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Questions tagged [complex-geometry]

Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry. For elementary questions about geometry in the complex plane, use the tags (complex-numbers) and (geometry) instead.

Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry.

3688 questions
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Explicit diffeomorphism between complex tori

Let $Im(\tau) > 0$ and $X_{\tau}$ be the complex torus given by $\mathbb{C}/\mathbb{Z}\oplus \tau\mathbb{Z}$. How do I go about constructing an explicit diffeomorphism (as real manifolds) between $X_{\tau}$ and $X_{\tau'}$, where both $\tau$ and…
Geometer
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geometry of Automorphism groups

It is proved that Aut group of a compact complex manifold is a lie transformation group. How do we show that a automorphism group of a tubular geometry is a lie group?
Ryan
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How to compute the number of branch points and ramification indices of the quoteint covers?

Take a ramified Galois cover $f:X\rightarrow Z$ of Riemann surfaces over $\mathbb{C}$ with Galois group $G$. If $H$ is a non-tirival normal subgroup of the Galois group $G$, this cover factors as $f:X\rightarrow Y\rightarrow Z$ and $X\rightarrow Y$…
Darius Math
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Hartog's theorem in the proof of the Kodaira embedding theorem

I'm reading Huybrechts's Complex Geometry, p.250, proof of the Kodaira Embedding theorem : (Here Kodaira embedding theorem is, "Let $X$ be a compact Kahler manifold. Then a line bundle $L$ on $X$ is positive if and only if $L$ is ample." And a line…
Plantation
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Why are hermitian metrics on (holomorphic) vector bundles only assumed to vary smoothly?

I am confused about something regarding hermitian metrics. I understand that since we can define a complex vector bundle over a smooth manifold, it makes sense to consider a hermitian metric that only varies smoothly. However, in the case that we…
trystero
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hermitean structure vs hermitean metric

Define a hermitean structure $H$ on a complex linear space $V$ as $H: V\times V \to \mathbb{C}$ s.t. i. $H(u,v)$ is $\mathbb{C}$-linear in $u$ for every $v \in V,$ ii. $H(u,v)=\overline{H(v,u)}$ iii. $H(u,u)> 0\quad \forall u\neq 0.$ Define a…
jj_p
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The definition of contraction of differential forms?

At P211 in 'Complex Geometry, An Introduction' by Huybrechts: The contraction of the curvature $F_{\nabla}\in\mathscr{A}^{1,1}(\mathrm{End}(T^{1,0}X))$ with the Kahler form $\omega$ yields an element…
WakeUp-X.Liu
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Are complex submanifolds necessarily closed?

In the excellent book From Holomorphic Functions to Complex Manifolds by Klaus Fritzsche and Hans Grauert, if I follow the definition and properties of analytic subsets and the definition of a complex submanifold $A$ of a complex manifold $X$, then…
Philippe Malot
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When does a real analytic space admit a compatible complex analytic structure?

A complex manifold can be viewed as a smooth manifold. A smooth manifold together with an integrable almost complex structure can be given a complex structure. Clearly a complex analytic space can be viewed as a real analytic space. My question is…
Thomas Kurbach
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The existence of a local orthonormal frame of a hermitian vector bundle

Let $X$ be a complex manifold. Let $E$ be a hermitian vector bundle with a given hermitian metric over $X$. On a local trivialization open subset, is there a smooth orthonormal local frame? is there a holomorphic orthonormal local frame?
jack lion
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Why $\Lambda^{p,q}M$ are not holomorphic bundles for $q\neq 0$?

I just read from a lecture note that $\Lambda^{p,q}M$ are not holomorphic bundles for $q\neq 0$. But, take $\Lambda^{0,1}M$ for example, locally we have $$ d\bar{z} = \frac{\partial \bar{z}}{\partial w}dw + \frac{\partial\bar{z}}{\partial…
string
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Is every complex manifold that's homeomorphic to $\mathbb C\mathbb P^n$ also isomorphic to $\mathbb C\mathbb P^n$?

Put another way, is the complex structure on $\mathbb C\mathbb P^n$ unique? I know that this is the case for $n\in\{1,2\}$, so I'm curious as to whether it's the case in general. If it's not known, is it believed to be true, and what progress has…
Niven
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From compatible Riemannian metric to Hermitian metric

By this notes p.42 It gives a hermitian metric by a compatible Riemannian metric $g$, and from p.23, it extends $g$ to $T_{\mathbb{C}}M$ complex bilinearly. I wonder if we extend $g$ via the sesquilinear convention, can we still get the same…
Danny
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Showing surjectivity of map $L^{n-k}$

I refer to the text Complex Geometry by David Huybrechts. In remark 3.2.7 iii) he stated that the surjectivity of the map $L^{n-k}:\mathcal{H}^{p,q}(X,g)\rightarrow \mathcal{H}^{n-q,n-p}(X,g)$ can be deduced from the fact that the dual Lefschetz…
Soby
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About the holomorphic tangent bundle of $\mathbb{CP}^n$

Are there global sections of $T{\mathbb{CP}^n}$, which vanish only in a finite number of points? If yes, how many? Here $T{\mathbb{CP}^n}$ is the holomorphic bundle. Any help would mean a lot, thanks.
unicornki
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