We study local biholomorphisms with finite orbits in some neighborhood of the origin since they are intimately related to holomorphic foliations with closed leaves. We describe the structure of the set of periodic points in dimension 2. As a consequence we show that given a finite-orbits local biholomorphism F, in dimension 2, there exists an analytic curve passing through the origin and contained in the fixed-point set of some non-trivial iterate of $F.$ As an application we obtain that at least one eigenvalue of the linear part of F at the origin is a root of unity. Moreover, we show that such a result is sharp by exhibiting examples of finite-orbits local biholomorphisms such that exactly one of the eigenvalues is a root of unity. These examples are subtle since we show they cannot be embedded in one-parameter groups.

We say that a group G of local (maybe formal) biholomorphisms satisfies the uniform intersection property if the intersection multiplicity $(\phi (V), W)$ takes only finitely many values as a function of G for any choice of analytic sets V and W of complementary dimension. In dimension $2$ we show that G satisfies the uniform intersection property if and only if it is finitely determined – that is, if there exists a natural number k such that different elements of G have different Taylor expansions of degree k at the origin. We also prove that G is finitely determined if and only if the action of G on the space of germs of analytic curves has discrete orbits.

In a previous paper [P. Mardešić and M. Resman. Analytic moduli for parabolic Dulac germs. Russian Math. Surveys, to appear, 2021, arXiv:1910.06129v2.] we determined analytic invariants, that is, moduli of analytic classification, for parabolic generalized Dulac germs. This class contains parabolic Dulac (almost regular) germs, which appear as first-return maps of hyperbolic polycycles. Here we solve the problem of realization of these moduli.

We prove that one-parameter families of real germs of conformal diffeomorphisms tangent to the involution x ↦−x are rigid in the parameter. We establish a connection between the dynamics in the Poincaré and Siegel domains. Although repeatedly employed in the literature, the dynamics in the Siegel domain does not explain the intrinsic real properties of these germs. Rather, these properties are fully elucidated in the Poincaré domain, where the fixed points are linearizable. However, a detailed study of the dynamics in the Siegel domain is of crucial importance. We relate both points of view on the intersection of the Siegel normalization domains.

Let $X$ be a projective manifold of dimension $n$. Suppose that $T_{X}$ contains an ample subsheaf. We show that $X$ is isomorphic to $\mathbb{P}^{n}$. As an application, we derive the classification of projective manifolds containing a $\mathbb{P}^{r}$-bundle as an ample divisor by the recent work of Litt.

Consider a pseudogroup on $( \mathbb{C} , 0)$ generated by two local diffeomorphisms having analytic conjugacy classes a priori fixed in $\mathrm{Diff} \hspace{0.167em} ( \mathbb{C} , 0)$. We show that a generic pseudogroup as above is such that every point has a (possibly trivial) cyclic stabilizer. It also follows that these generic groups possess infinitely many hyperbolic orbits. This result possesses several applications to the topology of leaves of foliations, and we shall explicitly describe the case of nilpotent foliations associated to Arnold’s singularities of type ${A}^{2n+ 1} $.

In this paper we study the classification of holomorphic flows on Stein spaces of dimension two. We assume that the flow has periodic orbits, not necessarily with a same period. Then we prove a linearization result for the flow, under some natural conditions on the surface.

Our main result says that the generic rank of the Baum–Bott map for foliations of degree $d,\ d\ge 2$, of the projective plane is $d^2+d$. This answers a question of Gómez-Mont and Luengo and shows that are no other universal relations between the Baum–Bott indexes of a foliation of $\mathbb P^2$ besides the Baum–Bott formula. We also define the Camacho–Sad field for foliations on surfaces and prove its invariance under the pull-back by meromorphic maps. As an application we prove that a generic foliation of degree $d\ge 2$ is not the pull-back of a foliation of smaller degree. In Appendix A we show that the monodromy of the singular set of the universal foliation with very ample cotangent bundle is the full symmetric group.

This paper gives a characterization of valuations that follow the singular infinitely near points of plane vector fields, using the notion of L'Hôpital valuation, which generalizes a well known classical condition. With that tool, we give a valuative description of vector fields with infinite solutions, singularities with rational quotient of eigenvalues in its linear part, and polynomial vector fields with transcendental solutions, among other results.