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Fixed point varieties on the space of lattices. (English) Zbl 0779.14004

Let \(k\) be an algebraically closed field and let \(K=k((\varepsilon))\), \({\mathfrak O}=k[[\varepsilon]]\), where \(\varepsilon\) is an indeterminate. Let \(V\) be a \(K\)-vector space of dimension \(n \geq 1\) with a given volume form \(\omega\). – Let \(X\) be the set of all \({\mathfrak O}\)-lattices \({\mathcal L}\) in \(V\) such that for some \({\mathfrak O}\)-basis \(f_ 1,\ldots,f_ n\) of \({\mathcal L}\), we have \(f_ 1\wedge\ldots\wedge f_ n=\omega\). – Let \(\tilde X\) be the set of sequences of lattices \({\mathcal L}_ 0\), \({\mathcal L}_ 1,\ldots,{\mathcal L_ n}\) in \(V\) such that \({\mathcal L}_ i\) is contained with \(k\)-codimension 1 in \({\mathcal L}_{i-1}\) for \(i=1,\ldots,n\), and \({\mathcal L}_ n =\varepsilon{\mathcal L}_ 0\). (This is an affine flag manifold.) Let \(\rho:\tilde X \to X\) be the map \(\rho({\mathcal L}_ 0,\ldots,{\mathcal L}_ n)={\mathcal L}_ 0\).
Let \(T\) be an endomorphism of \(V\). Let \(X^ T\) be the set of all \({\mathcal L} \in X\) such that \(T({\mathcal L}) \subset{\mathcal L}\); let \(\tilde X^ T\) be the set of all \(({\mathcal L}_ 0,\ldots,{\mathcal L}_ n) \in \tilde X\) such that \(T({\mathcal L}_ i) \subset{\mathcal L}_ i\) for all \(i\). We say that \(T\) is nil-elliptic if \(\lim_{m \to \infty}T^ m=0\) and the characteristic polynomial of \(T\) is irreducible over \(K\). – Let \(e_ 1,\ldots,e_ n\) be a basis of \(V\) such that \(e_ 1\wedge\cdots\wedge e_ n=\omega\). Let \(N:V \to V\) be the \(K\)-linear transformation given by \(N(e_ i)=e_{i- 1}\) for \(i=2,\ldots,n\) and \(N(e_ 1)= \varepsilon e_ n\). Let \(t\) be an integer \(\geq 1\) which is relatively prime to \(n\); we write \(t=rn+s\), where \(r\) is an integer \(\geq 0\) and \(0<s<n\). The \(t\)-th power \(N^ t\) of \(N\) is given by \(N^ t(e_ i)=\varepsilon^ re_{i-s}\) if \(s<i \leq n\), \(N^ t(e_ i)=\varepsilon^{r+1}e_{i-s+n}\) if \(1 \leq i \leq s\). We shall take \(T=N^ t\); this is a nil-elliptic endomorphism. We set \(Y=X^ T\), \(\tilde Y=\tilde X^ T\); as above, we have a morphism \(\rho':\tilde Y\to Y\). Main result:
(i) The Euler characteristics of \(Y\), \(\tilde Y\) are given respectively by \((n+t-1)!(n!)^{-1}(t!)^{-1}\), \(t^{n-1}\).
(ii) The varieties \(Y\), \(\tilde Y\) can be stratified into finitely many strata each of which is isomorphic to an affine space over \(k\). In particular, their étale cohomology vanishes in odd degrees.

MSC:

14F45 Topological properties in algebraic geometry
14N99 Projective and enumerative algebraic geometry
14F17 Vanishing theorems in algebraic geometry
14M15 Grassmannians, Schubert varieties, flag manifolds
14E99 Birational geometry
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