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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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What is a Holomorphic Vector Field?

On a smooth manifold $M$, a smooth vector field is an element of $\Gamma(M, TM)$ which is the space of all smooth sections of the bundle $TM \to M$. If $M$ is a complex manifold, then we have the holomorphic tangent space $T^{1,0}M$. We can form…
Michael Albanese
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Why the moduli space of complex structure in a compact complex manifold is of finite dimension

I admit the statement in the title might be much too unclear. I just heard from my teacher that we can form a finite dimension moduli space of all the complex structure in a compact complex manifold. He said it is a standart fact in complex geometry…
Honglu
  • 583
8
votes
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Necessary and sufficient condition for branch points on a Riemann surface.

I've been reading out of a book by V.B. Alekseev about Abel's theorem on the insolubility of the quintic, and I'm a bit troubled by its presentation on Riemann surfaces. My question is as follows: Suppose $X$ is the Riemann surface defined by the…
user127562
8
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Holomorphic vector fields on complex projective spaces

Question: is there a simple description of holomorphic vector fields on the complex projective space $\mathbb{C P}^n$ ? More precisely : for $n=1$, the holomorphic vector fields are of the form $P(z) \frac{d}{dz}$ in coordinates, where $P$ is a…
Hal
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Adjunction formula (Griffiths & Harris proof)

I'm having trouble understanding the proof of the adjunction formula on Griffiths & Harris book (p. 146). The formula states that if $V \subset M$ is a smooth analytic hypersurface then we have an isomorphism $N^*_V \simeq [-V]|_V$, where $N_V$ is…
Lucas Kaufmann
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7
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1 answer

Proof that the Nijenhuis tensor vanishes in a complex manifold

I'm in trouble proving that if $(M,J)$ is a complex manifold with $J$ a compatible almost complex structure then the Nijenhuis tensor of $J$ vanishes: in other words I would like to find that for any two vector fields $X,Y$ one…
fosco
  • 11,814
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The torus as a projective plane curve $x^3+y^3+z^3=0$

The homogeneous polynomial $F(x,y,z)=x^3+y^3+z^3$ clearly defines a smooth projective curve $X\subset\mathbb{P}^2$. It is easy to see that $\pi:X\rightarrow\mathbb{P}^1$ defined by $$\pi([x:y:z])=[x:y] \ , $$ is a well defined holomorphic map of…
Davide
  • 93
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Why is the Hodge-Deligne polynomial a polynomial?

Let $X$ be a compact complex manifold. Its Hodge-Deligne polynomial is then defined to be $\sum_{p, q \geq 0} (-1)^{p+q} h^{p, q}(X)$ where $h^{p, q}(X):= \mbox{dim}_{\mathbb{C}}H^{p, q}(X)$. The question now is: why is this a polynomial? More…
phil
  • 607
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constructing a holomorphic subbundle from a holomorphic section

Assume $X$ is a Riemann surface and $\pi: E \to X$ is a holomorphic vector bundle of rank $n$ on $X$. Let's say $f$ is a holomorphic section of $E$ over $X$. It is clear that if $f$ is a nonvanishing holomorphic function then $F := \bigcup_{x \in…
Acton
  • 815
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Why is Fubini-Study metric invariant under $SU(n+1)$ action on $\mathbb{P}^n \mathbb{C}$

In a lot of books on complex geometry, they say that the Fubini-Study metric on $\mathbb{P}^n \mathbb{C}$ is invariant under the action of $SU(n+1)$ on $\mathbb{P}^n \mathbb{C}$, but i can not see why. The Fubini Study metric $\omega_{FS}$ is…
Alicia Basilio
  • 149
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Why does the matrix of 1 forms associated to the Chern connexion is not nul ?

I am a bit confused about the following fact ( cf Griffiths and Harris, principles of algebraic geometry): Let $E$ be a hermitian vector bundle on a complex manifold M. Let $D$ be a connexion on E such that $D^{0,1}=\overline{\partial}$ and $D$…
Axel S
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How does the following define a hermitian structure?

Suppose we have a complex holomorphic line bundle $L$ over a real manifold $M$. We define a hermitian structure $h$ on $L \to M$ as a hermitian product $h_x$ on each fibre $E(x)$ which depends differentiable on $x$. Suppose that $L$ is generated…
user7090
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There exists a subspace not intersecting the analytic variety.

Given an analytic subvariety $V$ of dimension $k$ in $\mathbb{P}^n$ and a point $p$ not in it, how do I show that there is an $n-k-1$ plane $\mathbb{P}^{n-k-1}$ containing $p$ and not intersecting $V$? Is this some shady intersection theory stuff or…
Chertopkhanov on Malek Adel
  • 1,557
4
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2 answers

riemann surface and discontinuous group action

If G is a group that acts properly discontinuously on a Riemann surface X , than we can give to the quotient X/G a structure of Riemann surface such that the projection p:X→X/G is holomorphic. How can I prove that?
angy
  • 45
4
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2 answers

Compact complex surfaces with $h^{1,0} < h^{0,1}$

I am looking for an example of a compact complex surface with $h^{1,0} < h^{0,1}$. The bound that $h^{1,0} \leq h^{0,1}$ is known. In the Kähler case, $h^{p,q}=h^{q,p}$, so the example cannot be (for example) a projective variety or a complex torus.…
user31559
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