Strict Arakelov inequality for a family of varieties of general type

Abstract

<abstract><p>Let $ f:\, X\to Y $ be a semistable non-isotrivial family of $ n $-folds over a smooth projective curve with discriminant locus $ S \subseteq Y $ and with general fiber $ F $ of general type. We show the strict Arakelov inequality</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ {\deg f_*\omega_{X/Y}^\nu \over {{{\rm{rank\,}}}} f_*\omega_{X/Y}^\nu} < {n\nu\over 2}\cdot\deg\Omega^1_Y(\log S), $\end{document} </tex-math></disp-formula></p> <p>for all $ \nu\in \mathbb N $ such that the $ \nu $-th pluricanonical linear system $ |\omega^\nu_F| $ is birational. This answers a question asked by Möller, Viehweg and the third named author <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>.</p></abstract>

Strict Arakelov‐type inequalities for a family of curves

“…) is the slope of the pushfoward of the relative pluricanonical sheaf 𝑓 * 𝜔 ⊗𝑚 𝑋∕𝐵 . Note that Viehweg-Zuo's Arakelov inequality (2) still holds a family of semistable algebraic varieties (see Section 4 of [6] for the history of the Arakelov inequality). Furthermore, when 𝑚 = 1, they improved Equation (1) to the following Arakelov inequality:…”

“…Motivated by Theorem 1.1, Möller et al [7] asked whether the strictness of the Arakelov-type inequality (2) holds. Recently, Lu et al [6] gave a positive answer to this question, and generalize it to high-dimensional family under some extra conditions. Theorem 1.2 ([6, Theorem 1.5]).…”

“…In a series of distinguished papers [11–14], Viehweg–Zuo established the following Arakelov inequality: $$$$\begin{equation} \mu {\left(f_*\omega ^{\otimes m}_{X/B}\right)}\le \frac{m}{2}(2b-2+s), \end{equation}$$$$where $$m$$ is a positive integer and $$\mu \left(f_*\omega ^{\otimes m}_{X/B}\right)=\frac{\deg f_*\omega ^{\otimes m}_{X/B}}{\text{rank}\left(f_*\omega ^{\otimes m}_{X/B}\right)}$$ is the slope of the pushfoward of the relative pluricanonical sheaf $$f_*\omega ^{\otimes m}_{X/B}$$. Note that Viehweg–Zuo's Arakelov inequality (2) still holds a family of semistable algebraic varieties (see Section 4 of [6] for the history of the Arakelov inequality). Furthermore, when $$m=1$$, they improved Equation (1) to the following Arakelov inequality: $$$$\begin{equation} \deg f_*\omega _{X/B}\le \frac{g-u_f}{2}(2b-2+s_1), \end{equation}$$$$where …”

“…Recently, Lu et al. [6] gave a positive answer to this question, and generalize it to high‐dimensional family under some extra conditions. Theorem Let $$h: Y \rightarrow B$$ be a non‐isotrivial semistable family of $$n$$‐folds of general type, and $$A\subset h_*\omega ^{\otimes m}_{Y/B}$$ be a subbundle such that $$h^*A\rightarrow \omega ^{\otimes m}_{Y/B}$$ defines a birational $$B$$‐morphism $$\eta : Y\rightarrow \mathbb {P}_Y(A)$$.…”

“…Remark Note that the original Simpson's problem in [6] is to find $$\delta _n$$ such that $$$$\begin{equation*} \mu (A) \le \frac{mn}{2}(2b-2+s)-\delta _n. \end{equation*}$$$$Since we need the above inequality does not depend on base change, we modify it to the problem 1.1.…”