### 2-dimensionale Singularitäten und Differentialformen.

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Let $(\mathcal{S},0)$ 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 $\mathcal{S}$ 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 $E\xb7{E}_{i}\<0$ for any $i\in I$.

In this paper we study generic coverings of ${\u2102}^{2}$ 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 $n-1$.

We present a constructive proof of the fact that the set of algebraic Pfaff equations without algebraic solutions over the complex projective plane is dense in the set of all algebraic Pfaff equations of a given degree.

Let X be a germ of holomorphic vector field at the origin of Cn and vanishing there. We assume that X is a good perturbation of a “nondegenerate” singular completely integrable system. The latter is associated to a family of linear diagonal vector fields which is assumed to have nontrivial polynomial first integrals (they are generated by the so called “resonant monomials”). We show that X admits many invariant analytic subsets in a neighborhood of the origin. These are biholomorphic to the intersection...

We present a proof of embedded desingularization for closed subschemes which does not make use of Hilbert-Samuel function and avoids Hironaka's notion of normal flatness (see also [171 page 224). Given a subscheme defined by equations, we prove that embedded desingularization can be achieved by a sequence of monoidal transformations; where the law of transformation on the equations defining the subscheme is simpler then that used in Hironaka 's procedure. This is done by showing that desingularization...

We give an intersection theoretic proof of M. Soares’ bounds for the Poincaré-Hopf index of an isolated singularity of a foliation of ${\mathrm{\u2102\mathbb{P}}}^{n}$.

In this paper we prove that holomorphic codimension one singular foliations on $\u2102{\mathbb{P}}^{n},\phantom{\rule{0.277778em}{0ex}}n\ge 3$ have no non trivial minimal sets. We prove also that for $n\ge 3$, there is no real analytic Levi flat hypersurface in $\u2102{\mathbb{P}}^{n}$.

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...