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        Riemann surfaces(Riemannian surfaces) is one dimensional complex manifold. For questions about classical examples in complex analysis, complex geometry, surface topology.

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        103 views

        Embedding open connected Riemann Surfaces in $\mathbb{C}^2$

        This question arises in the context of a question asked on MSE: Are concrete Riemann surfaces Riemann domains over $\mathbb{C}$. Part of the answer to that question is the question above which is ...
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        votes
        2answers
        222 views

        Is every element of $Mod(S_{g,1})$ a composition of right handed Dehn twists?

        Let $S_{g,1}$ be the surface of genus $g \geq 1$ and $1$ boundary component. Let $Mod(S_{g,1})$ be the mapping class group in which we allow isotopies to rotate the action on the boundary (...
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        148 views

        Towards recognizing St. Venant geometrical invariant

        Using partial derivative notation we can express Gauss curvature $K$ in cartesian coordinates: $$\quad p= \partial w/ \partial x, q= \partial w/ \partial y; r=\frac{\partial ^2w}{{\partial {x} ^2} },...
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        votes
        3answers
        908 views

        References for Riemann surfaces

        I know this question has been asked before on MO and MSE (here, here, here, here) but the answers that were given were only partially helpful to me, and I suspect that I am not the only one. I am ...
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        votes
        0answers
        138 views

        Maps between moduli of curves

        Let $M_{g,n}$ be the moduli space of $n$-pointed curves, and $M_g[m]$ the moduli space of (unpointed) curves with $m$-level structure. Fix $m>0$. Is it true that for $n$ large enough, there is a ...
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        2answers
        106 views

        Simple Closed Hyperbolic Geodesics on Punctured Spheres

        Thinking of $\mathbb {CP^1}$ as the sphere $S^2\subset\mathbb R^3$, we can define the notion of a circle on it to be a subset that is got by a hyperplane section of $S^2$ inside $\mathbb R^3$. This ...
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        231 views

        Path lifting property of holomorphic unbranched map

        Suppose $X$ is a Riemann surface and $ a\in\ X $ suppose $ \phi\in\mathcal O_a $ is a holomorphic function germ at $a.$ According to the theorem 7.8 of Forster's book Lectures on Riemann surfaces on ...
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        2answers
        310 views

        Geodesic current supported on a pencil?

        Consider a geodesic current $\mu$ on a closed surface $\Sigma$, as defined by Bonahon ("The Geometry of Teichmüller space via geodesic currents"). These are $\pi_1(\Sigma)$-invariant measures on the ...
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        2answers
        632 views

        Are mapping class groups of orientable surfaces good in the sense of Serre?

        A group G is called ‘good’ if the canonical map $G\to\hat{G}$ to the profinite completion induces isomorphisms $H^i(\hat{G},M)\to H^i(G,M)$ for any finite $G$-module $M$. I’ve had multiple academics ...
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        2answers
        167 views

        Unramified map of Riemann surfaces

        Let $f:S \to T$ be a surjective, unramified, holomorphic map between connected Riemann surfaces. If $S$ is not compact is it always true that $f$ is a covering? This is of course true if $S$ is ...
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        votes
        1answer
        181 views

        non-existence of global coordinates

        Assume we have a smooth manifold, $M$, of dimension $n$. (An example of interest is the case when $M$ is a compact and orientable Riemann surface of genus $g$, but the question is intended to be broad....
        2
        votes
        1answer
        173 views

        Naive question on the Jacobian of a curve

        Let $X$ be a smooth, projective curve of genus $g \ge 2$. We know that the Jacobian $J(X)$ of the curve is a principally polarized abelian variety. The principal polarization is induced by the ...
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        1answer
        138 views

        How to classify a plane complex curve?

        Let $p_1, p_2, t_1, t_2, a \in \mathbb{C}$ be constants. Consider the following plane complex curve in $\mathbb{C}^2$ ($c_1, c_2$ are indetermniates) \begin{align} & {p_1}^2 {p_2}^2 c_1 {t_1}^2 ...
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        78 views

        Reference request: basics about modular curves

        Where can I find a reference (with carefully written proofs) for basic facts about modular curves? Namely: Congruence subgroups The open modular curve $Y_\Gamma$ admits the structure of a Riemann ...
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        2answers
        240 views

        Harder Narasimhan filtration for the endomorphism bundle

        Let $E$ be a vector bundle over a compact Riemann surface $X$, and let $$0=E_0\subsetneq E_1\subsetneq \ldots \subsetneq E_n=E$$ be its Harder-Narasimhan filtration: we have $V_i:=E_i/E_{i-1}$ ...

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