The arc space of horospherical varieties and motivic integration (Q2852242)

A formula for the stringy \(E\)-function of a \(\mathbb{Q}\)-Gorenstein horospherical variety, which generalizes the toric case, is obtained and a new smoothness criterion for such varieties is deduced. More precisely, the stringy \(E\)-function of a \(\mathbb{Q}\)-Gorenstein complex algebraic variety \(X\) is defined as \(E_{\text{st}}(X;u,v)=\sum_JE(D_J^0;u,v)\prod_{j\in J}\frac{uv-1}{(uv)^{\nu_j+1}-1}\), where \(D_J^0=\bigcap_{j\in J}D_j\setminus\bigcup_{j\notin J}D_j\) are the strata of the desingularization \(X'\) of \(X\) stratified by the components \(D_j\) of the exceptional divisor and their intersections, \(\nu_j\) are the discrepancies, and \(E(Z;u,v)=\sum_{i,p,q}(-1)^ih^{p,q}(H_c^i(Z;\mathbb{C}))u^pv^q\) is the \(E\)-polynomial \textit{V. V. Batyrev} [in: Integrable systems and algebraic geometry. Proceedings of the 41st Taniguchi symposium, Kobe, Japan, June 30--July 4, 1997, and in Kyoto, Japan, July 7--11 1997. Saito, M.--H. (ed.) et al., Integrable systems and algebraic geometry. Proceedings of the 41st Taniguchi symposium, Kobe, Japan, June 30--July 4, 1997, and in Kyoto, Japan, July 7--11 1997. Singapore: World Scientific. 1--32 (1998; Zbl 0963.14015)]. It does not depend on the chosen desingularization and can be expressed in terms of a certain motivic integral over the arc space \(X'(\mathbb{C}[[t]])\) (stringy motivic volume). (For motivic integration, see [\textit{J. Denef} and \textit{F. Loeser} [Invent. Math. 135, No. 1, 201--232 (1999; Zbl 0928.14004); \textit{M. Mustaţă}, Invent. Math. 145, No. 3, 397--424 (2001; Zbl 1091.14004); \textit{L. Ein} and \textit{M. Mustaţă}, Proc. Symp. Pure Math. 80, Pt. 2, 505--546 (2009; Zbl 1181.14019)].) For smooth \(X\) one has \(E_{\text{st}}(X;u,v)=E(X;u,v)\). A combinatorial formula for the stringy \(E\)-function of a toric variety was previously obtained by the first author [in: Integrable systems and algebraic geometry. Proceedings of the 41st Taniguchi symposium, Kobe, Japan, June 30--July 4, 1997, and in Kyoto, Japan, July 7--11 1997. Saito, M.--H. (ed.) et al., Integrable systems and algebraic geometry. Proceedings of the 41st Taniguchi symposium, Kobe, Japan, June 30--July 4, 1997, and in Kyoto, Japan, July 7--11 1997. Singapore: World Scientific. 1--32 (1998; Zbl 0963.14015)].NEWLINENEWLINENow let \(X\) be a horospherical \(G\)-variety, where \(G\) is a connected reductive group; this means that \(X\) contains a dense open orbit \(G/H\) and \(H\) contains a maximal unipotent subgroup of \(G\). Alike the toric case, horospherical varieties containing a given open orbit \(G/H\) are classified in combinatorial terms by colored fans \(\Sigma\) in the vector space \(N_{\mathbb{R}}\) spanned by the lattice \(N\) dual to the weight lattice of the torus \(A=P/H\), where \(P=N_G(H)\) is a parabolic subgroup [\textit{F. Knop}, in: Proceedings of the Hyderabad conference on algebraic groups held at the School of Mathematics and Computer/Information Sciences of the University of Hyderabad, India, December 1989. Ramanan, S. (ed.), Madras: Manoj Prakashan. 225--249 (1991; Zbl 0812.20023); \textit{D. A. Timashev}, Homogeneous spaces and equivariant embeddings. Encyclopaedia of Mathematical Sciences 138. Invariant Theory and Algebraic Transformation Groups 8. Berlin: Springer (2011; Zbl 1237.14057)]. If \(X\) is \(\mathbb{Q}\)-Gorenstein, then there is a piecewise linear function \(\omega_X\) on the support \(|\Sigma|=\bigcup_{\sigma\in\Sigma}\sigma\) of the fan which is linear on each cone \(\sigma\in\Sigma\) and takes value \(-1\) on the primitive lattice generator of each uncolored ray in \(\Sigma\) and a certain prescribed negative value on each color; it represents the canonical \(\mathbb{Q}\)-Cartier divisor. The following formula is proven: \(E_{\text{st}}(X;u,v)=E(G/H;u,v)\sum_{n\in|\Sigma|\cap N}(uv)^{\omega_X(n)}\). For complete locally factorial \(X\) the authors define the weighted Stanley--Reisner ring \(R^w_{\Sigma}\) as the usual Stanley--Reisner ring of the simplicial fan \(\Sigma\) with the degrees \(-\omega_X(e_i)\) assigned to the generators \(z_i\) of \(R^w_{\Sigma}\) corresponding to the primitive lattice generators \(e_i\) of the rays in \(\Sigma\). It is proved that \(E_{\text{st}}(X;u,v)=(-1)^rE(G/H;u,v)P(R^w_{\Sigma},uv)\), where \(P\) is the Poincaré series. It is also proved that if all closed \(G\)-orbits in \(X\) are projective, then the stringy Euler number \(e_{\text{st}}(X)=E_{\text{st}}(X;1,1)\) is greater than or equal to the usual Euler number \(e(X)=E(X;1,1)\) and the equality holds if and only if \(X\) is smooth. The authors conjecture that this smoothness criterion extends to arbitrary spherical varieties. Also they raise a problem to compute the stringy Euler function of an arbitrary spherical variety in terms of combinatorics of its colored fan.NEWLINENEWLINEThe proofs of main results are based on motivic integration and parametrization of \(G(\mathbb{C}[[t]])\)-orbits in the arc space in terms of lattice vectors: \((X(\mathcal{O})\cap(G/H)(\mathcal{K}))/G(\mathcal{O})\simeq|\Sigma|\cap N\), where \(\mathcal{O}=\mathbb{C}[[t]]\), \(\mathcal{K}=\mathbb{C}((t))\) [\textit{D. Luna} and \textit{Th. Vust}, Comment. Math. Helv. 58, 186--245 (1983; Zbl 0545.14010); \textit{D. Gaitsgory} and \textit{D. Nadler}, Mosc. Math. J. 10, No. 1, 65--137 (2010; Zbl 1207.22013); \textit{D. A. Timashev}, Homogeneous spaces and equivariant embeddings. Encyclopaedia of Mathematical Sciences 138. Invariant Theory and Algebraic Transformation Groups 8. Berlin: Springer (2011; Zbl 1237.14057)]. A special horospherical desingularization \(X'\) of \(X\) given by the ``decoloration'' and a regular subdivision of \(\Sigma\) is used; it is of the form \(X'=G\times^PY\), where \(Y\) is the toric \(A\)-variety corresponding to the subdivision of \(\Sigma\). In the proof of the smoothness criterion, a previously known criterion of smoothness for horospherical varieties in terms of their colored fans [\textit{B. Pasquier}, Bull. Soc. Math. Fr. 136, No. 2, 195--225 (2008; Zbl 1162.14030); \textit{D. A. Timashev}, Homogeneous spaces and equivariant embeddings. Encyclopaedia of Mathematical Sciences 138. Invariant Theory and Algebraic Transformation Groups 8. Berlin: Springer (2011; Zbl 1237.14057)] is used.