半dlt对的相对对数典范代数的有限生成
Finite generation of relative log canonical algebras of semi-dlt pairs
- Brown University（布朗大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文直接证明了半dlt对的相对对数典范代数的有限生成，避免Kollár粘合理论，并由此恢复并推广了相关MMP结果，为稳定对模空间固有性证明奠定基础。
AI中文摘要：
我们证明，在概形$T$上射影的半dlt对$(X,\Delta_X)$的相对对数典范代数是有限生成的，等价地，$(X,\Delta_X)$承认相对稳定模型，前提是：其正规化在$T$上承认对数典范模型；$(X,\Delta_X)$在$T$的稠密开子概形$T^0\subset T$上承认稳定模型，且其例外轨迹不包含导子的任何层；包含于导子中的对数中心在$T^0$中的像有交集。我们的主要贡献是一个直接证明，完全避免了Kollár的粘合理论。作为推论，我们恢复了Hacon--Xu和Birkar结果的半正规版本。此外，我们推导了slc对的某些MMP步骤的存在性，恢复了Ambro和Kollár的结果。最后，这些结果为稳定对的Kollár--Shepherd-Barron--Alexeev模空间固有性的简化证明奠定了基础，该证明将在后续工作中完成。
英文摘要：
We prove that the relative log canonical algebra of a semi-dlt pair $(X,Δ_X)$ projective over a scheme $T$ is finitely generated, equivalently, that $(X,Δ_X)$ admits a relative stable model, provided that the normalization admits a log canonical model over $T$, that $(X,Δ_X)$ admits a stable model over an open dense subscheme $T^0\subset T$ whose exceptional locus does not contain any stratum of the conductor, and that the log centers contained in the conductor have images meeting $T^0$. Our main contribution is a direct proof that avoids Kollár's gluing theory entirely. As a consequence, we recover demi-normal versions of results of Hacon--Xu and Birkar. Furthermore, we derive the existence of certain MMP steps for slc pairs, recovering results of Ambro and Kollár. Finally, these results lay the groundwork for a streamlined proof of the properness of the Kollár--Shepherd-Barron--Alexeev moduli space of stable pairs, to be completed in forthcoming work.
