关于对数多典范丛的正像层与非消失猜想的评述

关于对数平展层化的注记

Remarks on logarithmic étale sheafification

本文研究对数概形上预层的对数平展层化问题，给出预层成为全对数平展层的判定准则，明确结构层与对数结构为对数平展层的情形，并证明满足对数平展下降的对数概形上对数皮卡群是全对数平展拓扑中的栈。

Regularity on abelian varieties III: relationship with Generic Vanishing and applications

We describe the relationship between the notions of $M$-regular sheaf and $GV$-sheaf in the case of abelian varieties. The former is a natural strengthening of the latter, and we provide an algebraic criterion characterizing it among the larger class. Based on this we deduce new basic properties of both $M$-regular and $GV$-sheaves. In the second part we give a number of applications of generation criteria for $M$-regular sheaves to the study of Seshadri constants, Picard bundles, pluricanonical maps on irregular varieties, and semihomogeneous vector bundles. This second part of the paper is b…

奇异埃尔米特度量与Catanese、Fujita和Kawamata分解定理

Singular hermitian metrics and the decomposition theorem of Catanese, Fujita, and Kawamata

本文证明带半正曲率奇异埃尔米特度量且满足最小延拓性质的无挠层可分解为埃尔米特平坦丛与泛型丰沛层的直和，并将其应用于相对多重典范丛直接像，推广了Fujita、Catanese–Kawamata和Iwai的结果。

Remarks on the existence of Cartier divisors

Given an invertible sheaf, does it come from a Cartier divisor? This might fail in presence of embedded components. I give some examples and characterize those invertible sheaves that allow a Cartier divisor.

Remarks on the nonvanishing of cohomology groups for perverse sheaves on abelian varieties

It is shown that for a perverse sheaf $K$ on an abelian variety $X$ the integers $i$ for which the cohomology $H^i(X,K)$ does not vanish define an interval in the number line (under certain conditions on the field of definition of $K$)

A remark on subbundles of symplectic and orthogonal vector bundles over curves

We review the notions of symplectic and orthogonal vector bundles over curves, and the connection between principal parts and extensions of vector bundles. We give a criterion for a certain extension of rank 2n to be symplectic or orthogonal. We then describe almost all of its rank n vector subbundles using graphs of sheaf homomorphisms, and give criteria for the isotropy of these subbundles.

关于非阿基米德域上正规仿射阿迪空间的整结构层上同调中的挠率

On the torsion in the cohomology of the integral structure sheaf of affinoid adic spaces

本文证明了特征零非阿基米德域上正规仿射阿迪空间的整结构层上同调一致挠，源于Bartenwerfer的注记，部分回答了Hansen和Kedlaya的问题及《非阿基米德苏格兰书》相关问题。

Remarks on iterations of the $\mathbb A^1$-chain connected components construction

We show that the sheaf of $\mathbb A^1$-connected components of a Nisnevich sheaf of sets and its universal $\mathbb A^1$-invariant quotient (obtained by iterating the $\mathbb A^1$-chain connected components construction and taking the direct limit) agree on field-valued points. This establishes an explicit formula for the field-valued points of the sheaf of $\mathbb A^1$-connected components of any space. Given any natural number $n$, we construct an $\mathbb A^1$-connected space on which the iterations of the naive $\mathbb A^1$-connected components construction do not stabilize before the …

Remarks on rational points of varieties whose cotangent bundles are generated by global sections

In this short note, we will gives several remarks on rational points of varieties whose cotangent bundles are generated by global sections. For example, we will show that if the sheaf of differentials of a projective variety X over a number field k is ample and generated by global sections, then the set of k-rational points of X is finite.

A remark on the jet bundles over the projective line

This is a footnote of a recent interesting work of Cohen, Manin and Zagier, where they, among other things, produce a natural isomorphism between the sheaf of (n-1)-th order jets of the n-th tensor power of the tangent bundle of a Riemann surface equipped with a projective structure and the sheaf of differential operators of order n (on the trivial bundle) with vanishing 0-th order part. We give a different proof of this result without using the coordinates, and following the idea of this proof we prove: Take a line bundle L with $L^2 = T$ on a Riemann surface equipped with a projective struct…

Remarks on surfaces with $c_1^2 = 2χ-1$ having non-trivial 2-torsion

We shall show that any complex minimal surface of general type with c_1^2 = 2χ-1 having non-trivial 2-torsion divisors, where c_1^2 and χare the first Chern number of a surface and the Euler characteristic of the structure sheaf respectively, has the Euler characteristic χnot exceeding 4. Moreover, we shall give a complete description for the surfaces of the case χ=4, and prove that the coarse moduli space for surfaces of this case is a unirational variety of dimension 29. Using the description, we shall also prove that our surfaces of the case χ= 4 have non-birational bicanonical maps and no …

A remark on Ulrich and ACM bundles

I show that on any smooth, projective ordinary curve of genus at least two and a projective embedding, there is a natural example of a stable Ulrich bundle for this embedding: namely the sheaf $B^1_X$ of locally exact differentials twisted by $Ø_X(1)$ given by this embedding and in particular there exist ordinary varieties of any dimension which carry Ulrich bundles. In higher dimensions, assuming $X$ is Frobenius split variety I show that $B^1_X$ is an ACM bundle and if $X$ is also a Calabi-Yau variety and $p>2$ then $B^1_X$ is not a direct sum of line bundles. In particular I show that $B^1_…