arXiv subjects

Nivaldo Grulha

Publications and source records attributed to Nivaldo Grulha.

3 recordsLinked to original sources

arXiv2026

Divisorial persistent homology and volume asymptotics of analytic pairs

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.

math.AG↗

arXiv2026

Geometric Criteria for Morrey Admissibility via the Real Log-Canonical Threshold

We study local integrability of gradients of singular interaction kernels in aggregation equations. Suppose near the singularity that $|\nabla K|\asymp|\nabla f|\,|f|^{-(κ+1)}$, where $κ>0$ and $f$ is real-analytic with $f(0)=0$. A real log-resolution of $f$ and its Jacobian ideal gives the exact criterion $\nabla K\in L^p_{\mathrm{loc}}$ if and only if $p 1$, a suitable global truncation gives $\nabla K\in L^1(\mathbb R^n)$, yielding the kernel hypothesis in the Morrey-space well-posedness theorem of \cite{suleiman2023existence}.

math.AP↗

arXiv2026

On the Divisorial Geometry of Volume Asymptotics of Sublevel Sets

The log canonical threshold (LCT) is a fundamental invariant in birational geometry and singularity theory, measuring the complexity of an analytic singularity through discrepancy and valuation data on a log resolution. In this work we investigate the asymptotic behaviour of sublevel-set volumes associated with principal analytic ideals, equivalently with holomorphic function germs. Building on the classical theory of local zeta functions and Mellin asymptotics, we introduce the visible spectrum, the set of actual poles of the local zeta function, and show that it is determined by the asymptotic expansion of the volume. Conversely, we prove that this spectrum, together with its multiplicities and coefficients, can be recovered recursively from the volume asymptotics by an explicit reconstruction procedure. We also give complementary interpretations in terms of arc spaces, where the divisorial exponents appear both as ratios of vanishing orders along generic divisorial arcs and as normalized codimension growth rates of divisorial cylinders. Taken together, these results establish an explicit correspondence between the visible spectrum and the asymptotic expansion of sublevel-set volumes, providing an intrinsic metric characterization of the visible spectrum itself.

math.AG↗