arXiv subjects
Publications and source records attributed to Francesca Tovena.
We construct Atiyah classes using debar-closed forms. Under this point of view and using the Cech-Dolbeault cohomology, we provide several types of results about vanishing and localization of Atiyah classes and applications.
We first study the dynamics of the geodesic flow of a meromorphic connection on a Riemann surface, and prove a Poincaré-Bendixson theorem describing recurrence properties and $ω$-limit sets of geodesics for a meromorphic connection on $¶^1(\C)$. We then show how to associate to a homogeneous vector field $Q$ in ${\Bbb C}^n$ a rank 1 singular holomorphic foliation $\cal F$ of $¶^{n-1}(\C)$ and a (partial) meromorphic connection $\nabla^o$ along $\ca F$ so that integral curves of $Q$ are described by the geodesic flow of $\nabla^o$ along the leaves of $\ca F$, which are Riemann surfaces. The combination of these results yields powerful tools for a detailed study of the dynamics of homogeneous vector fields. For instance, in dimension two we obtain a description of recurrence properties of integral curves of $Q$, and of the behavior of the geodesic flow in a neighbourhood of a singularity, classifying the possible singularities both from a formal point of view and (for generic singularities) from a holomorphic point of view. We also get examples of unexpected new phenomena, we put in a coherent context scattered results previously known, and we obtain (as far as we know for the first time) a complete description of the dynamics in a full neighbourhood of the origin for a substantial class of 2-dimensional holomorphic maps tangent to the identity. Finally, as an example of application of our methods we study in detail the dynamics of quadratic homogeneous vector fields in $\C^2$.
This paper is devoted to the study of the embeddings of a complex submanifold $S$ inside a larger complex manifold $M$; in particular, we are interested in comparing the embedding of $S$ in $M$ with the embedding of $S$ as the zero section in the total space of the normal bundle $N_S$ of $S$ in $M$. We explicitely describe some cohomological classes allowing to measure the difference between the two embeddings, in the spirit of the work by Grauert, Griffiths, and Camacho-Movasati-Sad; we are also able to explain the geometrical meaning of the separate vanishing of these classes. Our results holds for any codimension, but even for curves in a surface we generalize previous results due to Laufert and Camacho-Movasati-Sad.
We describe a general construction providing index theorems localizing the Chern classes of the normal bundle of a subvariety inside a complex manifold. As particular instances of our construction we recover both Lehmann-Suwa's generalization of the classical Camacho-Sad index theorem for holomorphic foliations and our index theorem for holomorphic maps with positive dimensional fixed point set. Furthermore, we also obtain generalizations of recent index theorems of Camacho-Movasati-Sad and Camacho-Lehmann for holomorphic foliations transversal to a subvariety.
Let $M$ be a complex manifold and $S\subset M$ a (possibly singular) subvariety of $M$. Let $f\colon M\to M$ be a holomorphic map such that $f$ restricted to $S$ is the identity. We show that one can associate to $f$ a holomorphic section $X_f$ of a sheaf related to the embedding of $S$ in $M$ and that such a section reads the dynamical behavior of $f$ along $S$. In particular we prove that under generic hypotheses the canonical section $X_f$ induces a holomorphic action in the sense of Bott on the normal bundle of (the regular part of) $S$ in $M$ and this allows to obtain for holomorphic self-maps with non- isolated fixed points index theorems similar to Camacho-Sad, Baum-Bott and variation index theorems for holomorphic foliations. Finally we apply our index theorems to obtain information about topology and dynamics of holomorphic self-maps of surfaces with a compact curve of fixed points.
We describe a procedure for constructing formal normal forms of holomorphic maps with a hypersurface of fixed points, and we apply it to obtain a complete list of formal normal forms for 2-dimensional holomorphic maps tangential to a curve of fixed points.
We construct three sequences of regular surfaces of general type with unbounded numerical invariants whose canonical map is 2-to-1 onto a canonically embedded surface. Only sporadic examples of surfaces with these properties were previously known.
We study the behaviour of the topological fundamental group under totally ramified abelian covers (a special case of abelian Galois covers) of complex projective varieties of dimension at least 2.