Monodromy and normal forms

We discuss the history of the monodromy theorem, starting from Weierstrass, and the concept of monodromy group. From this viewpoint we compare then the Weierstrass , the Legendre and other normal forms for elliptic curves, explaining their geometric meaning and distinguishing them by their stabilizer in P SL(2,Z) and their monodromy. Then we focus on the birth of the concept of the Jacobian variety, and the geometrization of the theory of Abelian functions and integrals. We end illustrating the methods of complex analysis in the simplest issue, the difference equation $f(z) = g(z+1) - g(z)$ on $mathbb C$.

On the unirationality of higher dimensional Ueno-type manifolds

We prove the unirationality of the Ueno-type manifold $X_{4,6}$. $X_{4,6}$ is the minimal resolution of the quotient of the Cartesian product $E(6)^4$, where $E(6)$ is the equianharmonic elliptic curve, by the diagonal action of a cyclic group of order 6 (having a fixed point on each copy of $E(6)$). We collect also other results, and discuss several related open questions.

Higher dimensional Lemniscates: the geometry of $r$ particles in $n$-space with logarithmic potentials

We prove some basic theorems concerning lemniscate configurations in an Euclidean space of dimension $ n geq 3$. Lemniscates are defined as follows. Given m points $w_j $ in $mathbb R^n$, consider the function $F(x)$ which is the product of the distances $ |x-w_j|$: the singular level sets of the function $F$ are called lemniscates. We show via complex analysis that the critical points of $F$ have Hessian of positivity at least $(n-1)$. This implies that, if $F$ is a Morse function, then $F$ has only local minima and saddle points with negativity 1. The critical points lie in the convex span of the points $|w_j| $ (these are absolute minima): but we made also the discovery that $F$ can also have other local minima, and indeed arbitrarily many. We discuss several explicit examples. We finally prove in the appendix that all critical points are isolated.

Topological methods in moduli theory

One of the main themes of this long article is the study of projective varieties which are K(H,1)s, i.e. classifying spaces BH for some discrete group H. After recalling the basic properties of such classifying spaces, an important class of such varieties is introduced, the one of Bagnera-de Franchis varieties, the quotients of an Abelian variety by the free action of a cyclic group. Moduli spaces of Abelian varieties and of algebraic curves enter into the picture as examples of rational K(H,1)s, through Teichmueller theory. The main thrust of the paper is to show how in the case of K(H,1)s the study of moduli spaces and deformation classes can be achieved through by now classical results concerning regularity of classifying maps. The Inoue type varieties of Bauer and Catanese are introduced and studied as a key example, and new results are shown. Motivated from this study, the moduli spaces of algebraic varieties, and especially of algebraic curves with a group of automorphisms of a given topological type are studied in detail, following new results by the author, Michael Loenne and Fabio Perroni. Finally, the action of the absolute Galois group on the moduli spaces of such K(H,1) varieties is studied. In the case of surfaces isogenous to a product, it is shown how this yields a faifhtul action on the set of connected components of the moduli space: for each Galois automorphisms of order different from 2 there is a surface S such that the Galois conjugate surface of S has fundamental group not isomorphic to the one of S.

A characterization of varieties whose universal cover is a bounded symmetric domain without ball factors

We give two characterizations of varieties whose universal cover is a bounded symmetric domain without ball factors in terms of the existence of a holomorphic endomorphism s of the tensor product Totimes T of the tangent bundle T with the cotangent bundle T. To such a curvature type tensor s one associates the first Mok characteristic cone S, obtained by projecting on T the intersection of ker (s) with the space of rank 1 tensors. The simpler characterization requires that the projective scheme associated to S be a finite union of projective varieties of given dimensions and codimensions in their linear spans which must be skew and generate.

Quotients of products of curves, new surfaces with $p_g=0$ and their fundamental groups

The first main purpose of this paper is to contribute to the existing knowledge about the complex projective surfaces $S$ of general type with $p_g(S) = 0$ and their moduli spaces, constructing 19 new families of such surfaces with hitherto unknown fundamental groups. We also provide a table containing all the known such surfaces with K^2 <=7. Our second main purpose is to describe in greater generality the fundamental groups of smooth projective varieties which occur as the minimal resolutions of the quotient of a product of curves by the action of a finite group. We classify, in the two dimensional case, all the surfaces with q=p_g = 0 obtained as the minimal resolution of such a quotient, having rational double points as singularities. We show that all these surfaces give evidence to the Bloch conjecture.