On the geometry of varieties of invertible symmetric and skew-symmetric matrices
Let Sym\((n,K)\) and Sk\((n, K)\) denote the sets of \(n\times n\) invertible symmetric and skew-symmetric matrices over a field \(K\) respectively. Let GL\((n, K)\) be the set of \(n\times n\) invertible matrices over \(K\) and define an action GL\((n, K)\times\text{Sym}(n, K)\to\text{Sym}(n, K)\) by \(g\cdot A= gA^t g\). An action of GL\((n, K)\) on Sk\((n, K)\) is defined in an analogous way. The first results presented in this paper are that the homotopy type of Sym\((n,\mathbb{R})\) can be determined by fibering the orbits over the Grassmann variety and then showing that the fibres are contractible and that the Betti numbers of the orbits of Sk\((n, \mathbb{R})\) can be determined by fibering them over the sphere and using the associated Wang exact sequence. We also show how these results can be derived using Iwasawa decompositon and theorems of Borel and Ehresmann. Deligne showed that any complex algebraic variety \(X\) has a canonical mixed Hodge structure, i.e. a finite increasing filtration \(W\) on \(H^i (X; \mathbb{Q})\), called the weight filtration, and a finite decreasing filtration \(F\) on \(H^i (X; \mathbb{C})\), called the Hodge filtration such that the filtration induced by \(F\) on \(\text{Gr}^W_k H^i (X; \mathbb{C})\) is a Hodge structure of weight \(k\). Using the weight filtration an invariant for the variety can be defined known as the weight polynomial. The next result is that the weight polynomial of Sk\((n, \mathbb{C})\) is determined by applying a theorem of Dimca and Lehrer to the fibration \[ \text{Sp}(n, \mathbb{C}) \hookrightarrow\text{GL}(n, \mathbb{C})\to\text{Sk}(n, \mathbb{C}). \] Let Sym\((i, n, K)\) denote the variety of \(n\times n\) symmetric matrices over \(K\) with rank \(i\) and \(G(k, n, K)\) the Grassmannian variety of \(k\)-dimensional subspaces of \(K^n\). An inductive formula for the weight polynomial of Sym\((n, \mathbb{C})\) is found by using the fibration \[ \text{Sym}(i, \mathbb{C}) \hookrightarrow\text{Sym}(i, n, \mathbb{C})\to G(n-i, n, \mathbb{C}) \] which is then solved to give an explicit result. For general complex algebraic varieties a relationship between the number of \(\mathbb{F}_q\)-rational points of their reduction modulo \(q\) and the weight \(m\) Euler characteristic of their \(\mathbb{C}\)-rational points can be found using results of Deligne and the Grothendieck-Verdier fixed point theorem. We conclude by calculating the number of \(\mathbb{F}_q\)-rational points of the varieties considered in this article, thus giving an alternative method for computing their weight polynomials. We note that for these varieties there is also an interesting relationship between the compact Euler characteristics of the \(\mathbb{R}\)-rational points and the weight polynomials of the \(\mathbb{C}\)-rational points, namely \[ \chi_c (X (\mathbb{R}))= W_c (X (\mathbb{C}), i). \] At the moment, this relationship is not well understood.
scientific article; zbMATH DE number 950200