Minimal model theory for log surfaces in Fujiki's class $\mathcal C$

半 log canonical 丰度在紧 Kähler 三维流形上的失效

Abundance for semi log canonical compact Kähler threefolds

本文展示紧 Kähler 变量在三维时半 log canonical 丰度失效，构造了反例表明 K_X 非负但 κ 为零，而 K_X 不是半 ample。同时证明若起始为紧 Kähler 半 dlt 对，则丰度成立。

Log Minimal Model Program for the Kontsevich Space of Stable Maps $\bar{\mathcal M}_{0,0}(\mathbb P^{3}, 3)$

This work is inspired by conversations with Izzet Coskun and Joe Harris. We run the log minimal model program for the Kontsevich space of stable maps $\bar{\mathcal M}_{0,0}(\mathbb P^{3}, 3)$ and give modular interpretations to all the intermediate spaces appearing in the process. In particular, we show that one component of the Hilbert scheme $\mathcal H_{3,0,3}$ is the flip of $\bar{\mathcal M}_{0,0}(\mathbb P^{3}, 3)$ over the Chow variety. Finally as an easy corollary we obtain that $\bar{\mathcal M}_{0,0}(\mathbb P^{3}, 3)$ is a Mori dream space.

On accumulation points of volumes of log surfaces

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=\{…

The Torelli problem for Logarithmic bundles of hypersurface arrangements in the projective space

Let $ \mathcal{D} = \{D_{1}, \ldots, D_{\ell}\} $ be an arrangement of smooth hypersurfaces with normal crossings on the complex projective space $ \mathbb{P}^{n} $ and let $ Ω^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $ be the logarithmic bundle attached to it. Our aim is to study the injectivity of the correspondence $ \mathcal{D} \longrightarrow Ω^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $. In order to do that, we first show that $ Ω^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $ admits a resolution of length $ 1 $ depending on the degrees and on the equations of $ D_{1}, \ldots, D_{\ell} $. Then, we pro…

Moduli of log twisted $\mathcal{N} =1$ SUSY curves

The goal of the present paper is to construct a smooth compactification of the moduli superstack classifying pointed $\mathcal{N} =1$ SUSY (= $\text{SUSY}_1$) curves. This construction is based on the Abramovich-Jarvis-Chiodo compactification of the moduli stack classifying spin curves. First, we give a general framework of a theory of log superschemes (or more generally, log superstacks). Then, we introduce the notion of a pointed (stable) log twisted $\text{SUSY}_1$ curve; it may be thought of as a logarithmic and twisted generalization of the classical notion of a pointed $\text{SUSY}_1$ cu…

Logarithmically Regular Morphisms

We consider the stack $\mathcal{L}og_X$ parametrizing log schemes over a log scheme $X$, and weak and strong properties of log morphisms via $\mathcal{L}og_X$, as defined by Olsson. We give a concrete combinatorial presentation of $\mathcal{L}og_X$, and prove a simple criterion of when weak and strong properties of log morphisms coincide. We then apply this result to the study of logarithmic regularity, derive its main properties, and give a chart criterion analogous to Kato's chart criterion of logarithmic smoothness.

Logarithmic bundles of multi-degree arrangements in $\mathbf{P}^{n}$

Let $ \mathcal{D} = \{D_{1}, ..., D_{\ell}\} $ be a multi-degree arrangement with normal crossings on the complex projective space $ \mathbf{P}^{n} $, with degrees $ d_{1}, ..., d_{\ell} $; let $ Ω_{\mathbf{P}^{n}}^{1}(\log \mathcal{D}) $ be the logarithmic bundle attached to it. First we prove a Torelli type theorem when $ \mathcal{D} $ has a sufficiently large number of components by recovering them as unstable smooth irreducible degree-$ d_{i} $ hypersurfaces of $ Ω_{\mathbf{P}^{n}}^{1}(\log \mathcal{D}) $. Then, when $ n = 2 $, by describing the moduli spaces containing $ Ω_{\mathbf{P}^{2}…

对数Cartier变换的拓扑斯理论方法

A Topos-Theoretic Approach to the Logarithmic Cartier Transform

本文通过拓扑斯理论方法，将对数Cartier变换推广到对数光滑态射，构造晶体范畴并证明其诱导函子忠实满，从而推广了Ogus和Vologodsky的经典结果。

Weyl groups and cluster structures of families of log Calabi-Yau surfaces

Given a generic Looijenga pair $(Y,D)$ together with a toric model $ρ:(Y,D)\rightarrow(\overline{Y},\overline{D})$, one can construct a seed ${\bf s}$ such that the corresponding $\mathcal{X}$-cluster variety $\mathcal{X}_{\bf s}$ can be viewed as the universal family of the log Calabi-Yau surface $U=Y\setminus D$. In cases where $(Y,D)$ is positive and $\mathcal{X}_{\bf s}$ is not acyclic, we describe the action of the Weyl group of $(Y,D)$ on the scattering diagram $\mathfrak{D}_{\bf s}$. Moreover, we show that there is a Weyl group element ${\bf w}$ of order $2$ that either agrees with or a…

索引对数 Cartier 变换

Indexed Logarithmic Cartier Transform

本文推广 Cartier 变换到对数情形，通过引入索引结构细化拓扑斯与晶体，利用对数微分算子索引代数的 Azumaya 性质，建立索引晶体范畴的等价，解决 Frobenius 不平坦导致的本质满射问题。

Log smoothness and polystability over valuation rings

Let $\mathcal{O}$ be a valuation ring of height one of residual characteristic exponent $p$ and with algebraically closed field of fractions. Our main result provides a best possible resolution of the monoidal structure $M_X$ of a log variety $X$ over $\calO$ with a vertical log structure: there exists a log modification $Y\to X$ such that the monoidal structure of $Y$ is polystable. In particular, if $X$ is log smooth over $\mathcal{O}$, then $Y$ is polystable with a smooth generic fiber. As a corollary we deduce that any variety over $\mathcal{O}$ possesses a polystable alteration of degreee…

Lagrangianity for log extendable overconvergent $F$-isocrystals

In the framework of Berthelot's theory of arithmetic $\mathcal{D}$-modules, we prove that Berthelot's characteristic variety associated with a holonomic $\mathcal{D}$-modules endowed with a Frobenius structure has pure dimension. As an application, we get the lagrangianity of the characteristic variety of a log extendable overconvergent $F$-isocrystal.

On Base Change of Local Stability in Positive Characteristics

We prove that a pointed one dimensional family of varieties $\mathcal{X}\to {b\in B}$ in positive characteristics is locally stable iff the log pair $(\mathcal{X'}, \mathcal{X}'_{b'})$ arising from its base change to the perfectoid base $b'\in B_{perf}$ is log canonical.

关于 $\mathcal{O}_{K}$ 上对数 $p$-可除群的对数棱镜 Dieudonné 理论

Log prismatic Dieudonné theory for log $p$-divisible groups over $\mathcal{O}_{K}$

本文证明了赋予典范对数结构的完备离散赋值环上的对数 $p$-可除群，对应于绝对对数棱镜景上的 Dieudonné 晶体，并利用对数下降将其归约为经典棱镜对应。

Systèmes inductifs surcohérents de D-modules arithmétiques logarithmiques

Let $\mathcal{V}$ be a complete discrete valuation ring of unequal characteristic with perfect residue field, $\mathcal{P}$ be a smooth, quasi-compact, separated formal scheme over $\mathcal{V}$, $\mathcal{Z}$ be a strict normal crossing divisor of $\mathcal{P}$ and $\mathcal{P}^\sharp := (\mathcal{P}, \mathcal{Z})$ the induced smooth formal log-scheme over $\mathcal{V}$. In Berthelot's theory of arithmetic $\mathcal{D}$-modules, we work with the inductive system of sheaves of rings $\smash{\hat{\mathcal{D}}}_{\mathcal{P} ^\sharp} ^{(\bullet)} := (\smash{\hat{\mathcal{D}}}_{\mathcal{P}^\sharp}…

关于 $\overline{\mathcal{M}_{0,n}}$ 的 Grothendieck 类的超对数凹性的改进界

Improved Bounds on Ultra-Log Concavity of the Grothendieck Class of $\overline{\mathcal{M}_{0,n}}$

本文研究了稳定n点曲线模空间 $\overline{\mathcal{M}_{0,n}}$ 的Grothendieck类的超对数凹性，通过为其渐近公式误差项提供可计算界，证明了在特定范围内超对数凹不等式对足够大的n成立。

Branched projective structures on a Riemann surface and logarithmic connections

We study the set ${\mathcal P}_S$ consisting of all branched holomorphic projective structures on a compact Riemann surface $X$ of genus $g \geq 1$ and with a fixed branching divisor $S:= \sum_{i=1}^d n_i\cdot x_i$, where $x_i \in X$. Under the hypothesis that $n_i=1$, for all $i$, with $d$ a positive even integer such that $d \neq 2g-2$, we show that ${\mathcal P}_S$ coincides with a subset of the set of all logarithmic connections with singular locus $S$, satisfying certain geometric conditions, on the rank two holomorphic jet bundle $J^1(Q)$, where $Q$ is a fixed holomorphic line bundle on …

紧致Kähler品种的fundamental groups与 nef反canonical bundle

Fundamental groups of compact Kahler varieties with nef anti canonical bundle

本文研究了紧致Kähler品种的fundamental groups，证明在特定条件下，其fundamental group几乎为Abelian，并扩展了相关定理至log canonical对和klt对。

$\u200b\overline{\mathcal M}_{0,n}$\u200b上同调表示的递归算法与对数凹性

Recursive algorithm and log-concavity of representations on the cohomology of $\overline{\mathcal M}_{0,n}$

该研究提出了计算$\overline{\mathcal M}_{0,n}$上同调$\mathbb{S}_n$-表示的可编程递归算法，得到不变部分的显式公式并证明其庞加莱多项式渐近对数凹，还提出了等变对数凹性猜想。

Singularity, Sasaki-Einstein manifold, Log del Pezzo surface and $\mathcal{N}=1$ AdS/CFT correspondence: Part I

A five dimensional Sasaki-Einstein (SE) manifold provides a AdS/CFT pair for four dimensional $\mathcal{N}=1$ SCFT, and those pairs are very useful in studying field theory and AdS/CFT correspondence. The space of known SE manifolds is increased significantly in the last decade, and we initiated the study of various field theory properties through the geometric property of these new SE manifolds. There is an associated three dimensional log-terminal singularity $X$ for each SE manifold $L_X$, and for quasi-regular case, there is an associated two dimensional log Del Pezzo surface $(S,Δ)$. The …

$\overline{\mathcal{M}}_{0,n}$ 的特征多项式与对数凹性

Characteristic polynomial of $\overline{\mathcal{M}}_{0,n}$ and log-concavity

受色对称函数启发，本文引入对称群表示的特征多项式，证明其在几何表示中系数呈对数凹性，并重点推导了带点有理曲线模空间上同调特征多项式的渐近公式与对数凹性。