Ingrid Bauer, Fabrizio Catanese
Answering a question posed by Enriques, we construct a minimal smooth algebraic surface $S$ of general type over the complex numbers with $K^2 = 45$ and $p_g = 4$, and with birational canonical map. Our surface is a regular (q=0) ball quotient which is an etale quotient of a Hirzebruch covering of the plane. The canonical system $|K_S|$ has a fixed part and the degree of the canonical image is 19.
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We construct a canonical singular hermitian metric with semipositive curvature current on the canonical line bundle of a compact Kähler manifold with pseudoeffective canonical bundle.
We study the canonical stability index of nonsingular projective varieties of general type with either large canonical volume or large geometric genus. As applications of a general extension theorem established in the first part, we prove some optimal results in dimensions 4 and 5, which are parallel to some well-known results on surfaces and 3-folds.
Maximizing Slice-Volumes of Semialgebraic Sets using Sum-of-Squares Programming
针对半代数集仿射切片体积最大化问题，提出将其转化为无穷维线性规划，借助moment-Sum-of-Squares半定规划层级求解收敛上界，可通过拓扑结构、对称性及斯托克斯冗余约束优化效率，并在示例集上完成验证。
An effective upper bound for anti-canonical volumes of canonical $\mathbb{Q}$-Fano threefolds
本文证明了典范弱 $\mathbb{Q}$-Fano 三维流形的反典范体积上界为 $324$，给出了有效估计。
We prove that the canonical volume $K^3\geq {1/30}$ for all projective 3-folds of general type with $χ(\mathcal{O})\leq 0$. This bound is sharp.
We prove Weak Borisov--Alexeev--Borisov Conjecture in dimension three which states that the anti-canonical volume of an $ε$-klt log Fano pair of dimension three is bounded from above.
On infinite accumulation points of log canonical volumes
本文证明了对任意整数d≥2，d维射影对数典范簇的体积集存在无限聚点，揭示了体积地理学的新现象。
Canonical bundle formula and degenerating families of volume forms
本文利用L²度量与多重次调和奇点的赋值等价性，证明了典范丛公式在纤维积分意义下的解析刻画，从而证实一个民间猜想，并将其应用于识别Ohsawa测度奇点及部分回答Berndtsson关于半正定性定理的问题。
Let V be a smooth projective 3-fold of general type. Denote by $K^3$, a rational number, the self-intersection of the canonical sheaf of any minimal model of V. One defines $K^3$ as the canonical volume of $V$. Assume $p_g\ge 2$. We show that $K^3\ge {1/3}$, which is a sharp lower bound. Then we classify those V with small $K^3$ up to explicit tructure. We also give some new examples with $p_g=2$ which have maximal canonical stability index. Finally we give an application to certain algebraic 4-folds.
An effective upper bound for anti-canonical volumes of singular Fano threefolds
本文研究奇异 Fano 三体簇的反典范体积上界，证明对 $0<ε<1/3$ 的 $ε$-klt Fano $3$-维簇，其反典范体积至多为 $3200/ε^4$，并证明该阶 $O(1/ε^4)$ 不可改进。
Optimal upper bounds for anti-canonical volumes of singular toric Fano varieties
本文给出了d维1/q-lc环面Fano簇反典范体积的最优上界，并构造例子证明最优性，同时给出恰含一个内部格点的d维格单纯形体积的最优上界。
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。
We give an upper bound on the volume vol(P*) of a polytope P* dual to a d-dimensional lattice polytope P with exactly one interior lattice point, in each dimension d. This bound, expressed in terms of the Sylvester sequence, is sharp, and is achieved by the dual to a particular reflexive simplex. Our result implies a sharp upper bound on the volume of a d-dimensional reflexive polytope. Translated into toric geometry, this gives a sharp upper bound on the anti-canonical degree $(-K_X)^d$ of a d-dimensional toric Fano variety X with at worst canonical singularities.
On Pseudo-Effectivity and Volumes of Adjoint Classes in Kähler Families with Projective Central Fiber
本文研究了Kähler族中伴随类的伪有效性与体积的变形行为，证明了体积的局部不变性并验证了Siu的不变性猜想。
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}.
The Noether inequality for threefolds and three moduli spaces with minimal volumes
本文建立了几何亏格5≤p_g≤10的一般型射影三维簇的诺特不等式，解决三维簇诺特不等式所有剩余情形，并证明三个特定极小体积典范三维簇的粗模空间是不可约单有理簇且成员单连通。
Canonical Forms as Dual Volumes
该研究提出正几何典范形式的对偶体积表示，识别出由双曲多项式边界刻画的完全单调正几何类，计算了平面正几何的对应测度，建立了多领域新联系。
Anticanonical Volumes of Terminal and Canonical Gorenstein Fano Fourfolds
本文证明终端和典范 Gorenstein Fano 四折叠的反典范体积上界，并给出尖锐界及特殊情形下的改进。
In this article we prove the following version of the Weak-BAB conjecture for $3$-folds in char $p>5$: Fix a DCC set $I\subset [0, 1)$ and an algebraically closed field $k$ of characteristic $p>5$. Let $\mathfrak{D}$ be a collection of klt pairs $(X, Δ)$ satisfying the following properties: (1) $X$ is a projective $3$-fold, (2) $Δ$ is an $\mathbb{R}$-divisor with coefficients in $I$, (3) $K_X+Δ\equiv 0$, and (4) $-K_X$ is ample. Then the set $\{\mbox{vol}_X(-K_X) \ | \ (X, Δ)\in\mathfrak{D}\mbox{ for some }Δ\}$ is bounded from above.
Log canonical pairs with conjecturally minimal volume
本文构造了具有非零约化除子B且K_X+B ample的对数典范对(X,B)，其体积是已知最小的，并猜想该体积在每个维度上都是最小的，同时完全证明了Esser例子在所有维度上的期望性质。
We construct a surface with log terminal singularities and ample canonical class that has $K_X^2=1/48 983$ and a log canonical pair $(X,B)$ with a nonempty reduced divisor $B$ and ample $K_X+B$ that has $(K_X+B)^2 = 1/462$. Both examples significantly improve known records.
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
In this paper, we study the behavior of the sets of volumes of the form $\mathrm{vol}(X,K_X+B+M)$, where $(X,B)$ is a log canonical pair, and $M$ is a nef $\mathbb{R}$-divisor. After a first analysis of some general properties, we focus on the case when $M$ is $\mathbb{Q}$-Cartier with given Cartier index, and $B$ has coefficients in a given DCC set. First, we show that such sets of volumes satisfy the DCC property in the case of surfaces. Once this is established, we show that surface pairs with given volume and for which $K_X+B+M$ is ample form a log bounded family. These generalize results …
Characterizing terminal Fano threefolds with the smallest anti-canonical volume
本文刻画了反典范体积达到最小下界 1/330 的非有理 Q-阶乘终端法诺三维簇，证明其为 P(1,5,6,22,33) 中次数 66 的加权超曲面。
On the Iitaka volumes of log canonical surfaces and threefolds
该研究证明三维及以下对数典范对的Iitaka体积集合在系数集满足降链条件时也满足该条件，刻画了低维特定曲面的Iitaka体积集性质，部分证实了高维klt对的相关猜想。
We find an explicit upper bound for the anticanonical volume of Fano 4-folds with canonical singularities.
Let $V$ be a smooth projective 3-fold of general type. Denote by $K^{3}$, a rational number, the self-intersection of the canonical sheaf of any minimal model of $V$. One defines $K^{3}$ as a canonical volume of $V$. The paper is devoted to proving the sharp lower bound $K^{3}\ge {1/420}$ which can be reached by an example: $X_{46}\subseteq \mathbb{P}(4,5,6,7,23)$.
On volumes and the generic invariance of Fano type varieties
本文证明了当Fano型纤维的反典范子群体积在Zariski稠密集上为常数或纤维维数为2时，Fano型性质的一般不变性，并平行地证明了在约化模p下反典范子群体积为常数时Schwede-Smith猜想。
Volumes of foliations birationally bounded by algebraically integrable families
研究证明了由代数可积族理性有界所限定的log canonical叶体积集合满足DCC，并建立了弱半稳定映射家族的相对log canonical体积的变形不变性。
The minimal volume of log surfaces of general type with positive geometric genus
本文确定了一般型对数典范曲面在给定正几何亏格下的最小体积，并由此导出对数典范三维簇与稳定曲面的有效诺特型不等式以及一般型光滑射影曲面辛自同构群阶的一致上界。
A canonical Fano threefold has degree $\leq 72$
本文证明典范弱 Fano 三维簇的反典范体积上界为 72，且该上界最优。
We construct klt projective varieties with ample canonical class and the smallest known volume. We also find exceptional klt Fano varieties with the smallest known anticanonical volume. We conjecture that our examples have the smallest volume in every dimension, and we give low-dimensional evidence for that. In order to improve on earlier examples, we are forced to consider weighted hypersurfaces that are not quasi-smooth. We show that our Fano varieties are exceptional by computing their global log canonical threshold (or $α$-invariant) exactly; it is extremely large, roughly $2^{2^n}$ in dim…
We generalize Kawamata's product formula for volumes of canonical divisors to arbitrary divisors using Okounkov bodies
Let $\mathcal C\subset(0,1]$ be a set satisfying the descending chain condition. We show that any accumulation point of volumes of log canonical surfaces $(X, B)$ with coefficients in $\mathcal C$ can be realized as the volume of a log canonical surface with big and nef $K_X+B$ and coefficients in $\overline{\mathcal C}\cup\{1\}$, with at least one coefficient in $Acc(\mathcal C)\cup\{1\}$. As a corollary, if $\overline{\mathcal C}\subset\mathbb Q$ then all accumulation points of volumes are rational numbers, solving a conjecture of Blache. For the set of standard coefficients $\mathcal C_2=\{…
Sharp Volume Bounds in Picard Rank One and Polynomial Bounds for Log Calabi-Yau Surfaces
本文证明了对数 Calabi-Yau 曲面上积分 nef 且 big Weil 除子体积的多项式下界，并推导出有效双有理性阶，无需系数大性假设。
By Hacon-McKernan-Xu, there is a positive lower bound in each dimension for the volume of all klt varieties with ample canonical class. We show that these bounds must go to zero extremely fast as the dimension increases, by constructing a klt $n$-fold with ample canonical class whose volume is less than $1/2^{2^n}$. These examples should be close to optimal. We also construct a klt Fano variety of each dimension $n$ such that $H^0(X,-mK_X)=0$ for all $1\leq m < b$ with $b$ roughly $2^{2^n}$. Here again there is some bound in each dimension, by Birkar's theorem on boundedness of complements, an…
Positive Singularities and Volumes in Scattering Amplitudes
本论文围绕正几何，从对偶体积表示、圈级振幅奇点结构、朗道奇点与簇代数的猜想关系三方面展开研究，确立正几何为散射振幅的分析与几何组织提供统一语言。
We give an optimal upper bound for the anti-canonical volume of an $ε$-lc weak log del Pezzo surface. Moreover, we consider the relation between the bound of the volume and the Picard number of the minimal resolution of the surface. Furthermore we consider blowing up several points on a Hirzebruch surface in general position and give some examples of smooth weak log del Pezzo surfaces.
The arithmetic volume of hypersurfaces in toric varieties and Mahler measures
该研究确定了光滑射影环面簇中超曲面的典范算术体积，并由此证明了这类超曲面上的广义Hodge指标定理，为算术代数几何领域相关问题提供了新结果。
Characterizing terminal Fano threefolds with the smallest anti-canonical volume, II
本文证明具有最小反典范体积 1/330 且皮卡德数为 1 的 Q-阶乘终端法诺三维簇必为 P(1,5,6,22,33) 中的 66 次加权超曲面，并将同类刻画推广到 Iano-Fletcher 列表中的另外 11 个加权超曲面例子。
The minimal volume of surfaces of log general type with non-empty non-klt locus
该研究确定了 log 一般型曲面的极小体积为 1/825，并通过构造特定的 ample 模型揭示了其在模空间中的结构。
Let $X$ be a simple normal crossing (SNC) compact complex surface with trivial canonical bundle which includes triple intersections. We prove that if $X$ is $d$-semistable, then there exists a family of smoothings in a differential geometric sense. This can be interpreted as a differential geometric analogue of the smoothability results due to Friedman, Kawamata-Namikawa, Felten-Filip-Ruddat, Chan-Leung-Ma, and others in algebraic geometry. The proof is based on an explicit construction of local smoothings around the singular locus of $X$, and the first author's existence result of holomorphic…
We give an Eynard-Orantin type topological recursion formula for the canonical Euclidean volume of the combinatorial moduli space of pointed smooth algebraic curves. The recursion comes from the edge removal operation on the space of ribbon graphs. As an application we obtain a new proof of the Kontsevich constants for the ratio of the Euclidean and the symplectic volumes of the moduli space of curves.
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