Andrew Harder, Zhiyuan Li, Junliang Shen, Qizheng Yin
We identify the perverse filtration of a Lagrangian fibration with the monodromy weight filtration of a maximally unipotent degeneration of compact hyper-Kähler manifolds.
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Stable Degenerations of log Fano Fibration Germs
本文证明了Sun-Zhang提出的log Fano纤维芽稳定退化猜想，引入了H-不变量，证明存在唯一准单项估值使H-不变量最小，并证明其关联环是有限生成的，诱导出特殊退化到K-semistable极化log Fano纤维芽。
Parabolic automorphisms of hyperk{ä}hler manifolds: Orbits and Betti maps
本文研究了带有 lagrangian 纤维的不可约 holomorphically symplectic 流形的抛物自同构，证明了相关 Betti 映射具有最大秩，从而得出诱导平移为有限阶的纤维集合是稠密的。
We discuss various topics on degenerations and special Lagrangian torus fibrations of Calabi-Yau manifolds in the context of mirror symmetry. A particular emphasis is on Tyurin degenerations and the Doran-Harder-Thompson conjecture, which builds a bridge between mirror symmetry for Calabi-Yau manifolds and that for quasi-Fano manifolds. The proof of the conjecture is of interest in its own right and leads us to a few other related topics such as SYZ mirror symmetry, theta functions and geometric quantization. Inspired by the conjecture, we also propose a new construction of Landau-Ginzburg mod…
Degenerations, fibrations and higher rank Landau-Ginzburg models
本文研究了准Fano簇的退化与高阶Landau-Ginzburg模型的镜像关系，通过拼接低阶模型来揭示其镜像性质，并推导出更高余维数的Calabi-Yau纤维化。
This survey article begins with a discussion of the original form of the Strominger-Yau-Zaslow conjecture, surveys the state of knowledge concering this conjecture, and explains how thinking about this conjecture naturally leads to the program initiated by the author and Bernd Siebert to study mirror symmetry via degenerations of Calabi-Yau manifolds and log structures.
Geodesics in the space of relatively Kähler metrics
本文推导了纤维丛上相对Kähler度量的测地线方程，证明了具有纤维常标量曲率度量间唯一光滑测地线的存在性，并利用对数范数泛函的凸性证明了最优辛联络的唯一性与纤维丛的多重稳定性。
Degenerations and Fibrations of K3 Surfaces: Lattice Polarisations and Mirror Symmetry
本文研究K3曲面Tyurin退化及椭圆纤维化的格极化理论，建立兼容现有定义的格极化概念，并探讨两种设置间的镜像对称对应。
A conjecture of Kato says that the monodromy operator on the cohomology of a semi-stable degeneration of projective varieties is represented by an algebraic cycle on the special fiber of a normal crossing model of the fiber product degeneration. We prove this conjecture in the simple case of a semi-stable degeneration arising from a Lefschetz fibration.
Fibrations by Lagrangian tori for maximal Calabi-Yau degenerations and beyond
介绍在凯勒流形退化中构造拉格朗日环面纤维化的通用技术，该技术利用阿坎波空间边界及辛联络，证明极大卡拉比 - 丘退化附近纤维一般区域有符合期望渐近性质的拉格朗日环面纤维化。
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.
We show that some of centeral fibers of degenerations of hyperelliptic curves are realized as those trigonal curves. In particular, any hyperelliptic curve can be the central fiber of a degeneration of trigonal curves.
We introduce a new model for elliptic fibrations endowed with a Mordell-Weil group of rank one. We call it a Q$_7(\mathscr{L},\mathscr{S})$ model. It naturally generalizes several previous models of elliptic fibrations popular in the F-theory literature. The model is also explicitly smooth, thus relevant physical quantities can be computed in terms of topological invariants in straight manner. Since the general fiber is defined by a cubic curve, basic arithmetic operations on the curve can be done using the chord-tangent group law. We will use this model to determine the spectrum of singular f…
Non-archimedean SYZ fibrations via tropical contractions
本文通过热带收缩映射，证明了Batyrev-Borisov完全交的环退化中，对偶相交复形与Berkovich解析化的基本骨架同构，并构造了仿射环面纤维化。
We investigate a potential relationship between mirror symmetry for Calabi-Yau manifolds and the mirror duality between quasi-Fano varieties and Landau-Ginzburg models. More precisely, we show that if a Calabi-Yau admits a so-called Tyurin degeneration to a union of two Fano varieties, then one should be able to construct a mirror to that Calabi-Yau by gluing together the Landau-Ginzburg models of those two Fano varieties. We provide evidence for this correspondence in a number of different settings, including Batyrev-Borisov mirror symmetry for K3 surfaces and Calabi-Yau threefolds, Dolgachev…
Weight decomposition for toroidal abelian fibrations
该研究探讨有理乘N映射在特定阿贝尔簇退化周环群上的作用，结合傅里叶变换，计算单位截面类、证明相对周动机广义权分解并确定重言环，属代数几何领域的相关研究。
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 $π:X\rightarrow\mathbb{P}^n$ be a (holomorphic) Lagrangian fibration that is very general in the moduli space of Lagrangian fibrations. We conjecture that the singular fibres in codimension one must be semistable degenerations of abelian varieties. We prove a partial result towards this conjecture, and describe an example that provides further evidence.
Irrationality of degenerations of Fano varieties
本文研究了任意维数ε-lc Fano变种的无理度，证明了在特定条件下，中心纤维可通过有理映射到有界无理度的光滑射影变种，支持了一个关于log Fano纤维化猜想的泛化界限情况。
We describe explicitly the possible degenerations of a class of double Kodaira fibrations in the moduli space of stable surfaces. Using deformation theory we also show that under some assumptions we get a connected component of the moduli space of stable surfaces.
In this note, we show that for surfaces admitting suitable fibrations, any given degeneration X / Delta is bimeromorphic to a fiber space over Delta and we apply this result to the study of the degenerate fiber.
Kähler-Ricci shrinkers and Fano fibrations
本文用双有理代数几何证明Kähler-Ricci收缩孤立子是拟射影簇，并提出将其存在性与极化Fano纤维化K-稳定性及奇点切流相联系的统一猜想。
This is an extended example of the study of mirror symmetry via log schemes and the discrete Legendre transform on affine manifolds, introduced by myself and Bernd Siebert in "Mirror Symmetry via Logarithmic Degeneration Data I" (math.AG/0309070). In this paper, I consider the construction as it applies to the Batyrev-Borisov construction for complete intersections in toric varieties. Given a pair of reflexive polytopes with nef decompositions dual to each other, and given polarizations on the corresponding toric varieties, we construct examples of toric degenerations and their dual intersecti…
Semi-stable degenerations of Calabi-Yau manifolds and mirror P=W conjecture
本文将 Doran–Harder–Thompson 的双 Fano 情形推广到一般半稳定退化，用混合 Landau–Ginzburg 模型构造 Calabi–Yau 的拓扑镜像，并由镜像 P=W 猜想推出两类滤过的对应。
Inspired by the work of Gross on topological Mirror Symmetry we construct candidate Lagrangian torus fibration models for the 105 families of smooth Fano threefolds. We prove, in the case the second Betti number is one, that the total space of each fibration is homeomorphic to the expected Fano threefold, and show that the numerical invariants coincide for all 105. Our construction relies on a notion of toric degeneration for affine manifolds with singularities, and the correspondence we obtain between polytopes and Fano manifolds is compatible with that appearing in the work of Coates-Corti-K…
Xiao's Genus-Two Fibration: Branched Covers and Braid Monodromy
本文通过分支覆盖构造和辫子单值性方法，直接计算Xiao亏格二Lefschetz纤维化的几何单值性分解，得到类型$(4,3)$的有序正分解及七个因子的Artin坐标。
We study the collapsing behaviour of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the rescaled metrics become flat in the limit. The limit metric on the base minus the critical locus is locally isometric to an open dense subset of any Gromov-Hausdorff limit space of the Ricci-flat metrics. We then apply these results to study metric degenerations of families of pola…
Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we made a conjecture concerning the Gromov-Hausdorff limits of Calabi-Yau n-folds (with Ricci-flat Kähler metric) as one approaches a large complex structure limit point in moduli; a similar conjecture was made independently by Kontsevich, Soibelman and Todorov. Roughly stated, the conjecture says that, if the metrics are normalized to have constant diameter, then this limit is the base of the conjectural special lagrangian torus fibration associated with the large complex structure limit, namely an n-sphere, …
A classical way to construct a Lagrangian in a symplectic manifold $Σ$ is to let $Σ$ appear as a smooth fiber in a Lefschetz fibration. If this is possible the singularities of the fibration induce Lagrangian spheres in $Σ$ and these spheres, in turn, are representatives of the corresponding vanishing cycles in the homology of $Σ$. In this paper our aim is twofold: The first is to describe a generalization of the above mentioned construction to the "Morse-Bott" case. This leads, whenever such a degeneration exists, to the existence of Lagrangian sphere bundles rather than just spheres. In the …
Toric Kähler metrics seen from infinity, quantization and compact tropical amoebas
本文研究了紧致 toric 流形上 toric Kähler 度量空间在无穷远处的极限，揭示了 Kähler 极化在 Lagrangian 纤维上的退化，实现了全纯与实极化几何量子化的连续插值，并利用度量退化证明了紧致超曲面阿米巴极限可由热带阿米巴描述。
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