Hajime Tsuji
We prove that for every positive integer $m\geq 18(2^{9}\cdot 3^{7})!$ and every smooth projective 3-fold of general type X defined over complex numbers, $\mid mK_{X}\mid$ gives a birational rational map from X into a projective space.
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We prove that the anti-pluricanonical map $Φ_{|-mK_{X}|}$ is birational when $m\geq 16$ for 5-fold $X$ whose anticanonical divisor is nef and big.
We explicitly find lower bounds on the volume of threefolds and fourfolds of general type in order to have nonvanishing of pluricanonical systems and birationality of pluricanonical maps. In the case of threefolds of large volume, we also give necessary and sufficient conditions for the fourth canonical map to be birational.
On the pluricanonical map of projective 3-folds of general type with $P_3 \geq 2$
该数学代数几何领域的研究证明，第三多亏格不小于2的光滑一般型射影三维簇，当m不小于14时多典范映射φ_m双有理，且该下界最优。
Let $X$ be a minimal projective 3-fold of general type. The pluricanonical section index $δ(X)$ is defined to be the minimal integer $m$ so that $P_{m}(X)\geq 2$. According to Chen-Chen, one has either $1\leq δ(X)\leq 15$ or $δ(X)=18$. This note aims to intensively study those with maximal such index. A direct corollary is that the $57$th canonical map of every minimal 3-fold of general type is stably birational.
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.
We prove the finiteness of log pluricanonical representations for projective log canonical pairs with semi-ample log canonical divisor. As a corollary, we obtain that the log canonical divisor of a projective semi log canonical pair is semi-ample if and only if so is the log canonical divisor of its normalization. We also treat many other applications.
This is a sequal paper to math.AG/9909021. By using the theory of AZD originated by the author, I prove that for every smooth projective $n$-fold $X$ of general type and every \[ m\geq \lceil\sum_{\ell =1}^{n}\sqrt[\ell]{2} \ell\rceil +1, \] $\mid mK_{X}\mid$ gives a birational rational map from $X$ into a projective space, unless it has a nontrivial (relative dimension is positive) rational fiber space structure whose general fiber is birational to a variety of relatively low degree in a projective space.
On the pluricanonical map and the canonical volume of projective 4-folds of general type
本文研究了一般型非奇异射影4-重的多重典范映射双有理性的有效界及典范体积下界，证明了当m≥76m0+77时φ_m双有理，并给出Vol(V)的下界，改进了Meng Chen的早期结果。
This note mainly studies the generic finiteness of ϕ_m of a complex projective 3-fold of general type. A new result on the classification to bicanonical pencil for Gorenstein 3-folds is attached in the last section.
Freeness of Pluricanonical Linear System on Smooth Minimal n-folds of General Type
本文证明了一般型光滑极小n维簇上$|\frac{n^2+n+2}{2}K_X|$无基点，并给出Fujita无基点猜想在nef且大除数情形下的界限，同时证明了$K_X+mB$的基点集包含于增广基轨迹。
Finiteness of projective pluricanonical representation for automorphisms of complex manifolds
本文研究紧复流形双亚纯自同构群在多重典范映射像上的射影多重典范表示，在 Kodaira 维数等于维数减一的条件下证明其像为有限群，并建立相对维数 1 的典范丛公式，从而验证 Prokhorov–Shokurov 猜想的解析版本。
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 $m\geq ν_{n}$.
This paper aims to improve a theorem of Janos Kollar. For a given Complex projective threefold X of general type, suppose the plurigenus p_k(X)\ge 2, Kollar proved that the (11k+5)-canonical map is birational. Here we show that either the (7k+3)-canonical map or the (7k+5)-canonical map is birational and that the m-canonical map is stably birational for m\ge 13k+6. If P_k(X)\ge 3, then the m-canonical map is stably birational for m\ge 10k+8. In particular, the 12-canonical map is birational when p_g(X)\ge 2 and the 11-canonical map is birational when p_g(X)\ge 3.
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.
We extend Kollar's technique to look for an explicit function h(n) with phi_m birational onto its image for all integers $m\geq h(n)$ and for all n-dimensional nonsingular projective varieties of general type.
In this paper we study pluricanonical maps of minimal projective 3-folds of general type with geometric genus $1$, $2$ and $3$. We go in the direction pioneered by Enriques and Bombieri, and other authors, pinning down, for low projective genus, a finite list of exceptions to the birationality of some pluricanonical map. In particular, apart from a finite list of weighted baskets, we prove the birationality of $φ_{16}$, $φ_{6}$ and $φ_{5}$ respectively.
On generic finiteness of pluricanonical maps of threefolds of general type
本文证明了对满足P2(X)≥2的一般型极小三维流形X，|6K_X|定义通用有限映射，且该结果最优；同时证明对所有一般型极小三维流形X，当n≥38时|nK_X|定义通用有限映射。关键技术是曲面的新通用有限性准则和特殊解消下的有效比较不等式。
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.
For every smooth complex projective variety $W$ of dimension $d$ and nonnegative Kodaira dimension, we show the existence of a universal constant $m$ depending only on $d$ and two natural invariants of the very general fibres of an Iitaka fibration of $W$ such that the pluricanonical system $|mK_W|$ defines an Iitaka fibration. This is a consequence of a more general result on polarized adjoint divisors. In order to prove these results we develop a generalized theory of pairs, singularities, log canonical thresholds, adjunction, etc.
Nonsingular projective 3-folds $V$ of general type can be naturally classified into 18 families according to the {\it pluricanonical section index} $δ(V):=\text{min}\{m|P_m\geq 2\}$ since $1\leq δ(V)\leq 18$ due to our previous series (I, II). Based on our further classification to 3-folds with $δ(V)\geq 13$ and an intensive geometrical investigation to those with $δ(V)\leq 12$, we prove that $\text{Vol}(V) \geq \frac{1}{1680}$ and that the pluricanonical map $Φ_{m}$ is birational for all $m \geq 61$, which greatly improves known results. An optimal birationality of $Φ_m$ for the case $δ(V)=2$…
We prove an effective vanishing theorem for direct images of log pluricanonical bundles of projective semi-log canonical pairs. As an application, we obtain a semipositivity theorem for direct images of relative log pluricanonical bundles of projective semi-log canonical pairs over curves, which implies the projectivity of the moduli spaces of stable varieties. It is worth mentioning that we do not use the theory of variation of (mixed) Hodge structure.
We show that $|mK_X|$ defines a birational map and has no fixed part for some bounded positive integer $m$ for any $\frac{1}{2}$-lc surface $X$ such that $K_X$ is big and nef. For every positive integer $n\geq 3$, we construct a sequence of projective surfaces $X_{n,i}$, such that $K_{X_{n,i}}$ is ample, ${\rm{mld}}(X_{n,i})>\frac{1}{n}$ for every $i$, $\lim_{i\rightarrow+\infty}{\rm{mld}}(X_{n,i})=\frac{1}{n}$, and for any positive integer $m$, there exists $i$ such that $|mK_{X_{n,i}}|$ has non-zero fixed part. These results answer the surface case of a question of Xu.
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_…
In this article we prove the following boundedness result: Fix a DCC set $I\subset [0, 1]$. Let $\mathfrak{D}$ be the set of all log pairs $(X, Δ)$ satisfying the following properties: (i) $X$ is a projective surface defined over an algebraically closed field, (ii) $(X, Δ)$ is log canonical and the coefficients of $Δ$ are in $I$, and (iii) $K_X+Δ$ is big. Then there is a positive integer $N=N(I)$ depending only on the set $I$ such that the linear system $|\lceil m(K_X+Δ)\rceil|$ defines a birational map onto its image for all $m\geq N$ and $(X, Δ)\in\mathfrak{D}$
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)''.
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。
Given a fibration $f$ between two projective manifolds $X$ and $Y$, we discuss the effective generation of the higher direct images $R^{i}f_{\ast}(K^{m}_{X})$, where $K^{m}_{X}$ is the $m$-th tensor power of the canonical bundle of $X$. In particular, we answer two questions posed by Popa--Schnell in [PS14].
We show the existence of an anti-pluricanonical curve on every smooth projective rational surface X which has an infinite group G of automorphisms of either null entropy or of type Z . Z (semi-direct product), provided that the pair (X, G) is minimal. This was conjectured by Curtis T. McMullen (2005) and further traced back to Marat Gizatullin and Brian Harbourne (1987). We also prove (perhaps) the strongest form of the famous Tits alternative theorem.
Given a fibration $f$ between two projective manifolds $X$ and $Y$, we provide a sufficient condition such that the direct images $f_{\ast}(K_{X/Y}\otimes L\otimes\mathscr{I}(f,\|L\|))$ is nef, where $L$ is a holomorphic line bundle with non-negative relative Iitaka dimension and $\mathscr{I}(f,\|L\|)$ is the relative asymptotic multiplier ideal sheaf.
In this paper, I prove a very general extension theorem for log pluricanonical systems. The main application of this extension theorem is (together with Kawamata's subadjunction theorem) to give an optimal subadjunction theorem which relates the positivities of canonical bundle of the ambient projective manifold and that of the (maximal) center of log canonical singularities. This is an extension of the corresponding result in my previous work where I dealt with log pluricanonical systems of general type. This subadjunction theorem indicates an approach to solve the abundance conjecture for ca…
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.
Let $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12}(V):=\text{dim} H^0(V, 12K_V)>0$ and $P_{m_0}(V)>1$ for some positive integer $m_0\leq 24$. A direct consequence is the birationality of the pluricanonical map $φ_m$ for all $m\geq 126$. Besides, the canonical volume $\text{Vol}(V)$ has a universal lower bound $ν(3)\geq \frac{1}{63\cdot 126^2}$.
In this paper, we study pushforwards of log pluricanonical bundles on projective log canonical pairs $(Y,Δ)$ over the complex numbers, partially answering a Fujita-type conjecture due to Popa and Schnell in the log canonical setting. We show two effective global generation results. First, when $Y$ surjects onto a projective variety, we show a quadratic bound for generic generation for twists by big and nef line bundles. Second, when $Y$ is fibered over a smooth projective variety, we show a linear bound for twists by ample line bundles. These results additionally give effective non-vanishing s…
Let $V$ be a nonsingular projective 3-fold of general type. When the pluricanonical section index $δ(V)>12$, Chen-Chen \cite{Chen3} has a complete list of the possibility for the weighted basket ${\mathbb B}(V)$. However the possibility of ${\mathbb B}(V)$ could be infinite in the situation $δ(V)\leq 12$, which is the main challenge to the classification. In this paper we mainly study the case with $δ(V)=12$ and show that $Vol(V)\geq \frac{31}{48048}$, which improves the corresponding result of Chen--Chen \cite[Prop. 4.9(5)]{Chen3}.
We prove that for any smooth complex projective threefold of Kodaira dimension one, the $m$-th pluricanonical map is birational to the Iitaka fibration for every $m\geq5868$ and divisible by $12$.
In this paper, we show an extension type theorem for twisted pluricanonical sections on a family of smooth projective manifolds (the twisting line bundle being pseudo-effective and having a prescribed multiplier ideal on the central fiber).
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.
On bicanonical maps of threefolds of general type with large volumes
研究一般型三维流形大体积的双CANONICAL映射的维度以及大canonical体积的pluricanonical映射的纤维结构
We study the cohomology of reflexive rank 2 sheaves on smooth projective threefolds. Applications are given to the moduli space of reflexive sheaves.
Let X be a projective complex 3-fold, quasihomogeneous with respect to an action of a linear algebraic group. We show that X is a compactification of SL_2/G, G a discrete subgroup, or that X can be equivariantly transformed into the 3-dim. projective space, the quadric Q_3, or into certain quasihomogeneous bundles with very simple structure.
We study pluricanonical systems on smooth projective varieties of positive Kodaira dimension, following the approach of Hacon-McKernan, Takayama and Tsuji succesfully used in the case of varieties of general type. We prove a uniformity result for the Iitaka fibration of smooth projective varieties of positive Kodaira dimension, provided that the base of the Iitaka fibration is not uniruled, the variation of the fibration is maximal, and the generic fiber has a good minimal model.
In the present work we classify the relatively minimal 3-dimensional quasihomogeneous complex projective varieties under the assumption that the automorphism group is not solvable. By relatively minimal we understand varieties X having at most Q-factorial terminal singularities and allowing an extremal contraction X to Y, where dim Y <3.
In this paper, we will construct new examples of derived equivalent Calabi--Yau 3-folds with Picard number greater than one. We also study their mirror Calabi--Yau manifolds and find that they are given by Schoen's fiber products of suitable rational elliptic modular surfaces.
In two recent papers, we described some Siegel modular threefolds which admit a weak Calabi--Yau model. Not all of them admit a {\it projective} model. The purpose of this paper is to exhibit criterions for the projectivity, to treat several examples, and to compute their Hodge numbers.
On The Log Sarkisov Program For Foliations On Projective 3-Folds
本文证明了具有适当奇点的投影三重曲面上协秩为一的叶状结构的Sarkisov程序，并展示了两个具有秩一叶状结构的叶状Mori纤维空间之间的弱化理性刚性。
Let $k$ be an algebraically closed field of characteristic $p > 0$. Let $X$ be an irreducible smooth projective curve of genus $g$ over $k$. Fix an integer $n \geq 2$, and let $S^n(X)$ be the $n$-fold symmetric product of $X$. In this article we find the $S$-fundamental group scheme and Nori's fundamental group scheme of $S^n(X)$.
Let $p$ and $q$ be odd prime numbers. In this paper we study non-abelian pq-fold regular covers of the projective line, determine algebraic models for some special cases and provide a general isogeny decomposition of the corresponding Jacobian varieties. We also give a classification and description of the one-dimensional families of compact Riemann surfaces as before.
Let $X$ be a smooth projective variety of dimension $5$ and $L$ be an ample line bundle on $X$ such that $L^5>7^5$ and $L^d\cdot Z\geq 7^d$ for any subvariety $Z$ of dimension $1\leq d\leq 4$. We show that $\mathcal{O}_X(K_X+L)$ is globally generated.
There are many examples of 3-folds of general type with $χ(\mathcal {O})=1$ found by Iano-Fletcher and Reid about twenty years ago. Iano-Fletcher has ever proved $P_{12}(X)\ge 1$ and $P_{24}(X)\ge 2$ for all minimal 3-folds $X$ of general type with $χ(\mathcal {O}_X)=1$. In this paper, we improve on Fletcher's method. Our main result is that $ϕ_m$ is birational onto its image for all $m\ge 54$. To prove this we establish a general theory on birationality for general 3-folds which improve known results. Also we will show $P_m\geq 1$ for all $m\geq 14$ and $P_{18}\geq 3$.
We compute the expected dimension of the moduli space of torsion-free rank 2 sheaves at a point corresponding to a stable reflexive sheaf and give conditions for the existence of a perfect tangent-obstruction complex on a class of smooth projective threefolds; this class includes Fano and Calabi-Yau threefolds. We also explore both local and global relationships between moduli spaces of reflexive rank 2 sheaves and the Hilbert scheme of curves.
Conifold Transitions for Complete Intersection Calabi-Yau 3-folds in Products of Projective Spaces
本文证明了射影空间乘积上的通用完全相交 Calabi-Yau 3-流形可经历 conifold 转变成为 S^{3} × S^{3} 的连通和，从而在其上构造出具有平凡典范丛的复结构，推广了前人使用五次多项式的方法。
We construct new examples of Kobayashi hyperbolic hypersurfaces in the projective 4-space. They are generic projections of the triple symmetric product of a generic curve of genus at least 7, smoothly embedded in the projective 7-space.
Ample line bundles on certain toric fibered 3-folds
本文研究具有到射影直线的满射环面等变态射的射影非奇异环面3维流形，证明了其上的丰富线丛总是正规生成。
We supply a detailed proof of the result by P.S. Green and T. H$\ddot{\text{u}}$bsch that all complete intersection Calabi--Yau 3-folds in product of projective spaces are connected through projective conifold transitions (known as the standard web). We also introduce a subclass of small transitions which we call primitive small transitions and study such subclass. More precisely, given a small projective resolution $π: \widehat{X} \rightarrow X$ of a Calabi--Yau 3-fold $X$, we show that if the natural closed immersion $Def(\widehat{X}) \hookrightarrow Def(X)$ is an isomorphism then $X$ has on…
On the genus of a curve in a projective $3$-fold
本文在 Liu 的结果基础上，改进了 Castelnuovo 对曲线算术亏约的界，并在 Calabi-Yau 情况下推进了关于 Gopakumar-Vafa 不变量消失的猜想。
We address the following question: what are the cardinalities of maximal finite families of pairwise incident planes in a complex projective space? One proves easily that the span of the planes has dimension 5 or 6. Up to projectivities there is one such family spanning a 6-dimensional projective space - this is an elementary result. Maximal finite families of pairwise incident planes in a 5-dimensional projective space are considerably more misterious: they are linked to certain special (EPW) sextic hypersurfaces which have a non-trivial double cover, generically a hyperkaehler 4-fold. We pro…
Let $X$ be a smooth projective $n$-fold such that $q(X)=0$ and $L$ a globally generated, big line bundle on $X$ such that $h^0(K_X+(n-2)L) >0$. We give necessary and sufficient conditions for the adjoint systems $|K_X+kL|$ to be birational for $k \geq n-1$. In particular, for Calabi-Yau $n$-folds we generalize and prove parts of a conjecture of Gallego and Purnaprajna.
We will prove the following results for $3$-fold pairs $(X,B)$ over an algebraically closed field $k$ of characteristic $p>5$: log flips exist for $\Q$-factorial dlt pairs $(X,B)$; log minimal models exist for projective klt pairs $(X,B)$ with pseudo-effective $K_X+B$; the log canonical ring $R(K_X+B)$ is finitely generated for projective klt pairs $(X,B)$ when $K_X+B$ is a big $\Q$-divisor; semi-ampleness holds for a nef and big $\Q$-divisor $D$ if $D-(K_X+B)$ is nef and big and $(X,B)$ is projective klt; $\Q$-factorial dlt models exist for lc pairs $(X,B)$; terminal models exist for klt pair…
For any $n\geq 3$, we explicitly construct smooth projective toric $n$-folds of Picard number $\geq 5$, where any nontrivial nef line bundles are big.
A lifting principle for canonical stability indices of varieties of general type
本文证明了一般型光滑射影簇典范稳定性指标序列的提升原理，将 $r_n$ 与高维、大典范体积簇的指标联系起来，并给出多典范映射双有理性的等价体积判据。
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