Stable configurations of linear subspaces and quotient coherent sheaves (Q2501373)

Consider the product \(\mathcal{G}:=\prod_{i=1}^m \text{Gr}(k_i,V\otimes W)\) (resp. \(\mathcal{G}':=\prod_{i=1}^m \text{Gr}(V\otimes W,k_i)\)) of linear subspaces of \(V \otimes W\) of dimension \(k_i\) (resp. quotient linear spaces of \(V\otimes W\) of dimension \(k_i\)) where \(V\) and \(W\) are two fixed vector spaces over complex numbers. For a set of positive integers \(\theta_i\), we consider the ample line bundle \(L=\bigotimes_{i=1}^m \prod^*_i(\mathcal{O}_{\text{Gr}(k_i,V\otimes W)}(\theta_i))\) (resp. \(L'=\bigotimes_{i=1}^m \prod^*_i(\mathcal{O}_{\text{Gr}(V\otimes W,k_i)}(\theta_i))\)) which admits a unique \(\text{SL}(V)\)-linearization. There is a diagonal action of \(\text{SL}(V)\) on \(\mathcal{G}\) (resp. \(\mathcal{G}'\)) by operation on the factor \(V\). Note that for \(m=1\), \(\mathcal{G}'\) was studied by C. Simpson in order to construct the moduli space of coherent sheaves. In this paper, the author investigates the GIT stability with respect to this action. This leads the author to define the normalized total weighted dimension of the configuration \(K=\{K_i\}\in \mathcal{G}\) with respect to \(\theta=\{\theta_i\}\) by \[ \mu_{\theta}(K)=\frac{1}{\dim{V}}\sum_i \theta_i \dim{K_i} \] and the configuration is said \(\mu\)-semistable (resp. \(\mu\)-stable) with respect to the weights \(\theta_i\) if for every subspace \(H\) of \(K\), \(\mu_{\theta}(H) \leq \mu_{\theta}(K)\) resp. \(<).\) The GIT stability of \(K\) with respect to \(L\) is equivalent to its \(\mu\)-stability with respect to the same weights. One defines polystable configurations as direct sum of a finite number of stable subconfigurations of the same normalized total weighted dimension. \newline Assume now that \(\dim{W}=1\). Inspired \textit{S.K. Donaldson} [J. Differ. Geom. 59, No.~3, 479--522 (2001; Zbl 1052.32017)], \textit{H. Luo} [J. Differ. Geom. 49, No.~3, 577--599 (1998; Zbl 1006.32022)], \textit{X. Wang} [Math. Res. Letter No.~2--3, 393--411 (2002; Zbl 1011.32016)] and using the formalism of moment maps, the author introduces a notion of balanced metric on \(V\). Generalizing the work of B. Totaro and A. Klyachko, he proves that the existence of a hermitian balanced metric is equivalent to the polystability of the configuration. Another result of the paper is about a generalized Gelfand-MacPherson correspondence. Consider a configuration of vector subspaces \((V_1,\dots ,V_m)\in \text{Gr}(k_1,n)\times \cdots \times \text{Gr}(k_m,n)\), and denote \(U^0_{n,(k_1,..,k_m)}\) the space of matrices \(M\) of size \(n\times \sum_i k_i\) such that \(M\) and each of its \(n\times k_i\) block \(M_i\) are of maximum rank. On \(U^0_{n,(k_1,\dots ,k_m)}\) there are two groups acting, one is \(\text{SL}(n)\) and the other one is \[ G_{k_1,\dots ,k_m}=S(\text{GL}(k_1)\times\cdots \times \text{GL}(k_m)) \subset \text{SL}(k_1+\dots +k_m). \] Note that quotienting \(U^0_{n,(k_1,..,k_m)}\) by \(G_{k_1,\dots ,k_m}\) we get \(X=\text{Gr}(k_1,n)\times \dots \times \text{Gr}(k_m,n),\) while quotienting \(U^0_{n,(k_1,\dots ,k_m)}\) by \(\text{SL}(n)\) gives \(Y=\text{Gr}(n,k_1+\dots +k_m).\) Finally there is a homeomorphism between the orbit spaces \(X/\text{SL}(n)\) and \(Y/G_{k_1,\dots ,k_m}\). In the GIT setup, it is obtained, following the approach of M. Kapranov, a natural isomorphism between \(X^{\text{ss}}//\text{SL}(n)\) and \(Y^{\text{ss}}//G_{k_1,\dots ,k_m}\) for any choice of \(\theta\). Actually, it is expected an isomorphism between the Chow quotients of the two actions. Moreover, inspired from the lines of [\textit{I. V. Dolgachev} and \textit{Y. Hu}, Publ. Math., Inst. Hautes Étud. Sci. 87, 5--56 (1998; Zbl 1001.14018)], the author studies the dependence on \(\theta\) of the moduli. This allows him to give an example of a manifold (precisely \(X=\prod_{i=1}^m \text{Gr}(2,\mathbb{C}^4)\)) for which there are no top chambers in the \(\text{SL}(V)\)-ample cone of \(X\). Using the classical Grothendieck's embedding of the Quot scheme into the Grassmannian, some of the previous results can be extended to the case of configurations of coherent quotient sheaves. In the case of configurations of vector bundles, a notion of balanced metric is defined and its existence is a necessary and sufficient condition for the polystability of the configuration in the Quot scheme, which leads to a partial generalisation of [\textit{X. Wang}, Math. Res. Letter No. 2--3, 393--411 (2002; Zbl 1011.32016)] and [\textit{D.H. Phong} and \textit{J. Sturm}, Commun. Anal. Geom. 11, No. 3, 565--597 (2003; Zbl 1098.32012)].