%0 Generic %1 singh2019moduli %A Singh, Anoop %D 2019 %K connections %T Moduli space of logarithmic connections singular over a finite subset of a compact Riemann surface %U http://arxiv.org/abs/1912.01288 %X Let $S$ be a finite subset of a compact connected Riemann surface $X$ of genus $g 2$. Let $M_lc(n,d)$ denote the moduli space of pairs $(E,D)$, where $E$ is a holomorphic vector bundle over $X$ and $D$ is a logarithmic connection on $E$ singular over $S$, with fixed residues in the centre of $gl(n,\C)$, where $n$ and $d$ are mutually corpime. Let $L$ denote a fixed line bundle with a logarithmic connection $D_L$ singular over $S$. Let $M'_lc(n,d)$ and $M_lc(n,L)$ be the moduli spaces parametrising all pairs $(E,D)$ such that underlying vector bundle $E$ is stable and $(\bigwedge^nE, D) (L,D_L)$ respectively. Let $M'_lc(n,L) M_lc(n,L)$ be the Zariski open dense subset such that the underlying vector bundle is stable. We show that there is a natural compactification of $M'_lc(n,d)$ and $M'_lc(n,L)$ and compute their Picard groups. We also show that $M'_lc(n,L)$ and hence $M_lc(n,L)$ do not have any non-constant algebraic functions but they admit non-constant holomorhic functions. We also study the Picard group and algebraic functions on the moduli space of logarithmic connections singular over $S$, with arbitrary residues.

@misc{singh2019moduli, abstract = {Let $S$ be a finite subset of a compact connected Riemann surface $X$ of genus $g \geq 2$. Let $\cat{M}_{lc}(n,d)$ denote the moduli space of pairs $(E,D)$, where $E$ is a holomorphic vector bundle over $X$ and $D$ is a logarithmic connection on $E$ singular over $S$, with fixed residues in the centre of $\mathfrak{gl}(n,\C)$, where $n$ and $d$ are mutually corpime. Let $L$ denote a fixed line bundle with a logarithmic connection $D_L$ singular over $S$. Let $\cat{M}'_{lc}(n,d)$ and $\cat{M}_{lc}(n,L)$ be the moduli spaces parametrising all pairs $(E,D)$ such that underlying vector bundle $E$ is stable and $(\bigwedge^nE, \tilde{D}) \cong (L,D_L)$ respectively. Let $\cat{M}'_{lc}(n,L) \subset \cat{M}_{lc}(n,L)$ be the Zariski open dense subset such that the underlying vector bundle is stable. We show that there is a natural compactification of $\cat{M}'_{lc}(n,d)$ and $\cat{M}'_{lc}(n,L)$ and compute their Picard groups. We also show that $\cat{M}'_{lc}(n,L)$ and hence $\cat{M}_{lc}(n,L)$ do not have any non-constant algebraic functions but they admit non-constant holomorhic functions. We also study the Picard group and algebraic functions on the moduli space of logarithmic connections singular over $S$, with arbitrary residues.}, added-at = {2020-06-03T08:54:34.000+0200}, author = {Singh, Anoop}, biburl = {https://www.bibsonomy.org/bibtex/266aed176262a4af84fc8de6a6943895f/simonechiarello}, description = {Moduli space of logarithmic connections singular over a finite subset of a compact Riemann surface}, interhash = {ab72870c77c23ef4010f65a82909a4af}, intrahash = {66aed176262a4af84fc8de6a6943895f}, keywords = {connections}, note = {cite arxiv:1912.01288Comment: 17 pages}, timestamp = {2020-06-03T08:54:34.000+0200}, title = {Moduli space of logarithmic connections singular over a finite subset of a compact Riemann surface}, url = {http://arxiv.org/abs/1912.01288}, year = 2019 }