Relative Severi inequality for fibrations of maximal Albanese dimension over curves

Let $f: X \to B$ be a relatively minimal fibration of maximal Albanese dimension from a variety $X$ of dimension $n \ge 2$ to a curve $B$ defined over an algebraically closed field of characteristic zero. We prove that $K_{X/B}^n \ge 2n! \chi_f$, which was conjectured by Barja in [2]. Via the strategy outlined in [5], it also leads to a new proof of the Severi inequality for varieties of maximal Albanese dimension. Moreover, when the equality holds and $\chi_f > 0$, we prove that the general fiber $F$ of $f$ has to satisfy the Severi equality that $K_F^{n-1} = 2(n-1)! \chi(F, \omega_F)$. We also prove some sharper results of the same type under extra assumptions.Comment: Comments are welcom

Relative crystalline representations and pp-divisible groups in the small ramification case

Let $k$ be a perfect field of characteristic $p > 2$, and let $K$ be a finite totally ramified extension over $W(k)[\frac{1}{p}]$ of ramification degree $e$. Let $R_0$ be a relative base ring over $W(k)\langle t_1^{\pm 1}, \ldots, t_m^{\pm 1}\rangle$ satisfying some mild conditions, and let $R = R_0\otimes_{W(k)}\mathcal{O}_K$. We show that if $e < p-1$, then every crystalline representation of $\pi_1^{\text{\'et}}(\mathrm{Spec}R[\frac{1}{p}])$ with Hodge-Tate weights in $[0, 1]$ arises from a $p$-divisible group over $R$.Comment: 19 pages; changed the title; added section 6 and more detail

Making Hanzi learnable for nonbackground beginning learners : an action research study in a primary school in Australia

Hanzi plays an important role in Chinese language. However, many learners find it hard to learn and to recognise, especially young nonbackground beginning learners. This study aimed at making Hanzi learnable to nonbackground beginning learners in Western Sydney, Australia. To achieve this goal, a suitable Hanzi pedagogy should be established and refined; and proper scaffolding strategies should be used in assisting students’ Hanzi learning. The following three research questions were posed in this study: 1. Which Hanzi teaching pedagogy is suitable for nonbackground beginning learners in Western Sydney public schools? 2. What scaffolding strategies should be used to assist students’ Hanzi learning? 3. What activities are suitable for Hanzi learning in terms of its pronunciation, form, and meaning? To answer these questions, an activity-based Hanzi teaching pedagogy was established and tested in a two-cycle action research project and refined after Cycle 1. The data shows that activity-based Hanzi teaching effectively engaged students and helped their Hanzi learning. Students learned Hanzi well through activities, and they remembered and recognised the meaning of Hanzi weeks after learning. In Cycle 1, oral language was integrated with activity-based Hanzi teaching, but the data shows that oral language was of limited help in Hanzi learning. The data shows that Hanzi can be more efficiently taught without integrating oral language as the pictographic and ideographic nature of Hanzi determines that Hanzi should be taught in a direct and systematic way. Cycle 2 further demonstrated the effectiveness of the activity-based Hanzi teaching when the teaching focus shifted from oral language to Hanzi itself. Thus, it is suitable to use activity-based Hanzi teaching to teach Hanzi directly and collectively through a series of activities. Useful scaffolding strategies such as questioning, giving feedback, and engaging learners’ prior knowledge were identified in the two-cycle action research. Some activities such as chanting, and calligraphy writing used in activity-based Hanzi teaching were found useful in Hanzi writing, form, and meaning recognition. However, students tended to forget the pronunciation of Hanzi after a while, even when the related activities were completed successfully at the time. Further studies are invited to improve this activity-based Hanzi teaching pedagogy