AI 中文总结
本文通过第一jet丛和切几何，证明了极化Calabi-Yau三维簇的Chern数二次不等式，改进一致界，并研究切次数与切簇正规化态射。
AI 中文摘要
我们通过第一jet丛、切几何和射影对偶性，为极化Calabi-Yau三维簇的数值地理学带来了新的见解。设$X$为一个具有非常丰沛极化$H$的Calabi-Yau三维簇。我们利用第一jet丛的射影对偶上的混合交截，证明了关于次数$d=H^3$和Chern数的二次不等式。将该不等式与超平面截面和切簇估计相结合，得到了改进的一致界$-4d-80\le h^{1,1}(X)-h^{2,1}(X)\le\frac{173}{66}d$。对于$\mathbb{P}^6$中的非退化嵌入，我们将次数的上界从$41$改进到$39$，并将切次数表示为$d$的二次多项式。我们还证明了，对于每个$m\ge2$，与$|mH|$相关的切-关联态射是切簇的正规化态射。对于$m=1$，我们猜想在$N\ge7$的$\mathbb{P}^N$中完全嵌入的切次数为$1$，并验证了若干族，包括四个二次曲面的一般交和一般的GPK$^3$三维簇。
英文摘要
We study the numerical geography and tangent geometry of very amply polarized Calabi--Yau threefolds $(X,H)$ through the positivity of the first jet bundle $J^1(H)$. Writing $d=\int_X H^3$, $c=\int_Xc_2(X) H$, and $e=\int_X c_3(X)$, we exploit two different positivity properties of this single bundle. Mixed intersections on $\mathbb{P}(J^1(H)^*)$ give $e\ge-5d-c-c^2/(4d)$, while a volume estimate for a perturbed tautological class gives $e\ge40d\left[(1-\frac{c}{10d})^{3/2}-1\right]$; in particular $e+6c\ge0$, improving Sun's inequality $e+10c\ge0$. As consequences, we obtain the uniform Hodge bounds $-4d-80\le h^{1,1}(X)-h^{2,1}(X)\le173d/66$, the lower bound $\mathrm{deg}X^\vee\ge78$ for the dual hypersurface, and, in the critical case $X\subset\mathbb{P}^6$, the upper bound $d\le34$. We also prove that, for every $m\ge2$, the tangent-incidence morphism associated with $|mH|$ is the normalization of the tangent variety, conjecture tangent birationality for complete embeddings $X\subset\mathbb{P}^N$ with $N\ge7$, and verify it for several families.
