Etant donnés () des -modules non triviaux de dimensions respectives (avec ) et un -homomorphisme, nous montrons que l’hyperdéterminant de est nul sauf si les modules sont irréductibles et si l’homomorphisme est la multiplication des polynômes homogènes à deux variables.
We prove that the exceptional complex Lie group has a transitive action on the hyperplane section of the complex Cayley plane . Although the result itself is not new, our proof is elementary and constructive. We use an explicit realization of the vector and spin actions of . Moreover, we identify the stabilizer of the -action as a parabolic subgroup (with Levi factor ) of the complex Lie group . In the real case we obtain an analogous realization of .
Let be a desingularization of a normal surface . The group Pic is provided with an order relation , defined by . for any effective exceptional divisor . Comparing to the usual order relation we define the ceiling of which is an exceptional divisor. This notion allows us to improve the usual vanishing theorem and we deduce from it a numerical criterion for rationality and a genus formula for a curve on a normal surface; the difficulty lies in the case of a Weil divisor which is not a Cartier...
A formula of Matsuo Oka (1990) expresses the Milnor number of a germ of a complex analytic map with a generic principal part in terms of the Newton polyhedra of the components of the map. In this paper this formula is generalized to the case of the index of a 1-form on a local complete intersection singularity (Theorem 1.10, Corollaries 1.11, 4.1). In particular, the Newton polyhedron of a 1-form is defined (Definition 1.6). This also simplifies the Oka formula in some particular cases (Propositions...
Using the Berline-Vergne integration formula for equivariant cohomology for torus actions, we prove that integrals over Grassmannians (classical, Lagrangian or orthogonal ones) of characteristic classes of the tautological bundle can be expressed as iterated residues at infinity of some holomorphic functions of several variables. The results obtained for these cases can be expressed as special cases of one formula involving the Weyl group action on the characters of the natural representation of...
We extend the methods developed in our earlier work to algorithmically compute the intersection cohomology Betti numbers of reductive varieties. These form a class of highly symmetric varieties that includes equivariant compactifications of reductive groups. Thereby, we extend a well-known algorithm for toric varieties.
These notes present some fundamental results and examples in the theory of algebraic group actions, with special attention to the topics of geometric invariant theory and of spherical varieties. Their goal is to provide a self-contained introduction to more advanced lectures.