Let $(𝒮,0)$ be a germ of complex analytic normal surface. On its minimal resolution, we consider the reduced exceptional divisor $E$ and its irreducible components $E_i$, $i\in I$. The Nash map associates to each irreducible component $C_k$ of the space of arcs through $0$ on $𝒮$ the unique component of $E$ cut by the strict transform of the generic arc in $C_k$. Nash proved its injectivity and asked if it was bijective. As a particular case of our main theorem, we prove that this is the case if $E·E_i\<0$ for any $i\in I$.

In this paper we study generic coverings of ${ℂ}^2$ branched over a curve s.t. the total space is a normal analytic surface, in terms of a graph representing the monodromy of the covering, called monodromy graph. A complete description of the monodromy graphs and of the local fundamental groups is found in case the branch curve is $\{x^n=y^m\}$ (with $n\le m$) and the degree of the cover is equal to $n$ or $n-1$.

Let f be a complex polynomial. We relate the behaviour of f “at infinity” to the sheaf of vanishing cycles of the family $\overline{f}$ of projective closures of fibres of f. We show that the absence of such cycles: (i) is equivalent to a condition on the asymptotic behaviour of gradient of f known as Malgrange’s Condition, (ii) implies the $C^{\infty }$-triviality of f. If the support of sheaf of vanishing cycles of $\overline{f}$ is a finite set, then it detects precisely the change of the topology of the fibres of f. Moreover, in...

To a germ $f:({ℂ}^n,0)\to (ℂ,0)$ with one-dimensional singular locus one associates series of isolated singularities $f_N:=f+l^N$, where l is a general linear function and $N\in$. We prove an attaching result of Iomdin-Lê type which compares the homotopy types of the Milnor fibres of $f_N$ and f. This is a refinement of the Iomdin-Lê theorem in the general setting of a singular underlying space.

Let $(V,0)$ be a germ of a complete intersection variety in ${ℂ}^{n+k}$, $n\>0$, having an isolated singularity at $0$ and $X$ be the germ of a holomorphic vector field having an isolated zero at $0$ and tangent to $V$. We show that in this case the homological index and the GSV-index coincide. In the case when the zero of $X$ is also isolated in the ambient space ${ℂ}^{n+k}$ we give a formula for the homological index in terms of local linear algebra.

Some relations between normal complex surface singularities and symplectic fillings of the links of the singularities are discussed. For a certain class of singularities of general type, which are called hypersurface K3 singularities in this paper, an inequality for numerical invariants of any minimal symplectic fillings of the links of the singularities is derived. This inequality can be regarded as a symplectic/contact analog of the 11/8-conjecture in 4-dimensional topology.