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Register a new userTopological methods in moduli theory
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Authors
Fabrizio Catanese
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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.
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Kodaira fibred surfaces are a remarkable example of projective classifying spaces, and there are still many intriguing open questions concerning them, especially the slope question. The topological characterization of Kodaira fibrations is emblematic of the use of topological methods in the study of moduli spaces of surfaces and higher dimensional complex algebraic varieties, and their compactifications. The paper contains some new results but is mostly a survey paper, dealing with fibrations, questions on monodromy and factorizations in the mapping class group, old and new results on Variation of Hodge Structures, especially a recent answer given (in joint work with Dettweiler) to a long standing question posed by Fujita. In the landscape of our tour, Galois coverings, deformations and rigid manifolds (new results obtained with Ingrid Bauer) projective classifying spaces, the action of the absolute Galois group on moduli spaces, stand also in the forefront. These questions lead to interesting algebraic surfaces, for instance the BCDH surfaces, hypersurfaces in Bagnera-de Franchis varieties, Inoue-type surfaces.
We give a characterizaton of smooth ample Hypersurfaces in Abelian Varieties and also describe an irreducible connected component of their moduli space: it consists of the Hypersurfaces of a given polarization type, plus the iterated univariate coverings of normal type (of the same polarization type). The above manifolds yield also a connected component of the open set of Teichmuller space consisting of Kahler complex structures.
An increasingly important area of interest for mathematicians is the study of Abelian differentials. This growing interest can be attributed to the interdisciplinary role this subject plays in modern mathematics, as various problems of algebraic geometry, dynamical systems, geometry and topology lead to the study of such objects. It comes as a natural consequence that we can employ in our study algebraic, analytic, combinatorial and dynamical perspectives. These lecture notes aim to provide an expository introduction to this subject that will emphasize the aforementioned links between different areas of mathematics. We will associate to an Abelian differential a flat surface with conical singularities such that the underlying Riemann surface is obtained from a polygon by identifying edges with one another via translation. We will focus on studying these objects in families and describe some properties of the orbit as we vary the polygon by the action of $GL_2^{+}(mathbb{R})$ on the plane.
We construct a compactification of the moduli spaces of abelian differentials on Riemann surfaces with prescribed zeroes and poles. This compactification, called the moduli space of multi-scale differentials, is a complex orbifold with normal crossing boundary. Locally, our compactification can be described as the normalization of an explicit blowup of the incidence variety compactification, which was defined in [BCGGM18] as the closure of the stratum of abelian differentials in the closure of the Hodge bundle. We also define families of projectivized multi-scale differentials, which gives a proper Deligne-Mumford stack, and our compactification is the orbifold corresponding to it. Moreover, we perform a real oriented blowup of the unprojectivized moduli space of multi-scale differentials such that the $mathrm{SL}_2(mathbb R)$-action in the interior of the moduli space extends continuously to the boundary.
We consider the family $mathrm{MP}_d$ of affine conjugacy classes of polynomial maps of one complex variable with degree $d geq 2$, and study the map $Phi_d:mathrm{MP}_dto widetilde{Lambda}_d subset mathbb{C}^d / mathfrak{S}_d$ which maps each $f in mathrm{MP}_d$ to the set of fixed-point multipliers of $f$. We show that the local fiber structure of the map $Phi_d$ around $bar{lambda} in widetilde{Lambda}_d$ is completely determined by certain two sets $mathcal{I}(lambda)$ and $mathcal{K}(lambda)$ which are subsets of the power set of ${1,2,ldots,d }$. Moreover for any $bar{lambda} in widetilde{Lambda}_d$, we give an algorithm for counting the number of elements of each fiber $Phi_d^{-1}left(bar{lambda}right)$ only by using $mathcal{I}(lambda)$ and $mathcal{K}(lambda)$. It can be carried out in finitely many steps, and often by hand.
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