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Results tagged with complex-geometry
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user 15242
Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.
3
votes
1
answer
2k
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question about the developing map
I'm having some trouble finding literature on the developing map.
All the sources I could find on it seem to refer to thurston's definition in either:
http://www.ucl.ac.uk/~ucahhjr/Notes/Essay.pdf
or …
- 10.7k
22
votes
1
answer
3k
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Why is Oka's coherence theorem a deep result?
This is a very naive question.
Let $X$ be a complex manifold. Let $\mathcal{O}_X$ be the structure sheaf of $X$, a sheaf of rings whose sections over opens $U\subset X$ are just the holomorphic funct …
- 10.7k
7
votes
1
answer
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Relating the holomorphic Euler characteristic of a family of algebraic varieties to properti...
Let $f : X\rightarrow Y$ be a proper flat morphism (of schemes) with connected fibers over a smooth projective curve $Y$ over $\mathbb{C}$. Let $X_{y_0}$ denote a smooth fiber over $y_0\in Y$.
If $f$ …
- 10.7k
5
votes
0
answers
277
views
Is there a geography of Hodge numbers for minimal general type algebraic surfaces?
Let $X$ be a minimal smooth projective surface of general type (over $\mathbb{C}$). Let's call such a surface MSPGT. Such a surface has two chern numbers $c_1^2$ and $c_2$. It is known that they are b …
- 10.7k
7
votes
3
answers
433
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Are non-isomorphic covers of riemann surfaces also generally nonisomorphic as riemann surfaces?
Suppose you've got a Riemann surface $E$, and two topological covers $X,Y\rightarrow E$. Suppose $X,Y$ are nonisomorphic topological covers of $E$, then would you expect $X,Y$ as Riemann surfaces (wit …
- 10.7k
7
votes
1
answer
2k
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Universal covering map from $\mathcal{H}$ to $\mathbb{C}\setminus \mathbb{Z}\oplus i\mathbb{...
It's a consequence of the uniformization theorem for simply connected Riemann surfaces that the universal cover of $\mathbb{C}\setminus(\mathbb{Z}\oplus i\mathbb{Z})$ ($\mathbb{C}$ punctured at all th …
- 10.7k
19
votes
1
answer
3k
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Questions about the "universal elliptic curve" over the affine $j$-line punctured at 0 and 1728
So my question refers to families of elliptic curves over the $\mathbb{A}^1_\mathbb{C}\setminus\{0,1728\}$ whose fiber above a point $j$ has $j$-invariant equal to $j$ (I understand it's not universal …
- 10.7k
10
votes
2
answers
494
views
Copies of topological fundamental groups inside etale fundamental groups given by different ...
Let $X$ be a smooth curve over a number field $K$ (not necessarily proper). Fix an algebraic closure $\overline{K}$ of $K$.
Let $i,i' : \overline{K}\hookrightarrow\mathbb{C}$ be two abstract embeddin …
- 10.7k
3
votes
2
answers
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Elementary question about Isotopy (in the definition of a Teichmuller space)
Disclaimer - I don't have much experience in topology/complex geometry, so I apologize if what I'm asking is too elementary for this site.
Let $S$ be some orientable surface obtained by removing fini …
- 10.7k
6
votes
0
answers
252
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Picard-Lefschetz formula for the quotient of a degenerating family of curves by a cyclic group
$\newcommand{\cD}{\mathcal{D}}\newcommand{\cX}{\mathcal{X}}$(This is a slight rephrasing and modification of the original question)
Let $D\subset\mathbb{C}$ be the complex unit disk. Let $X$ be a comp …
- 10.7k
5
votes
1
answer
373
views
Lengths of closed geodesics on a flat vs hyperbolic punctured torus
Let $T$ be a torus (oriented closed surface of genus 1), $p\in T$, and $T^* := T - \{p\}$.
Let $\mu$ denote a flat structure on $T$. This can be obtained for example by choosing a uniformization $p_f …
- 10.7k
8
votes
0
answers
401
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rings of modular functions on the upper half plane
Let $\Gamma_1\le SL_2(\mathbb{Z})$ be a noncongruence subgroup of finite index.
Let $\Gamma_2\le SL_2(\mathbb{Z})$ be another subgroup of finite index.
Let $M_0(\Gamma_i)$ denote the ring of modular …
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