Chen Jiang, Haidong Liu
We show the boundedness of B-pluricanonical representations of lc log Calabi-Yau pairs in dimension $2$. As applications, we prove the boundedness of indices of slc log Calabi-Yau pairs up to dimension $3$ and that of non-klt lc log Calabi-Yau pairs in dimension $4$.
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We prove that there is a fixed constant r=r_n, such that if X is a variety of general type, then the rth pluricanonical map is birational.
We prove that the r-th pluricanonical maps of threefolds of general type have birationally bounded fibers if $r\geqslant 2$. Similarly, we prove that the r-th pluricanonical maps of fourfolds of general type have birationally bounded fibers if $r\geqslant 4$. We extend these results to higher dimensions in terms of constants arising naturally from the birational geometry of varieties of general type.
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 $(X, Δ)\in\mathfrak{D}$
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。
We show the birational boundedness of anti-canonical irreducible hypersurfaces which form 3-fold plt pairs. We also treat a collection of Du Val K3 surfaces which is birationally bounded but unbounded.
Boundedness of polarized log Calabi-Yau fibrations
该文研究带对数Calabi-Yau纤维化结构的对数对的有界性，证明在对数典范除子Iitaka体积有界、一般纤维属有界极化对数Calabi-Yau对族的条件下，纤维化全空间在crepant双有理等价下有界。
Boundedness of polarized log Calabi-Yau fibrations
本文研究基与一般纤维有界的极化对数Calabi-Yau纤维化的有界性问题，改进arXiv:2504.05243的结果至实际有界性并证实相关猜想，还给出该文献双有理有界性结果的更具概念性的新证明。
We prove a boundedness result for klt pairs $(X,B)$ such that $K_X+B\equiv 0$ and $B$ is big. As a consequence we obtain a positive answer to the Effective Iitaka Fibration Conjecture for klt pairs with big boundary.
Boundedness of polarized log Calabi-Yau fibrations with bounded bases
研究基和一般纤维有界的对数 Calabi - Yau 纤维化有界性，通过固定自然不变量证明余维数一有界，一般纤维不规则性消失时全空间有界，并应用于稳定极小模型和纤维化 Calabi - Yau 簇。
On the boundedness of elliptic Calabi-Yau 4-folds
本研究解决了一大类椭圆Calabi-Yau 4-fold的有界性问题，证明非等平凡纤维化的此类4-fold属于有限代数族，还为Calabi-Yau 3-fold中间Betti数有界性提供部分证据并研究相关4-fold的指标。
We prove that rationally connected Calabi--Yau 3-folds with kawamata log terminal (klt) singularities form a birationally bounded family, or more generally, rationally connected $3$-folds of $ε$-CY type form a birationally bounded family for $ε>0$. Moreover, we show that the set of $ε$-lc log Calabi--Yau pairs $(X, B)$ with coefficients of $B$ bounded away from zero is log bounded modulo flops. As a consequence, we deduce that rationally connected klt Calabi--Yau $3$-folds with mld bounded away from $1$ are bounded modulo flops.
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 $4$.
We study the birational boundedness of special fibers of log Calabi-Yau fibrations and Fano fibrations. We show that for a locally stable family of Fano varieties or polarised log Calabi-Yau pairs over a curve, if the general fiber satisfies some natural boundedness conditions, then every irreducible component of the special fiber is birationally bounded.
Boundedness of complements for log Calabi-Yau threefolds
本研究针对系数集为[0,1]的对数Calabi-Yau型簇，证明三维簇对若存在局部klt的ℝ-补集，则对应有限正整数集中的某个n有n-补集，同时证得ℝ-互补曲面对的补集有界性。
Birational boundedness of rationally connected log Calabi-Yau pairs with fixed index
本文研究固定维数的有理连通 klt 对数 Calabi–Yau 对，证明在反对数典范类的固定 Cartier 指数与 nef 条件下，相应射影簇的集合模 flops 有界。
We show some inductive statements for the index conjecture for log canonical Calabi-Yau pairs. Using it, we show that boundedness of log canonical index for log canonical Calabi Yau pairs with rational DCC coefficients in dimension 3. We also show boundedness of log canonical index for klt log Calabi-Yau pairs in dimension 4 with $B\neq 0$.
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。
Boundedness of elliptic Calabi-Yau threefolds
该研究证明椭圆Calabi-Yau三维簇构成有界族，同时明确在固定多重典范形式增长率与Iitaka纤维化多截面次数的条件下，Kodaira维数为2的极小终端三维簇也满足该有界性结论，且两个假设均为必要条件。
Boundedness questions for Calabi-Yau threefolds
本文研究光滑 Calabi–Yau 三维簇由三次上积形式和第二陈类线性形式决定后的模空间有界性，借助刚性有理曲面分析，并在 Picard 数 2 时证明一般有界性。
On boundedness and moduli spaces of K-stable Calabi-Yau fibrations over curves
本文证明曲线上极化 Calabi–Yau 纤维化在固定一般纤维体积与 Iitaka 体积时具有有界性，并据此构造绝热意义下的分离粗模空间，确立成员的一致 K-稳定性及光滑成员的 cscK 度量。
Boundedness of Stein degrees in positive characteristics
本文在足够大的特征下证明了相对维数1的Shokurov猜想，并由此得出对数Calabi-Yau纤维化中水平边界Stein次数的有界性。
Bershadsky, Cecotti, Ooguri and Vafa constructed a real valued invariant for Calabi-Yau manifolds, which is called the BCOV invariant. In this paper, we extend the BCOV invariant to such pairs $(X,D)$, where $X$ is a compact Kähler manifold and $D$ is a pluricanonical divisor on X with simple normal crossing support. We also study the behavior of the extended BCOV invariant under blow-up. The results in this paper lead to a joint work with Fu proving that birational Calabi-Yau manifolds have the same BCOV invariant.
In this paper we study boundedness properties and singularities of log Calabi-Yau fibrations, particularly those admitting Fano type structures. A log Calabi-Yau fibration roughly consists of a pair $(X,B)$ with good singularities and a projective morphism $X\to Z$ such that $K_X+B$ is numerically trivial over $Z$. This class includes many central ingredients of birational geometry such as Calabi-Yau and Fano varieties and also fibre spaces of such varieties, flipping and divisorial contractions, crepant models, germs of singularities, etc.
Discreteness of volumes of divisors on Calabi-Yau type varieties
本文研究 Calabi-Yau 型簇上除子体积的离散性，证明其属于仅依赖维数和奇点的固定离散集，并由此证明 Birkar 猜想的有界性结果。
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.
We prove the general diagram method theorem valid for the quite large class of 3-folds with Q-factorial singularities (see Basic Theorem 1.3.2 and also Theorem 2.2.6). This gives the generalization of our results about Fano 3-folds with Q-factorial terminal singularities (Preprint alg-geom/9311007). As an application, we get the following result about 3-dimensional Calabi-Yau manifolds X: Assume that the Picard number ρ(X) > 93. Then one of two cases (i) or (ii) holds: (i) there exists a small extremal ray on X; (ii) There exists a nef element h such that h^3=0 (thus, the nef cone NEF(X) and t…
In this paper we investigate various properties of generalised pairs in families, especially boundedness of several kinds. We show that many statements for usual pairs do not hold for generalised pairs. In particular, we construct an unexpected counter-example to boundedness of generalised lc models with fixed appropriate invariants. We also show that the DCC of Iitaka volumes and existence of nef reduction maps fail in families of generalised pairs. In a positive direction we show boundedness of bases of log Calabi-Yau fibrations with their induced generalised pair structure under natural ass…
Stein degree on log Calabi-Yau fibrations
本文证明了关于 log Calabi-Yau 纤维化上除子的 Stein 度有界性的猜想，给出了仅依赖于相对维数和系数下界的纤维不可约分量数上界。
Log Calabi-Yau pairs of birational complexity zero
本文研究指数一、双有理复杂度零的对数 Calabi-Yau 配对，提出基于对偶复形与对数典范位置有理性的刻画猜想，证明其补集可由代数环面仿射开子集除子式覆盖，并建立二维情形的仿射有界性。
In this paper, we continue the study of boundedness questions for (simply connected) smooth Calabi-Yau threefolds commenced in arXiv:1706.01268. The diffeomorphism class of such a threefold is known to be determined up to finitely many possibilities by the integral middle cohomology and two integral forms on the integral second cohomology, namely the cubic cup-product form and the linear form given by cup-product with the second chern class. The question addressed in both papers is whether knowledge of these cubic and linear forms determines the threefold up to finitely many families, that is …
A Pluripotential-Theoretic Approach to Finiteness of Polarized Calabi--Yau Manifolds
本文提出多势理论框架，通过m-Hessian容量等估计连接复势理论与几何代数有界性，以研究极化Calabi--Yau流形的有限性，并指出关键待发展步骤。
Boundedness of some fibered K-trivial varieties
本文证明具有阿贝尔簇或本原辛簇纤维化的固定维数Calabi-Yau簇是双有理有界的，并证明带Lagrangian纤维化的本原辛簇及纤维化Calabi-Yau 3-流形仅有有限多个形变类，进而在相关猜想下限制hyperkähler簇的形变类数量。
On multiplicities of fibers of Fano fibrations
本文将birational几何中的多个猜想，包括Shokurov关于log Calabi-Yau纤维化基的奇点猜想和有理连通Calabi-Yau品种的有界猜想，归结为Fano纤维化过曲线的纤维乘积猜想。
A Finiteness Theorem for Quartic K3-Fibred Calabi--Yau Threefolds in Scrolls
本文针对堆卷中以四次K3曲面为纤维的Calabi-Yau三维簇，应用Reid-Shepherd-Barron-Tai准则分类了n=1,2,3的情况，共得到14个形变族，补充了相关显式分类数据。
A Picard rank bound for base surfaces of elliptic Calabi-Yau 3-folds
本文通过双有理几何新策略，为可作为椭圆Calabi-Yau三维流形基的紧致曲面建立了Picard秩的显式上界，并阐明了有界性的来源。
Spectrum bounds in geometry
本文基于椭圆纤维化Calabi-Yau三维流形拓扑类型有限性的已有结果，探讨了其对F理论紧致化无质量谱中几何量有界性的影响，并给出了显式界限。
This work consists of two parts. In the first part we develop new techniques to compute Koszul cohomology groups for several classes of varieties. As applications we prove results on projective normality and syzygies for algebraic surfaces. From more general results we obtain in particular the following: a) Mukai's conjecture (and stronger variants of it) regarding projective normality and normal presentation for surfaces with Kodaira dimension 0, and uniform bounds for higher syzygies associated to adjoint linear series, b) effective bounds along the lines of Mukai's conjecture regarding proj…
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