On the moduli of logarithmic connections on elliptic curves (Q2152867)

Let \(C\) be a complex elliptic curve. Consider pairs \((E, \nabla)\) where \(E \to C\) is a rank \(2\) vector bundle and \(\nabla: E \to E\otimes \Omega^{1}_{C}(D)\) is a logarithmic connection with (reduced) polar divisor \(D = t_1 + \dots t_n\) with \(n\geq1\). Fix the following: \begin{itemize} \item[1.] The eigenvalues \((\nu_{i}^{+}, \nu_{i}^{-})\) of \(\mathrm{Res}_{t_i}(\nabla)\), for each \(i = 1, \dots n\), such that: \begin{itemize} \item[(a)] \(\nu_{1}^{\epsilon_1} + \dots + \nu_{n}^{\epsilon_n} \notin \mathbb{Z}\) for any \(\epsilon_i \in \{+,-\}\); \item[(b)] and \(\nu_i^{+} \neq \nu_{i}\), for \(i = 1 \dots n\); \end{itemize} \item[2.] A trace connection \((L, \zeta)\), i.e. \(\mathrm{det}(E) = L\) and \(\mathrm{tr}(\nabla) = \zeta\). \end{itemize} \textit{M.-A. Inaba} [J. Algebr. Geom. 22, No. 3, 407--480 (2013; Zbl 1314.14063)] constructed the moduli space \(\mathfrak{Con}^{\nu}\) consisting of pairs \((E, \nabla)\) up to isomorphism and showed that it is a smooth, irreducible quasi-projective variety of dimension \(2n\), equipped with an algebraic symplectic structure. In the current article, the Fassarella, Loray and Muniz study the geometry of this moduli space. Consider the forgetful map \(\pi: (E, \nabla) \to (E, \mathbf{p})\) which associates to a connection an underlying quasi-parabolic bundle. Given a choice of signs \(\epsilon_i \in \{+,-\}\) for each \(i = 1, \dots, n\), the parabolic data \(\mathbf{p}^{\epsilon}(\nabla) = (p_1^{\epsilon_1}(\nabla),\dots, p_n^{\epsilon_n}(\nabla))\) consists of the \(\nu_i^{\epsilon_i}\)- eigenspace \(p_i^{\epsilon_i} \subset E|_{t_i}\) for \(\mathrm{Res}_{t_i}(\nabla)\) at each pole. These are well-defined since by assumption \(\nu_{i}^{+} \neq \nu_{i}^{-}\). Denote by \(\mathfrak{Con}^{\nu}_{st}\) the open subset of \(\mathfrak{Con}^{\nu}\) formed by pairs \((E, \nabla)\) which admit a \(\mu\)-stable parabolic bundle \((E, \mathbf{p}(\nabla))\) for some \(\epsilon\) and some weight vector \(\mu\). Let \(\mathcal{Z}_n := \mathfrak{Con}^{\nu}\backslash \mathfrak{Con}^{\nu}_{st}\) denote the complement. The authors show that \(\mathcal{Z}_n\) is empty for \(n\) even and has four irreducible components which are isomorphic to \(\mathbb{C}^n\) for \(n\) odd. Assuming \(\nu^{+}_{i} - \nu^{-}_{i} \in \{0,1, -1 \}\) for \(i \in \{1, \dots n\}\), Theorem A shows that \(\mathfrak{Con}^{\nu}_{st}\) admits an open covering given by open subsets isomorphic to \(S^n\), where \(S\) is the complement of the diagonal in \(\mathbb{P}^1 \times \mathbb{P}^1\). Using the forgetful map \(\pi\), the authors investigate \(\mathfrak{Con}^{\nu}\) via an underlying quasi-parabolic structure. The authors show that there exists an open subset of \(\mathfrak{Con}^{\nu}\), denoted \(\mathrm{Con}^{\nu}\) where the underlying vector bundle is \(E_1\), the unique indecomposable vector bundle of degree \(1\) with given determinant. Let \(\mathbb{P}\mathrm{Higgs}\) be the projectivization of the space of Higgs fields on \(E_1\). In Theorem B, the authors show that the moduli space \(\mathrm{Con}^{\nu}\) has compactification \(\overline{\mathrm{Con}^{\nu}} = \mathbb{P}(\mathcal{E})\), where the boundary divisor is isomorphic to \(\mathbb{P}\mathrm{Higgs}\). Furthermore, they show that the inclusion \(\mathbb{P}\mathrm{Higgs} \hookrightarrow \mathbb{P}(\mathcal{E})\) is determined, up to automorphisms of \(\mathbb{P}(\mathcal{E})\), by \((\nu_1, \dots \nu_n)\). Theorem C shows that \((\nu_1, \dots \nu_n)\) is detected by the symplectic structure of \(\mathrm{Con}^{\nu}\). Recall that a global section of \(E_1\) plays the role of a cyclic vector for connection \(\nabla\), which yields a second order ODE on \(C\). By [\textit{F. Loray} and \textit{M.-H. Saito}, Int. Math. Res. Not. 2015, No. 4, 995--1043 (2015; Zbl 1349.14047)] the Apparent map, \(\mathrm{App}\) assigns to \(\nabla\) the apparent singular points of this equation. The authors conclude the paper by using \(\mathrm{App}\) to study the birational geometry of \(\mathrm{Con}^{\nu}\) (Theorem D).