arXiv subjects

Lê Dũng Tráng

Publications and source records attributed to Lê Dũng Tráng.

6 recordsLinked to original sources

arXiv2026

On the topology of complex map-germs and a general Lê-Greuel formula

Consider a singular holomorphic map-germ $f: (X,\underline{0}) \to (\mathbb C,0)$ where $X$ is a singular complex analytic variety in $\mathbb C^N$, and another holomorphic map-germ $g: (X,\underline{0}) \to (\mathbb C,0)$ which is "sufficiently good" relatively to $f$. We use stratified Morse theory to determine up to homeomorphism, the topology of the Milnor fiber $F_f$ out from the slice $F_{g,f}$ and the Morse data of a Morsification of the restriction of $g$ to $F_f$. This generalizes classical results for the case where $X$ is non-singular, and it provides a general formula comparing the Euler characteristics of $F_f$ and $F_{g,f}$. Restricting to the case where the singularity of $X$ at $\underline{0}$ is isolated, the formula for the difference of the Euler characteristics becomes algebraic and easily computable, generalizing in two directions the classical Lê-Greuel formula for the Milnor number of isolated complete intersection germs (ICIS): Firstly, $X$ can have an isolated singularity, and secondly $f$ can have arbitrary critical set. This unifies several known formulae in this vein: i) Lê-Greuel for ICIS of arbitrary codimension; ii) the formula relating the Milnor number of a curve with that of a function on it, and an extension of it for surfaces; iii) the formula for determinantal singularities; iv) and the one for the image Milnor number. All of these are special cases of our general formula.

math.AG↗

arXiv2024

Thom condition and Monodromy

We give the definition of the Thom condition and we show that given any germ of complex analytic function $f:(X,x)\to(\mathbb{C},0)$ on a complex analytic space $X$, there exists a geometric local monodromy without fixed points, provided that $f\in\mathfrak m_{X,x}^2$, where $\mathfrak m_{X,x}$ is the maximal ideal of $\mathcal O_{X,x}$. This result generalizes a well-known theorem of the second named author when $X$ is smooth and proves a statement by Tibar in his PhD thesis. It also implies the A'Campo theorem that the Lefschetz number of the monodromy is equal to zero. Moreover, we give an application to the case that $X$ has maximal rectified homotopical depth at $x$ and show that a family of such functions with isolated critical points and constant total Milnor number has no coalescing of singularities.

math.AG↗

arXiv2015

Vanishing polyhedron and collapsing map

In this paper we give a detailed proof that the Milnor fiber $X_t$ of an analytic complex isolated singularity function defined on a reduced $n$-equidimensional analytic complex space $X$ is a regular neighborhood of a polyhedron $P_t \subset X_t$ of real dimension $n-1$. Moreover, we describe the degeneration of $X_t$ onto the special fiber $X_0$, by giving a continuous collapsing map $Ψ_t: X_t \to X_0$ which sends $P_t$ to $\{0\}$ and which restricts to a homeomorphism $X_t \backslash P_t \to X_0 \backslash \{0\}$.

math.CV↗

arXiv2011

A remark on vanishing cycles with two strata

Suppose that the critical locus $Σ$ of a complex analytic function $f$ on affine space is, itself, a space with an isolated singular point at the origin $\0$, and that the Milnor number of $f$ restricted to normal slices of $Σ-\{\0\}$ is constant. Then, the general theory of perverse sheaves puts severe restrictions on the cohomology of the Milnor fiber of $f$ at $\0$, and even more surprising restrictions on the cohomology of the Milnor fiber of generic hyperplane slices.

math.AG↗

arXiv2005

Hypersurface Singularities and Milnor Equisingularity

Suppose that $f$ defines a singular, complex affine hypersurface. If the critical locus of $f$ is one-dimensional at the origin, we obtain new general bounds on the ranks of the homology groups of the Milnor fiber, $F_{f, \mathbf 0}$, of $f$ at the origin, with either integral or $\mathbb Z/p\mathbb Z$ coefficients. If the critical locus of $f$ has arbitrary dimension, we show that the smallest possibly non-zero reduced Betti number of $F_{f, \mathbf 0}$ completely determines if $f$ defines a family of isolated singularities, over a smooth base, with constant Milnor number. This result has a nice interpretation in terms of the structure of the vanishing cycles as an object in the perverse category.

math.AG↗

arXiv2005

Hypersurface Singularities and the Swing

Suppose that $f$ defines a singular, complex affine hypersurface. If the critical locus of $f$ is one-dimensional, we obtain new general bounds on the ranks of the homology groups of the Milnor fiber of $f$. This result has an interesting implication on the structure of the vanishing cycles in the category of perverse sheaves.

math.AG↗