On the formal arc space of a reductive monoid (Q2800422)

For an algebraic variety \(X\) over a field \(k\), the arc space \(\mathcal{L}X\), introduced by J. Nash, bears essential information about its singular locus \(\mathrm{Sing}(X)\). About its own singular locus, how it should be defined, and what properties would it have is not known much (see [\textit{A. J. Reguera}, Am. J. Math. 131, No. 2, 313--350 (2009; Zbl 1188.14010)]). By the theorem of Grinberg-Kazhdan-Drinfeld [\textit{M. Grinberg} and \textit{D. Kazhdan}, Geom. Funct. Anal. 10, No. 3, 543--555 (2000; Zbl 0966.14002)]; [\textit{V. Drinfeld}, ``On the Grinberg-Kazhdan formal arc theorem'', Preprint, \url{arXiv:0203263}] the formal neighborhood of a closed arc, whose image is not contained in \(\mathrm{Sing}(X)\), is isomorphic to \(Y_y \hat{\times} \mathbb{D}^{\infty}\) for some scheme of finite type \(Y\) with a point \(y \in Y(k)\), its formal neighborhood \(Y_y\) is called a finite dimensional formal model of \(\mathcal{L}X\) at \(x\), and \(\mathbb{D}\) is the formal disc. One can try to define the intersection complex \(IC_{\mathcal{L}X}\) using these local finite dimensional models, in a way that it could be seen as a measure how singular the arc space is.NEWLINENEWLINEIn the article under review, after modifying the usual definition of the intersection complex for a scheme of finite type, is shown that the trace of Frobenius function on the intersection complex is well defined on the space of non-degenerate closed arcs (see bellow). The main result is the calculation of this function in the cases when \(X\) is toric variety, or \(X\) belongs to a class of reductive monoids, called \(L\)-monoids. In the latter case also is proved a conjecture of Braverman and Kazhdan, claiming that this function is the generating series of the local unramified \(L\)-function for the irreducible representation of the dual group with highest weight determining the isomorphism class of \(X\) [\textit{A. Braverman} et al., in: GAFA 2000. Visions in mathematics -- Towards 2000. Proceedings of a meeting, Tel Aviv, Israel, August 25--September 3, 1999. Part I. Basel: Birkhäuser. 237--278 (2000; Zbl 1004.11026)] and [\textit{B. C. Ngô}, Contemp. Math. 614, 337--343 (2014; Zbl 1298.11041)]NEWLINENEWLINELet \(X\) be an integral scheme over \(k\), take \(X^o = X \setminus Z\) to be smooth open dense subset, and call \(\mathcal{L}^{o}X(k) = \mathcal{L}X(k) \setminus \mathcal{L}Z(k)\) the space of the non-degenerate arcs with respect to \(X^o\). Let \(i_r: U_r \hookrightarrow X\) be the natural inclusion for any connected component \(U_r\) of \(X^o\). The intersection complex of \(X\) is defined as \(IC_X = \bigoplus_r i_{r,!*}\mathbb{Q}_l \). The normalization is different from the usual one [\textit{A. A. Beilinson} et al., Faisceaux pervers. Astérisque 100, 172 p. (1982; Zbl 0536.14011)], and does not have the expected properties with the Verdier duality, but is needed when working with infinite dimensional schemes. If \(k\) is finite, the trace of Frobenius operator on a stalk of \(IC_X\) defines a function denoted again \(IC_X : X(k) \rightarrow \mathbb{Q}_l\), which takes value 1 on the \(k\)-points of \(X^o\), and satisfies the following property: if the formal neighborhood of a closed arc is represented in two ways as a product, using two schemes of finite type \((Y,y)\) and \((Y',y')\), then \(IC_Y(y) = IC_Y'(y')\). Thus we have a well defined function on the space of non-degenerate arcs \(IC_{\mathcal{L}X} : \mathcal{L}^oX(k) \rightarrow \mathbb{Q}_l\).NEWLINENEWLINESuppose \(X\) is affine normal variety with an open embedding of reductive group \(G \hookrightarrow X\) and action \(G \times G \curvearrowright X\) which extends the action on \(G\) by left and right multiplication. This makes \(X\) a monoid, and let us take \(X^o = G\). Then it is possible to construct a finite dimensional formal model at the points of \(\mathcal{L}^oX\) constructing a moduli problem for some type of bundles over a smooth projective curve. Take the stack \([G\setminus X / G]\), which on any test scheme \(S\) is the groupoid of pairs of left principle \(G\)-bundles \((E, E')\) with a \(G\)-equivariant section \(\alpha: S \rightarrow X \bigwedge^{G \times G} (E \times E')\), called \(X\)-morphism from \(E\) to \(E'\).NEWLINENEWLINELet \(C\) be a smooth projective and geometrically connected curve over \(k\) with a fixed principal \(G\)-bundle on it (suppose it is the trivial bundle \(E_0\)). Take the stack of all maps \(\mathrm{Map}(C, [G \setminus X /G])\), and for a test scheme \(S\), call such \(\xi: C \times S \rightarrow [G \setminus X /G]\) non-degenerate if \(\xi^{-1}([G \setminus G /G])\) is open, with surjective projection on \(S\). As there exists canonical ``left'' map of \(\mathrm{Map}(C, [G \setminus X /G])\) to the stack \(\mathrm{Bun}_G\) of principal \(G\)-bundles on \(C\), define \(M\) to be the fiber of this map over \(E_0\), but in the open substack of non-degenerate maps.NEWLINENEWLINELet assume \(C(k) \neq \emptyset\) and take a point \(v \in C(k)\) with formal neighborhood \(C_v \simeq \mathbb{D}\). If \(\widetilde{M}\) is the stack of classifying pairs \((\phi, \xi)\), where \(\phi \in M\) corresponds to a \(G\)-torsor, and \(\xi\) is a trivialization of its restriction to \(C_v\), then there is canonical projection \(\pi: \widetilde{M} \rightarrow M\). It could be defined also morphism \(h: \widetilde{M} \rightarrow \mathcal{L}^{o}X\), such that at any \(\tilde{m} = (\phi, \xi) \in \widetilde{M}(k)\) with \(Im(\phi) \subset X_{\mathrm{reg}}\), the induced morphism on formal neighborhoods is formally smooth. Moreover, for any \(x \in \mathcal{L}^{o}X(k)\) there exists \(\tilde{m}\) as above, such that \(h(\tilde{m}) = x\).NEWLINENEWLINENext, take \(X\) to be affine normal toric variety containing as an open dense subset torus \(T\), determined by a strictly convex rational polyhedral cone \(\sigma\). It is generated over \(\mathbb{R}\) by all cocharacters \(\lambda \in \mathrm{Hom}(\mathbb{G}_m, T)\) for which \(\lim_{t \rightarrow 0} \lambda(t)\) exists in \(X\). All such \(\lambda\) form a finitely generated saturated monoid \(\Lambda_X \subset \mathrm{Hom}(\mathbb{G}_m, T)\).NEWLINENEWLINEFrom now on we take \(\mathcal{L}^{o}X\) to be defined with respect to the open \(X^o = T\). Then any morphism of affine toric varieties \(X \rightarrow Y\) induces morphism of their arc spaces, and morphism \(\mathcal{L}^{o}X \rightarrow \mathcal{L}^{o}Y\) as well.NEWLINENEWLINEThe main result in this case claims that if \(k\) is finite, \(IC_{\mathcal{L}^{o}X}\) can be represented as a formal power series \(\sum_{\lambda \in \Lambda}m_{\lambda}e^{\lambda} \in \mathbb{Q}_l[[\Lambda]]\), and it is \(\mathcal{L}T\)-invariant. Moreover, \(IC_{\mathcal{L}^{o}X} = \prod_{\lambda \in \Lambda}(1-e^{\lambda})^{-1}\). To prove it has to be taken the global analogue \(M_X\) of the arc space of affine toric variety \(X\), so that we could think that \(M_X\) is attached rather to the monoid \(\Lambda_X\) than to \(X\) itself. It is shown that \(M_X\) has unique stratification, the strata being characterized in terms of the so called \(\Lambda_X\)-valued divisors, so that the components of \(M_X\) are closures of strata. Then over \(k\) a finite field is given a formula, presenting \(IC_{M_X}\) as formal power series.NEWLINENEWLINEIn the last section is discussed the case of an \(L\)-monoid \(X\), that is, a normal affine embedding of reductive group \(G \hookrightarrow X\), with a determinant map \(G \rightarrow \mathbb{G}_m\) whose kernel is simply connected. Then the left and right action of \(G\) will define a monoid structure on \(X\) [\textit{E. B. Vinberg}, in: Lie groups and Lie algebras: E. B. Dynkin's seminar. Providence, RI: American Mathematical Society. 145--182 (1995; Zbl 0840.20041)]. Define \(\mathcal{L}^oX\) with respect to the open \(G\), and when \(k\) is finite, there is a unique way to decompose \(IC_{\mathcal{L}^{o}X} = \sum_{m \in \mathbb{N}} \psi_n\), such that \(\psi_m\) has support \(\{g \in \mathcal{L}^{o}X: \mathrm{val}(\det(g)) = m\}\). To prove it, as in the toric case, it is considered the global analogue \(M\) of the arc space, and is obtained a formula for the function \(IC_M\) on \(M(k)\) when \(k\) is finite field.