# Tagged Questions

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

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### Vector bundle morphisms and determinants

Let $F,E$ be two locally free sheaves of equal rank on a complex manifold $X$. Assume that we have an injection $0\to F\to E$ such that the induced map $0\to \det F\to \det E$ is an isomorphism. Is ...
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### Possible subbundles on $\mathbb{CP}^1$

I have the vector bundle $E:= \mathcal{O}(1)^{\oplus k}$ on the projective line, $\mathbb{CP}^1$. I want to know what are the possible subbundles of this, and why? We know that the Birkhoff-...
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### Complex Elliptic Surfaces without sections

Is there a description of smooth complex projective surfaces without sections? While working on a problem a surface $X$ showed up with the following property: it is a non-ruled surface that has an ...
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### Why there is no biholomorphism between complex plane and unit disk? [closed]

Why there is no biholomorphism between complex plane and the unit disk?
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### Choice of Fundamental Domain of Torus (Dehn Twists?)

So I would like to consider a lattice $\Lambda \subseteq \mathbb{C}$ generated by $(1,\tau)$ with $\tau$ in the upper-half complex plane. This lattice $\Lambda$ will remain fixed. If you choose the ...
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### Show that $|1-h \lambda| <1$ is a disc

From the stability region of Euler, show that $|1-h \lambda| <1$ is a disc, where $\lambda$ is imaginary. I am wondering why it is a disc with center at $h \lambda = -1$
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### How much algebraic geometry is there in complex geometry (for example, Demailly)?

I wonder how much of modern algebraic geometry (schemes, etc) is there in complex (algebro-analytic) geometry. What I mean is complex algebraic (analytic) geometry in the sense of Demailly, Lasarsfeld ...
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### Vector bundle as locally free coherent sheaves

I am studying coherent sheaves and was looking for a geometric motivation. Hence, in wikipedia and although here is stated that it can be seen as a generalization of vector bundles, which is quite ...
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### Is there an irreducible projective hypersurface such that its complement has zero Euler characteristic?

We know that, if $f=X_0X_1...X_n \in \mathbb{C}[X_0,...,X_n]$ and $Z(f)\subset \mathbb{CP}^n$, then the Euler characteristic of its complement is zero, i.e. $$\chi(\mathbb{CP}^n\setminus Z(f))=0.$$ ...
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### Which line bundle has transition function $\psi_{12}([z_1,z_2])=\frac{z_1/z_2}{|z_1/z_2|}$?

We know that the complex line bundles over $\Bbb{CP}^1$ are classified by the integers. Each is isomorphic to one of the bundles $\mathcal{O}(n)$ for $n\in\Bbb Z$, where $\mathcal{O}(-1)$ is the ...
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### Analytic proof of Serre vanishing theorem

Consider the following equivalent statement of Serre vanishing theorem (replacing ampleness condition on the line bundle with postivity condition). Let $X$ be a compact complex manifold. Let $L$ be ...
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### Imaginary lines and tangents

My geometry text seems to say that the tangent to a circle from an interior point in the real plane is imaginary. Further...it seems that when a double cone is intersected by a plane with an angle ...
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### valid definition of complex geodesic

Let $X$ be a complex manifold, and let $Y$ be a complex submanifold of $X$. If $X$ has an hermitian structure(on its tangent bundle), we can consider the Chern connection $\nabla$ on the holomorphic ...
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### Functions that maps unit circle into unit circle

This and This problems discuses the characterization of Analytic functions which maps unit circle on to itself. I would like to know the characterization of functions which map unit circle into ...
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### Horrocks-Mumford bundle

Let $E$ be the Horrocks-Mumford bundle, which is a rank-2 vector bundle on $\mathbb P^4$ with $c_1(E)=5$ and $c_2(E)=10$, defined by some combinatorical construction (see Okonek, Schneider, Schindler, ...
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### One of Hermitian metric's properties?

We now define a Hermitian manifold is a complex manifold in which unmixed components of metric tensor vanish $g_{ij}=g_{\bar{i}\bar{j}}=0$. Is this a propert of a Hermitian manifold? Or is it an extra ...
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### first chern class of holomorphic tangent bundle $T\mathbb{C}P^n$

Let $L$ be tautopological bundle of $\mathbb{C}P^n$ and $L^{-1}$ be its duality. Because $L$ is a subbundle of $\underline{\mathbb{C}}^{n+1}$, $\underline{\mathbb{C}}=L\otimes L^{-1}$ is a subbundle ...
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### Algebraic surface with infinitely many exceptional curves

I am learning about the classification of Projective Algebraic Surfaces (in fact, Compact Complex Surfaces) and I am troubled with the following point. If I understood correctly every surface $X$ ...
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### Is there an elementary way to see that there is only one complex manifold structure on $R^2$?

Is there an elementary way to see that there is only one complex manifold structure on $\mathbb{R}^2$? (Up to biholomorphism, naturally.) Elementary in the sense of not appealing to the ...
### What does $dz^2$ mean?
I'm reading a paper ("La Formule de Verlinde" by Christoph Sorger) and at a certain point, the author switches from algebro geometric language to complex geometric language. He uses the symbol $dz^2$, ...
$X$ is a compact Kähler manifold or smooth projective variety. is there an example that a primitive class $0\neq [\omega]$ of $H^{p+q}(X, \mathbb{C})$ is wedge product of other two primitive classes: \$...