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

Numerical trivial automorphisms of Enriques surfaces in arbitrary characteristic

We extend to arbitrary characteristic some known results about automorphisms of complex Enriques surfaces that act trivially on the cohomology or the cohomology modulo torsion.

Torsion order of smooth projective surfaces

To a smooth projective variety $X$ whose Chow group of $0$-cycles is $\mathbf Q$-universally trivial one can associate its torsion index $\mathrm{Tor}(X)$, the smallest multiple of the diagonal appearing in a cycle-theoretic decomposition à la Bloch-Srinivas. We show that $\mathrm{Tor}(X)$ is the exponent of the torsion in the Néron-Severi-group of $X$ when $X$ is a surface over an algebraically closed field $k$, up to a power of the exponential characteristic of $k$.

Remarks on very basic slc-trivial fibrations

We study very basic slc-trivial fibrations. We show that restricting on any lc center of a very basic slc-trivial fibration, its moduli part is numerically trivial if and only if it is $\mathbb Q$-linearly trivial. We then prove that abundance conjecture for very basic slc-trivial fibrations holds true in dimension two when the moduli part is $\mathbb Q$-Cartier. As an application, we prove that the log canonical ring of a projective plt pair with Kodaira dimension 3 is finitely generated.

Universal triviality of the Chow group of 0-cycles and the Brauer group

We prove that a smooth proper universally CH_0-trivial variety X over a field k has universally trivial Brauer group. This fills a gap in the literature concerning the p-torsion of the Brauer group when k has characteristic p.

关于具有平凡局部基本群的孤立辛奇点的注记

A remark on isolated symplectic singularities with trivial local fundamental group

本文提出了一种简单方法构造任意维度的孤立辛奇点，其局部基本群为平凡群，回应了Beauville的问题。

关于数值平凡典范类的正则叶状结构的一些注记

Some remarks on regular foliations with numerically trivial canonical class

本文研究复射影流形上典范类数值平凡的正则叶状结构，分类了余维数二情形，并利用代数性准则处理小秩情形，最后在特殊情形下证实了广义Bondal猜想。

$p$-进 étale 上同调中的挠元：注记与猜想

Torsion in $p$-adic étale cohomology: remarks and conjectures

本文研究了非固有光滑形式概形的整 $p$-进 étale 上同调挠元问题，发现仿射情形下挠元可表示为特殊纤维的函子且挠子群有限，并提出了预测更广泛刚性解析簇挠元的猜想。

One remark on the b-semiampleness of the moduli part

In this short note we reduce the b-semiampleness conjecture for lc-trivial fibrations to the b-semiampleness conjecture for klt-trivial fibrations.

Fourier-Mukai transforms of slope stable torsion-free sheaves on Weierstrass elliptic threefolds

We focus on a class of Weierstrass elliptic threefolds that allows the base of the fibration to be a Fano surface or a numerically $K$-trivial surface. We define the notion of limit tilt stability, and show that the Fourier-Mukai transform of a slope stable torsion-free sheaf satisfying a vanishing condition in codimension 2 (e.g. a reflexive sheaf) is a limit tilt stable object. We also show that the inverse Fourier-Mukai transform of a limit tilt semistable object of nonzero fiber degree is a slope semistable torsion-free sheaf, up to modification in codimension 2.

p-torsion of Genus Two Curves Over Prime Fields of Characteristic p

Consider the Jacobian of a hyperelliptic genus two curve defined over a prime field of characteristic p and with complex multiplication. In this paper we show that the p-Sylow subgroup of the Jacobian is either trivial or of order p.

The Brauer group of Kummer surfaces and torsion of elliptic curves

We relate the Brauer group of a Kummer surface to the Brauer group of the corresponding abelian surface. For many pairs of elliptic curves over the rational numbers we prove that the Kummer surface attached to their product has trivial Brauer group.

A construction of numerical Campedelli Surfaces with \Z/6 torsion group

We produce a family of numerical Campedelli surfaces with \Z/6 torsion by constructing the (Gorenstein codimension 5) canonical ring of the étale six to one cover using serial unprojection. In Section 2 we develop the necessary algebraic machinery. Section 3 contains the numerical Campedelli surface construction, while Section 4 contains remarks and open questions.

曲面的torsion birational motives与unramified cohomology

Torsion birational motives of surfaces and unramified cohomology

该研究探讨了曲面的torsion birational motives与unramified cohomology的关系，并扩展了Kahn的结果，同时给出了违反积分Hodge猜想的例子。

Refined unramified cohomology of schemes

We introduce the notion of refined unramified cohomology of algebraic schemes and prove comparison theorems that identify some of these groups with cycle groups. This recovers for cycles of low codimensions on smooth projective varieties previous results of Bloch--Ogus, Colliot-Thélène--Voisin, Kahn, Voisin, and Ma. We combine our approach with the Bloch--Kato conjecture, proven by Voevodsky, to show that on a smooth complex projective variety, any homologically trivial torsion cycle with trivial Abel--Jacobi invariant has coniveau 1. This establishes a torsion version of a conjecture of Janns…

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。

Some remarks on Hodge symmetry

I make some remarks on Hodge symmetry, and prove for instance that if $k$ is a perfect field of characteristic $p>0$ and $X/k$ smooth, proper and Hodge-Witt scheme, and Hodge de Rham sequence of $X$ degenerates at $E_1$ and $X$ has torsion-free crystalline cohomology, then Hodge symmetry holds for $X$.

Odd torsion Brauer elements and arithmetic of diagonal quartic surfaces over number fields

We use recent advances in the local evaluation of Brauer elements to study the role played by {\it odd} torsion elements of the Brauer group in the arithmetic of diagonal quartic surfaces over {\it arbitrary} number fields. We show that over a local field if the order of the Brauer element is odd and coprime to the residue characteristic then the evaluation map it induces on the local points is constant. Over number fields we give a sufficient condition on the coefficients of the equation, which is mild and easy to check, so that the odd torsion does not obstruct weak approximation. We also no…

Double coverings of arrangement complements and $2$-torsion in Milnor fiber homology

We prove that the mod $2$ Betti numbers of double coverings of a complex hyperplane arrangement complement are combinatorially determined. The proof is based on a relation between the mod $2$ Aomoto complex and the transfer long exact sequence. Applying the above result to the icosidodecahedral arrangement ($16$ planes in the three dimensional space related to the icosidodecahedron), we conclude that the first homology of the Milnor fiber has non-trivial $2$-torsion.

Linear relations among the values of canonical heights from the existence of non-trivial endomorphisms

We study the interplay between canonical heights and endomorphisms of an abelian variety $A$ over a number field $k$. In particular we show that whenever the ring of endomorphisms defined over $k$ is strictly larger than $\Z$ there will be $\Q$-linear relations among the values of a canonical height-pairing evaluated at a basis modulo torsion of $A(k)$.

Picard group of the forms of the affine line and of the additive group

We obtain an explicit upper bound on the torsion of the Picard group of the forms of the affine line and their regular completions. We also obtain a sufficient condition for the Picard group of the forms of the affine line to be non trivial and we give examples of non trivial forms of the affine line with trivial Picard groups.