizations (in both fibers and bases) of the weakly boundedness in Paršin-Arakelov's reformulation of the geometric Shafarevich conjecture. As a consequence, we deduce the following results. Hyperbolicity: We give an alternative proof (using the theory of degeneration of Hodge structure) to the hyperbolicity (in Paršin-Arakelov's reformulation, i.e. Viehweg's hyperbolicity conjecture) of the base of a family with maximal variation whose general fibers a…

Boundedness properties of automorphism groups of forms of flag varieties

We call a flag variety admissible if its automorphism group is the projective general linear group. (This holds in most cases.) Let $K$ be a field of characteristic $0$, containing all roots of unity. Let the $K$-variety $X$ be a form of an admissible flag variety. We prove that $X$ is either ruled, or the automorphism group of $X$ is bounded, meaning that, there exists a constant $C\in\mathbb{N}$ such that if $G$ is a finite subgroup of $Aut_K(X)$, then the cardinality of $G$ is smaller than $C$.

Birational boundedness of low dimensional elliptic Calabi-Yau varieties with a section

We prove that there are finitely many families, up to isomorphism in codimension one, of elliptic Calabi-Yau manifolds $Y\rightarrow X$ with a rational section, provided that $\dim(Y)\leq 5$ and $Y$ is not of product-type. As a consequence, we obtain that there are finitely many possibilities for the Hodge diamond of such manifolds. The result follows from log birational boundedness of klt pairs $(X, Δ)$ with $K_X+Δ$ numerically trivial and not of product-type, in dimension at most $4$.

Complements on log canonical Fano varieties

In this paper, we generalise the theory of complements to log canonical log fano varieties and prove boundedness of complements for them in dimension less than or equal to 3. We also prove some boundedness results for the canonical index of sdlt log Calabi-Yau varieties in dimension 2.

Unboundedness of fiber invariants of canonically fibred varieties of general type

We answer an open question concerning the boundedness of canonical fiber spaces in high dimensions and prove the following: for any set of integers $n\geq 3$, $0<d<n$ and $N>0$, there exists a nonsingular projective $n$-fold $X$ of general type so that $X$ is canonically fibred by $d$-dimensional varieties $F$ with $p_g(F)\geq N$. This disproves the desired boundedness parallel to Beauville's boundedness theorem in the surface case.

Singularities of linear systems and boundedness of Fano varieties

We study log canonical thresholds (also called global log canonical threshold or $α$-invariant) of $\mathbb{R}$-linear systems. We prove existence of positive lower bounds in different settings, in particular, proving a conjecture of Ambro. We then show that the Borisov-Alexeev-Borisov conjecture holds, that is, given a natural number $d$ and a positive real number $ε$, the set of Fano varieties of dimension $d$ with $ε$-log canonical singularities forms a bounded family. This implies that birational automorphism groups of rationally connected varieties are Jordan which in particular answers a…

具有有理截面的椭圆卡拉比-丘簇的有界性

Boundedness of elliptic Calabi-Yau varieties with a rational section

本文证明了在典范类挠指数有界且椭圆纤维化具有有理截面的条件下，固定维数的 klt 椭圆卡拉比-丘簇在余维一内有界，并将三维情形推广到有理连通的 ε-lc 对。

具有大量微分形式的代数簇的丰度

Abundance for varieties with many differential forms

假设低维极小模型纲领成立，证明具有温和奇点且存在大量带 pluricanonical 丛系数自反微分形式的代数簇$X$满足丰度猜想，推出埃尔米特半正典范除子几乎半丰富等结论，且$K_X$数值维数为1时结果无条件成立，还处理 Calabi-Yau 簇上 nef 线丛半丰富性问题。

The Kodaira dimension of the moduli space of Prym varieties

We study the enumerative geometry of the moduli space R_g of Prym varieties of dimension g-1 (also known as the space of admissible double covers). Our main result is that the compactification of R_g is of general type as soon as g>13. We achieve this by computing the class of two types of cycles on R_g: one defined in terms of Koszul cohomology of Prym curves, the other defined in terms of Raynaud theta divisors associated to certain vector bundles on curves. We formulate a Prym-Green conjecture on syzygies of Prym-canonical curves. In the appendix we show that even though R_g has non-canonic…

On the bicanonical map of irregular varieties

From the point of view of uniform bounds for the birationality of pluricanonical maps, irregular varieties of general type and maximal Albanese dimension behave similarly to curves. In fact Chen-Hacon showed that, at least when their holomorphic Euler characteristic is positive, the tricanonical map of such varieties is always birational. In this paper we study the bicanonical map. We consider the natural subclass of varieties of maximal Albanese dimension formed by primitive varieties of Albanese general type. We prove that the only such varieties with non-birational bicanonical map are the n…

Invariance of Plurigenera and boundedness for Generalized Pairs

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 spherical Fano varieties

We prove that for any e>0, there exists only finitely many e-log terminal spherical Fano varieties of fixed dimension. We also introduce an invariant of a spherical subgroup H in a reductive group G which measures how nice an equivariant Fano compactification G/H there exists.

Boundedness of log terminal Fano pairs of bounded index

We prove a conjecture of Batryev which states that the family of all Fano varieties with kawamata log terminal singularities and fixed index, forms a bounded family.

关于特征 $p>0$ 下 Fano 簇的典范丛公式与有效双有理性

On the canonical bundle formula and effective birationality for Fano varieties in char $p>0$

本文研究了正特征下 Fano 型簇的双有理几何与有界性问题，证明了有界性在正规化下不变，给出了更易用的典范丛公式，并证实了具有良好奇点的弱 Fano 簇的有效双有理性。

广义对数典范曲面的有界性失效

Failure of Boundedness for Generalised Log Canonical Surfaces

本文通过构造反例证明了具有固定适当不变量且Kodaira维数任意的曲面广义对数典范模型不满足有界性，从而回答了Birkar与第一作者提出的问题。

Boundedness of $\mathbb{Q}$-Fano varieties with degrees and alpha-invariants bounded from below

We show that $\mathbb{Q}$-Fano varieties of fixed dimension with anti-canonical degrees and alpha-invariants bounded from below form a bounded family. As a corollary, K-semistable $\mathbb{Q}$-Fano varieties of fixed dimension with anti-canonical degrees bounded from below form a bounded family.

Boundedness of families of canonically polarized manifolds: A higher dimensional analogue of Shafarevich's conjecture

We show that the number of deformation types of canonically polarized manifolds over an arbitrary variety with proper singular locus is finite, and that this number is uniformly bounded in any finite type family of base varieties. As a corollary we show that a direct generalization of the geometric version of Shafarevich's original conjecture holds for infinitesimally rigid families of canonically polarized varieties.

On higher dimensional extremal varieties of general type

Relations among fundamental invariants play an important role in algebraic geometry. It is known that an $n$-dimensional variety of general type with nef canonical divisor and canonical singularities, whose image $Y$ under the canonical map is of maximal dimension, satisfies $K_X^n \ge 2 (p_g-n)$. We investigate the very interesting extremal situation $K_X^n=2(p_g-n)$, which appears in a number of geometric situations. Since these extremal varieties are natural higher dimensional analogues of Horikawa surfaces, we name them Horikawa varieties. These varieties have been previously dealt with in…

A remark on uniform boundedness for Brauer groups

The Tate conjecture for divisors on varieties over number fields is equivalent to finiteness of $\ell$-primary torsion in the Brauer group. We show that this finiteness is actually uniform in one-dimensional families for varieties that satisfy the Tate conjecture for divisors -- e.g. abelian varieties and $K3$ surfaces.

Boundedness results for periodic points on algebraic varieties

We prove that if f is a self-map of an algebraic variety over a field K, then under certain conditions on X, f and K the set of possible periods of K-valued periodic points of f is finite.