  # "Topological, Algebraic, Differential Methods in Classification and Moduli Theory"

## Final Report Summary - TADMICAMT (Topological, Algebraic, Differential Methods in Classification and Moduli Theory)

The project n. 340258, entitled TADMICAMT: Topological, Algebraic, Differential Methods in Classification and Moduli Theory
is a mathematical project focusing on different methodologies which brought remarkable progress in the Classification Theory
of algebraic varieties and compact complex manifolds, and in the Theory of Moduli, which is here seen as the finer aspect
of classification, with targeted goal the one of finding parameters for normal forms of the algebraic varieties which are
being classified.

Classification theory is still a wide ongoing project, inspired by the classification of algebraic surfaces achieved by Castelnuovo and Enriques
in 1914, when they established the so called P 12 theorem for complex algebraic surfaces (incidentally, one contribution of our project
was to fully extend this result to the case of other algebraically closed fields).
There are some basic questions which are still open, as the existence of minimal models, and the equality between numerical Kodaira dimension and Kodaira dimension, but one can assume that these questions shall be answered in the positive in a not too distant future.

One goal of the project was to explain and clarify the roles of the several existing Moduli Theories, some studying Hilbert schemes and applying Geometric Invariant Theory or similar methods, others using transcendental methods,
fixing a basic differentiable manifold, and studying the possible complex structures, and the so called Teichmueller space.

A striking result which was obtained by Bauer, Catanese, Grunewald, was to show that for each automorphism g in the Galois group
of the field of rational numbers (not in the class of complex conjugation) there are algebraic surfaces which are g-conjugate
(this means that the equations of the second are obtained by applying g to the coefficients of the equations of the first) whose fundamental groups are not isomorphic. This result, an avatar of which was found by Serre in the 60's, goes in the direction
of solving the questions raised by Grothendieck in his program on children drawings.

This result ties in with two of the 4 themes of research of the project: Moduli of curves with symmetries, and Uniformization.

Catanese, Loenne and Perroni introduced a new homological invariant for the action of a finite group G on an algebraic curve C
(the case where the action is free had been previously solved by Dunfield and Thurston, in this case the simple invariant was
just an element of the second homology of the group)
and showed that, if the genus of the quotient curve C/G is large enough, then the invariant detects a connected component of the moduli space.

Catanese and Di Scala strengthened some old results by Kazhdan, showing that two locally symmetric manifolds which are Galois conjugate have the same universal cover, while Catanese showed that this phenomenon does not extend to other
projective classifying spaces, as for instance the Kodaira fibred surfaces.

We refer for more details to a long survey article by Catanese, `Topological methods in moduli theory', published on the Bulletin of Mathematical Sciences,
and to the Takagi lectures by Catanese, published by the Japanese Journal of mathematics.

`Topological methods in moduli theory' refers to results which say that if two manifolds are homotopy equivalent (or satisfy even some weaker topological condition), then they belong to the same connected component of the moduli space.
A particular class, of the so called Inoue-Type varieties, was introduced by Bauer and Catanese, and recent results
of Catanese and Lee made further progress, allowing maps of strictly positive degree to classifying spaces.

Some unexpected breakthrough came through work of Catanese and Dettweiler, who gave a complete proof of an important result announced by Fujita in 1979 on the subject of Variations of Hodge Structures, and who answered a question posed by Fujita in 1982 on semiampleness of direct sheaves.

These themes were discussed in several major Conferences, including the ones organized in the framework
of the TADMICAMT project, for which we refer to the webpage: