Parabolic bundles over the projective line and the Deligne-Simpson problems (Q2794439)

The original version of the Deligne-Simpson problem is: Given conjugacy classes \(C_1, \dots, C_k\) of complex matrices in GL\((n, \mathbb{C})\), do there exist \(A_i \in C_i, 1\leq i \leq k\) such that \(A_1.A_2. \dots .A_k = Id\)? Using the theory of parabolic bundles on a compact Riemann surface, the problem can be restated as follows. Let \(D \subset \mathbb{P}^1\) be a divisor, \(D= x_1+ \dots + x_k\). Let \(\nabla\) be a logarithmic connection on a vector bundle \(E\) of rank \(n\) over \(\mathbb{P}^1\) (singular on \(D\)) with residue \(\mathrm{Res}_{x_i}\nabla \in \mathrm{End} (E_{x_i})\) for all \(i\). Then the restatement of the Deligne-Simpson problem is: For some \(D \subset \mathbb{P}^1\) and \(E\), does there exist a logarithmic connection \(\nabla\) on \(E\) with residues \(\mathrm{Res}_{x_i}\nabla \in C_i\)? A parabolic (vector) bundle of weight type \((D,w)\) is a vector bundle \(E\) together with a flag \(E_{x_i}= E_{i o} \supseteq \dots \supseteq E_{i w_i}= 0\) of length \(w_i\) at \(x_i\). Let \(\alpha_0 = \) rk\((E), \alpha_{i j} =\dim E_{i j}\), then \(\alpha = (\alpha_0, \alpha_{i j})\) is called the dimension vector of \(E\). Given a tuple of complex numbers \(\xi = (\xi_{i j})_{1\leq i\leq k, 1\leq j \leq w_i}\), a \(\xi\)-parabolic connection on a parabolic bundle \(E\) is a logarithmic connection \(\nabla\) satisfying \((\mathrm{Res}_{x_i}\nabla - \xi_{i j})(E_{i j-1} )\subset E_{i j}\). Semisimple conjugacy classes \(C_1, \dots, C_k \in \) GL\((n,\mathbb{C})\) determine a \(\xi\)-parabolic bundle \(E\) with \(\xi\) the vector of eigenvalues of \(C_1, \dots, C_k\). The authors define a stack \(\mathcal{Y}\) to be almost very good if codim \(\mathcal{Y}^{m+n} > n, \forall n>0\) where \(\mathcal{Y}^m = \{y\in \mathcal{Y} | \dim \mathrm{Aut}(y) > m\}, m=\min \dim \mathrm{Aut} y\). The authors show that the moduli stack Bun\(_{D,w,\alpha}(\mathbb{P}^1)\) of parabolic bundles over \(\mathbb{P}^1\) of weight type \((D,w)\) and dimension vector \(\alpha\) is almost very good if \(\alpha\) satisfies certain conditions \((C)\). The moduli stack of solutions of Deligne-Simpson problem can be defined as the stack Conn\(_{D,w,\alpha,\xi}(\mathbb{P}^1)\) of \(\xi\)-parabolic connections on parabolic bundles over \(\mathbb{P}^1\) of weight type \((D,w)\) and dimension vector \(\alpha\). By presenting Conn\(_{D,w,\alpha,\xi}(\mathbb{P}^1)\) as a twisted cotangent bundle over Bun\(_{D,w,\alpha}(\mathbb{P}^1)\), the authors show that if Bun\(_{D,w,\alpha}(\mathbb{P}^1)\) is almost very good and \(\Sigma_{i, j} \xi_{i j}(\alpha_{i j-1} - \alpha_{i j})\) is an integer, then Conn\(_{D,w,\alpha,\xi}(\mathbb{P}^1)\) is a non-empty, irreducible, locally complete intersection of dimension \(2 p(\alpha) -1\). The paper also has results on the additive Deligne-Simpson problem. The Deligne-Simpson problem and its additive analogue (for \(gl(n, \mathbb{C}))\) have been studied by several authors including I. Biswas, Crawley-Boevey, Shaw, Katz, Kostov and Simpson.