稳定流形中的轨道与范畴熵：有限 Albanese 态射簇的应用
Orbits in stability manifolds and categorical entropy: applications to varieties with finite Albanese morphisms
- Waseda University（早稻田大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明具有有限 Albanese 态射的光滑射影簇满足 Gromov--Yomdin 性质，通过轨道逃逸原理和有限平展覆盖归约，并构造正范畴熵自等价，其存在性等价于非一般型。
AI中文摘要：
本文证明了具有有限 Albanese 态射的光滑射影簇满足 Gromov--Yomdin 性质。我们的方法是研究 Bridgeland 稳定流形上自等价作用下的稳定性条件轨道。对于导出范畴的一个有限生成集，我们考虑其所有对象都半稳定的轨迹。利用该轨迹，我们提出了轨道逃逸原理，该原理为自等价满足 Gromov--Yomdin 等式提供了充分条件。结合简单半齐次丛在任意数值稳定性条件下的稳定性，该原理直接给出了阿贝尔簇情形的结果。随后，我们通过适当的有限平展覆盖，将具有有限 Albanese 态射的簇情形归约到阿贝尔情形。最后，我们在每个阿贝尔簇上构造了具有正范畴熵的自等价，并证明了具有有限 Albanese 态射的光滑射影簇存在这样的自等价当且仅当它不是一般型。
英文摘要：
This paper proves the Gromov--Yomdin property for smooth projective varieties with finite Albanese morphisms. Our approach is to study the orbits of stability conditions under the action of autoequivalences on Bridgeland stability manifolds. For a finite generating set of the derived category, we consider the locus on which all its objects are semistable. Using this locus, we formulate an orbit escape principle, which provides a sufficient condition for an autoequivalence to satisfy the Gromov--Yomdin equality. Together with the stability of simple semihomogeneous bundles with respect to arbitrary numerical stability conditions, this principle readily yields the result for abelian varieties. We then reduce the case of varieties with finite Albanese morphisms to the abelian case by passing to suitable finite étale covers. Finally, we construct autoequivalences with positive categorical entropy on every abelian variety and show that a smooth projective variety with a finite Albanese morphism admits such an autoequivalence if and only if it is not of general type.
