Browsing Singularity Theory and Algebraic Geometry by Title
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Combinatorial duality for Poincaré series, polytopes and invariants of plumbed 3manifolds
(201806)Assume that the link of a complex normal surface singularity is a rational homology sphere. Then its SeibergWitten invariant can be computed as the ‘periodic constant’ of the topological multivariable Poincaré series (zeta ... 
Criterion for logarithmic connections with prescribed residues
(Manucripta Mathematica, 20170401)A theorem of Weil and Atiyah says that a holomorphic vector bundle $E$ on a compact Riemann surface $X$ admits a holomorphic connection if and only if the degree of every direct summand of $E$ is zero. Fix a finite subset ... 
Equisingularity in OneParameter Families of Generically Reduced Curves
(International Mathematics Research Notices, 20160101)We explore some equisingularity criteria in oneparameter families of generically reduced curves. We prove the equivalence between Whitney regularity and Zariski’s discriminant criterion. We prove that topological triviality ... 
Equivariant motivic integration and proof of the integral identity conjecture for regular functions
(Mathematische Annalen, 20191202)We develop DenefLoeser’s motivic integration to an equivariant version and use it to prove the full integral identity conjecture for regular functions. In comparison with Hartmann’s work, the equivariant Grothendieck ... 
Euler reflexion formulas for motivic multiple zeta functions
(Journal of Algebraic Geometry, 20170514)We introduce a new notion of $\boxast$product of two integrable series with coefficients in distinct Grothendieck rings of algebraic varieties, preserving the integrability of and commuting with the limit of rational ... 
Examples of varieties with index one on C1 fields
(Journal of Number Theory, 20190416)Let K be the fraction field of a Henselian discrete valuation ring with algebraically closed residue field k. In this article we give a sufficient criterion for a projective variety over such a field to have index 1. 
General têteàtête graphs and Seifert manifolds
(20180210)Têteàtête graphs and relative têteàtête graphs were introduced by N. A’Campo in 2010 to model monodromies of isolated plane curves. By recent work of Fdez de Bobadilla, Pe Pereira and the author, they provide a way ... 
Globally subanalytic CMC surfaces in $\mathbb{R}^3$ with singularities
(Proceedings A of the Royal Society of Edinburgh, 20200302)In this paper we present a classification of a class of globally subanalytic CMC surfaces in $\mathbb{R}^3$ that generalizes the recent classification made by Barbosa and do Carmo in 2016. We show that a globally subanalytic ... 
Homogeneous singularity and the Alexander polynomial of a projective plane curve
(20171210)The Alexander polynomial of a plane curve is an important invariant in global theories on curves. However, it seems that this invariant and even a much stronger one the fundamental group of the complement of a plane curve ... 
Hölder equivalence of complex analytic curve singularities
(Bulletin of the London Mathematical Society, 20180806)We prove that if two germs of irreducible complex analytic curves at $0\in\mathbb{C}^2$ have different sequence of characteristic exponents, then there exists $0<\alpha<1$ such that those germs are not $\alpha$H\"older ... 
A jacobian module for disentanglements and applications to Mond's conjecture
(Revista Matemática Complutense, 2019)Let $f:(\mathbb C^n,S)\to (\mathbb C^{n+1},0)$ be a germ whose image is given by $g=0$. We define an $\mathcal O_{n+1}$module $M(g)$ with the property that $\mathscr A_e$$\operatorname{codim}(f)\le \dim_\mathbb C M(g)$, ... 
A LêGreuel type formula for the image Milnor number
(Hokkaido Mathematical Journal, 201902)Let $f\colon (\mathbb{C}^n,0)\to (\mathbb{C}^{n+1},0)$ be a corank 1 finitely determined map germ. For a generic linear form $p\colon (\mathbb{C}^{n+1},0)\to(\mathbb{C},0)$ we denote by $g\colon (\mathbb{C}^{n1},0)\to ... 
Logarithmic connections on principal bundles over a Riemann surface
(arxiv, 2017)Let $E_G$ be a holomorphic principal $G$bundle on a compact connected Riemann surface $X$, where $G$ is a connected reductive complex affine algebraic group. Fix a finite subset $D\, \subset\, X$, and for each $x\,\in\, ... 
Mixed têteàtête twists as monodromies associated with holomorphic function germs
(20180401)Têteàtête graphs were introduced by N. A’Campo in 2010 with the goal of modeling the monodromy of isolated plane curves. Mixed têteàtête graphs provide a generalization which define mixed têteàtête twists, which ... 
Moderately Discontinuous Algebraic Topology for Metric Subanalytic Germs
(20191031)We have developed both a homology theory and a homotopy theory in the context of metric subanalytic germs (see Definition 2.1). The former is called MD homology and is covered in Chapter 2, which contains a paper that is ... 
Monodromies as têteàtête graphs
(20180508) 
Multiplicity and degree as bi‐Lipschitz invariants for complex sets
(Journal of Topology, 20180829)We study invariance of multiplicity of complex analytic germs and degree of complex affine sets under outer biLipschitz transformations (outer biLipschitz homeomorphims of germs in the first case and outer biLipschitz ... 
Multiplicity of singularities is not a biLipschitz invariant
(Mathematische Annalen, 20200117)It was conjectured that multiplicity of a singularity is biLipschitz invariant. We disprove this conjecture constructing examples of biLipschitz equivalent complex algebraic singularities with different values of multiplicity.