Remark on nefness in higher codimension

A remark on the codimension of the Green-Griffiths locus of generic projective hypersurfaces of high degree

We show that for every smooth generic projective hypersurface $X\subset\mathbb P^{n+1}$, there exists a proper subvariety $Y\subsetneq X$ such that $\operatorname{codim}_X Y\ge 2$ and for every non constant holomorphic entire map $f\colon\mathbb C\to X$ one has $f(\mathbb C)\subset Y$, provided $°X\ge 2^{n^5}$. In particular, we obtain an effective confirmation of the Kobayashi conjecture for threefolds in $\mathbb P^4$.

关于叶状结构的第二陈类的注记

Remarks on the second Chern class of a foliation

本文研究了叶状结构的第二陈类及其奇点方案的度数，提出对于射影空间上的叶状结构，其奇点方案的度数下界为d+1，并探讨了等号成立的条件。

Some remarks on the minimal model program for log canonical pairs

We prove that the target space of an extremal Fano contraction from a log canonical pair has only log canonical singularities. We also treat some related topics, for example, the finite generation of canonical rings for compact Kähler manifolds, and so on. The main ingredient of this paper is the nefness of the moduli parts of lc-trivial fibrations. We also give some observations on the semi-ampleness of the moduli parts of lc-trivial fibrations. For the reader's convenience, we discuss some examples of non-Kähler manifolds, flopping contractions, and so on, in order to clarify our results.

关于小次数与小余维数光滑射影簇的若干注记

Some remarks on smooth projective varieties of small degree and codimension

本文针对小次数小余维光滑射影簇研究，给出余维2的高维Fano或Calabi-Yau簇为完全交的Ballico-Chiantini定理的快速证明，并将光滑射影簇为完全交的次数界从约0.63·n^(1/2)提升至约0.79·n^(2/3)。

On nef subvarieties

The goal of this work is to study positivity of subvarieties with nef normal bundle in the sense of intersection theory. After Ottem's work on ample subschemes, we introduce the notion of a nef subscheme, which generalizes the notion of a subvariety with nef normal bundle. We show that restriction of a pseudoeffective (resp. big) divisor to a nef subvariety is pseudoeffective (resp. big). We also show that ampleness and nefness are transitive properties. We define the weakly movable cone as the cone generated by the pushforward of cycle classes of nef subvarieties via proper surjective maps. T…

Hodge index theorem for arithmetic cycles of codimension one

In this note, we will give a partial answer for arithmetic analogues of Grothendieck's standard conjectures due to H. Gillet and C. Soule. (Remark : I changed the title of this note.)