The Noether map II

Let be a faithful representation of a finite group . In this paper we proceed with the study of the image of the associated Noether map

In our 2005 paper it has been shown that the Noether map is surjective if is a projective -module. This paper deals with the converse. The converse is in general not true: we illustrate this with an example. However, for -groups (where is the characteristic of the ground field ) as well as for permutation representations of any group the surjectivity of the Noether map implies the projectivity of .

     

The combinatorial map color theorem

This paper is written in the spirit of the author's book: Map Color Theorem (1974). We try to develop the Map Color Theorem in a combinatorial way, circumventing the unwieldy embedding theory. Similar (but not identical) generalizations have recently and independently been developed by Alpert (in press) and by Stahl (in press). The first nine theorems are one-dimensional versions of known facts from the theory of two-dimensional compact manifolds. Theorems 10 to 13 are to my knowledge completely new results.     

The character map in deformation quantization

The third author recently proved that the Shoikhet–Dolgushev L_∞-morphism from Hochschild chains of the algebra of smooth functions on a manifold to differential forms extends to cyclic chains. Localization at a solution of the Maurer–Cartan equation gives an isomorphism, which we call character map, from the periodic cyclic homology of a formal associative deformation of the algebra of functions to de Rham cohomology. We prove that the character map is compatible with the Gauss–Manin connection, extending a result of Calaque and Rossi on the compatibility with the cap product. As a consequence, the image of the periodic cyclic cycle 1 is independent of the deformation parameter and we compute it to be the A-roof genus of the manifold. Our results also imply the Tamarkin–Tsygan index theorem.     

The image of the transfer map

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The period map for cubic fourfolds

The period map for cubic fourfolds takes values in a locally symmetric variety of orthogonal type of dimension 20. We determine the image of this period map (thus confirming a conjecture of Hassett) and give at the same time a new proof of the theorem of Voisin that asserts that this period map is an open embedding. An algebraic version of our main result is an identification of the algebra of SL (6,ℂ)-invariant polynomials on the representation space Sym ³(ℂ⁶)^* with a certain algebra of meromorphic automorphic forms on a symmetric domain of orthogonal type of dimension 20. We also describe the stratification of the moduli space of semistable cubic fourfolds in terms of a Vinberg-Dynkin diagram.     

The time operator of the cusp map

We construct the natural time operator for the discrete dynamical system generated by the cusp map and give explicitly the age eigenfunctions which provide a shift representation of the evolution allowing for probabilistic prediction. In contradistinction, the construction of the spectral decomposition of the cusp map is not an easy problem.     

The boundary map and the connecting orbits

The purpose of this article is to show that the image of the homological boundary map attached to a filtration for an attractor-repeller pair of a smooth flow on a compact manifold is a submodule of the Alexander cohomology of certain order of the connecting set (some restrictions have to be imposed in order to have a valid argument). In particular, this gives an affirmative answer to a conjecture in Conley index theory which states that if the boundary map is not zero in two dimensions, the connecting set cannot be contractible.