Zhi Jiang
This paper is devoted to study the birational properties of the Albanese map. I generalize a theorem of Kawamata to tell when the Albanese map is surjective and when it is an algebraic fiber space.
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A footnote to a theorem of Kawamata
本文补充Kawamata关于拟Albanese映射双有理性的定理，证明在相同对数非正则度与对数Kodaira维数条件下，该映射在余维1处为真，验证Iitaka猜想。
We give an effective upper bound of |Bir(X)| for the birational automorphism group of an irregular n-fold (with n = 3) of general type in terms of the volume V = V(X) under an ''albanese smoothness and simplicity'' condition. To be precise, |Bir(X)| < d_3 V^{10}. An optimum linear bound |Bir(X)|-1 < (1/3)(42)^3 V is obtained for those 3-folds with non-maximal albanese dimension. For all n > 2, a bound |Bir(X)| < d_n V^{10} is obtained when alb_X is generically finite, alb(X) is smooth and Alb(X) is simple.
We show that any fibration of a 'special' compact K{ä}hler manifold X onto an Abelian variety has no multiple fibre in codimension one. This statement strengthens and extends previous results of Kawamata and Viehweg when $κ$(X) = 0. This also corrects the proof given in [2], 5.3 which was incomplete.
Let $(X,Δ)$ be a pair. We study how the condition $κ(K_X + Δ)=0$ causes surjectivity or birationality of the Albanese map and the Albanese morphism of $X$ in both characteristic $0$ and characteristic $p > 0$. In particular in characteristic $0$ we generalize Kawamata's result to the cases of log canonial pairs, and in characteristic $p>0$ we generalize a result of Hacon-Patakfalvi to the cases of log pairs. Moreover we show that if $X$ is a normal projective threefold in characteristic $p>0$, the coefficients of the components of $Δ$ are $\le 1$ and $-(K_X+Δ)$ is semiample, then the Albanese …
In this paper, we collect basic properties of the Albanese dimension and explain how to generalize the main theorem of [F2](math.AG/0204262). This paper is a supplement and a generalization of [F2]. We also prove an inequality of irregularities for algebraic fiber spaces in the appendix, which is an exposition of Fujita-Kawamata's semi-positivity theorem.
We give a criterion for a divisorial sheaf on a log terminal variety to be Cohen-Macaulay. The log canonical case and applications to moduli are also considered.
Using the techniques of J.-P. Demailly and Th. Peternell in math.AG/0208021 a weak version of Kawamata-Viehweg vanishing is proven for pseudo-effective line bundles on compact Kaehler manifolds. It is a weak version, since one has to adjust the line bundle $K_X + L$ with the upper regularized multiplier ideal sheaf $I_+(L)$ of $L$, and this is already not optimal in the nef case.
A proof of the Kawamata-Morrison Cone Conjecture for holomorphic symplectic varieties of K3^[n] or generalized Kummer deformation type
本文针对K3^[n]型及广义Kummer变形型的射影不可约全纯辛流形，证明了Kawamata–Morrison丰锥猜想的相应版本。
The purpose of this paper is to show how the generic vanishing theorems of M. Green and the second author can be used to settle several questions and conjectures concerning the geometry of irregular complex projective varieties. First, we prove a conjecture of Arbarello and DeConcini characterizing principally polarized abelian varieties whose theta divisors are singular in codimension one. Secondly, we study the holomorphic Euler characteristics of varieties of general type having maximal Albanese dimension: we verify a conjecture of Kollar for subvarieties of abelian varieties, but show that…
Let $a_X:X\rightarrow \mathrm{Alb}\, X$ be the Albanese map of a smooth complex projective variety. Roughly speaking in this note we prove that for all $i \geq 0$ and $α\in \mathrm{Pic}^0\, X$, the cohomology ranks $h^i(\mathrm{Alb}\, X, \,{a_X}_* ω_X\otimes P_α)$ are derived invariants. In the case of varieties of maximal Albanese dimension this proves conjectures of Popa and Lombardi-Popa -including the derived invariance of the Hodge numbers $h^{0,j}$ -- and a weaker version of them for arbitrary varieties. Finally we provide an application to derived invariance of certain irregular fibrati…
The main purpose of the following article is to give a proof of Y. Kawamata's celebrated subadjunction theorem in the spirit of our previous work on Bergman kernels. We will use two main ingredients : an $\displaystyle L^{2\over m}$--extension theorem of Ohsawa-Takegoshi type (which is also a new result) and a more complete version of our former results.
On twisted Kawamata's semi-positivity and finite generation of generalized canonical rings
本文提出了一种扭曲的 Kawamata 半正定性定理，并应用于广义对 canonical 环的有限生成性问题。
We present a generalization of the multiplier ideal version of inversion of adjunction, often known as the restriction theorem, to centers of arbitrary codimension. We approach inversion of adjunction from the subadjunction point of view. Let X be a smooth complex projective variety and let Z be an exceptional log-canonical center of an effective Q-divisor D on some dense open subset of X that contains the generic point of Z. Any subvariety of X can be expressed as such a center for some D. We define an adjoint ideal that measures how non-klt (X, D) is outside the generic point of Z. Our main …
We show that any pseudo-effective divisor on a normal surface decomposes uniquely into its "integral positive" part and "integral negative" part, which is an integral analog of Zariski decompositions. By using this decomposition, we give three applications: a vanishing theorem of divisors on surfaces (a generalization of Kawamata-Viehweg and Miyaoka vanishing theorems), Reider-type theorems of adjoint linear systems on surfaces (including a log version and a relative version of the original one) and extension theorems of morphisms defined on curves on surfaces (generalizations of Serrano and P…
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)), R…
The Frobenius--stable version of the Grauert--Riemenschneider vanishing theorem fails
本文证明Grauert–Riemenschneider消失定理的Frobenius稳定版本在任意正特征的三fold以及特征p∈{2,3,5}的terminal 3-fold上失效，并引入F_p-理性概念，证明至多4维的klt奇点是F_p-理性的，进而应用于K-平凡3-fold上的Kawamata–Viehweg消失定理的Frobenius稳定版本。
In this article our main result is a more complete version of the statements obtained in {\rm [6]}. One of the important technical point of our proof is an $\displaystyle L^{2\over m}$ extension theorem of Ohsawa-Takegoshi type, which is derived from the original result by a simple fixed point method. Moreover, we show that these techniques combined with an appropriate form of the"invariance of plurigenera" can be used in order to obtain a new proof of the celebrated Y. Kawamata subadjunction theorem.
In this paper, we introduce a coherent subsheaf of Saito's $S$-sheaf, which is a combination of the $S$-sheaf and the multiplier ideal sheaf. We construct its $L^2$-Dolbeault resolution, which generalizes MacPherson's conjecture on the $L^2$ resolution of the Grauert-Riemenschneider sheaf. We also prove various vanishing theorems for the $S$-sheaf (Saito's vanishing theorem, Kawamata-Viehweg vanishing theorem and some new ones like Nadel vanishing theorem) transcendentally. Finally, we discuss some applications of our results on the relative version of Fujita's conjecture (e.g. Kawamata's conj…
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