解析对的除子持续性与渐近同调
Divisorial persistent homology and volume asymptotics of analytic pairs
AI总结：
该研究为解析对引入除子渐近同调（DAH），它定义协变函子区分三层渐近信息，建立多种性质，证明正规解析曲面芽的刚性定理，通过例子表明DAH能检测特殊渐近拓扑信息，且高维中除子临界性与同调持续性可能不同。
AI中文摘要：
我们为解析对\((X,\mathcal{I})\)引入了“除子渐近同调”（DAH），这是一种由内在解析能量\(K_{\mathcal{I}}=\sum_j|f_j|^2\)的收缩子水平集内豪斯多夫测度的渐近浓度所支配的同调理论。DAH定义了一个到持久分次阿贝尔群范畴的协变函子，并区分了三层渐近信息：实对数典范阈值、内在除子谱和同调谱，后者支配持续性并记录可允许圈实现的渐近浓度率。可允许滤过是一个驯服的持久模，其临界值恰好是同调谱的有限多个元素。我们建立了函子性、相对长正合序列、迈耶 - Vietoris序列和双有理不变性。对于正规解析曲面芽，我们证明了一个刚性定理，将临界DAH群与临界加权对偶分辨率图的同调等同起来。例子表明DAH能检测到实对数典范阈值不可见的渐近拓扑信息，并且在高维中除子临界性和同调持续性可能会发散。
英文摘要：
A log-resolution of an analytic pair $(X,I)$ attaches to every prime divisor $E$ of its total transform a vanishing order $ν_E$, a Jacobian order $a_E$, and hence a divisorial exponent $γ_E=(a_E+1)/(2ν_E)$. The smallest exponent is one half of the real log canonical threshold and governs the volume of the sublevel sets $\{\sum |f_j|^2 \le \varepsilon\}$; the multiplicity of the logarithmic correction is a second intrinsic invariant. We organize all the exponents into two persistence modules and study what survives when the resolution is changed. The first module lives on $X$: each compact subanalytic chain receives a valuative threshold, the infimum of the normalized log discrepancy $A(v)/(2v(I))$ over the divisorial valuations reaching it; its superlevel sets filter the chain complex. We prove that the resulting module is the persistent homology of the complements of the sublevel sets of the local threshold; tameness, Mayer-Vietoris, independence of the resolution, and invariance under bi-Lipschitz subanalytic homeomorphisms preserving the energy up to equivalence follow. The second lives on the resolution: the dual complex of the total transform, filtered by the exponents. An admissible blow-up changes each sublevel complex by a stellar subdivision or by attaching a cone over a contractible subcomplex, the only input being a mediant inequality for the exponent of the new divisor. Under an explicit factorization hypothesis (unconditional for surface germs, expected from relative weak factorization in the complex algebraic case) the persistent homology is independent of the resolution. For normal surface singularities it is the persistent homology of the weighted dual graph, computed for the $A_n$ points. The two modules share the first critical value and the list of exponents, have opposite persistence directions, and are different shadows of the same data.
