# 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.

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### Relative differential operators and a Proposition due to Kodaira

The following appears in Claire Voisin's Hodge Theory an Complex Algebraic Geometry I. Consider a family $\phi: \mathcal X \to B$ of complex manifolds, and assume that $X_0, 0 \in B$ is a Kähler ...
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### There is no linear mapping which maps complex ellipsoid onto the unit disc in $\mathbb{C}^2$

I am trying to solve the following question: For each real $p\geq 1$ consider the set $$D_p=\{(z,w)\in \mathbb{C}^2:|z|^{2p}+|w|^2<1\}.$$ Then there is no linear operator on $\mathbb{C}^2$ which ...
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### Prerequesites to study algebraic analysis and current landscape in field of research

A couple of weeks ago I became aware of the existence of algebraic analysis, specifically the area described in here https://en.wikipedia.org/wiki/Algebraic_analysis and also the theory of D-modules, ...
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### Easier proof that the Grassmannian is a complex manifold

$G_r(\mathbb C^3,2)$ is the topological space of 2-dimensional complex linear subspaces of $\mathbb C^3$. Prove that $G_r(\mathbb C^3,2)$ is a complex manifold. I have a solution to this problem, but ...
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### Relation between invariant meromorphic and invariant holomorphic functions

Given an affine space $\mathbb{C}^n$ (more generally a Stein space), and an action of a complex Lie group $G$ on it. Is there a relation between (sheaves of) invariant holomorphic and invariant ...
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### Distinct ellipsoids are not biholomorphic in $\mathbb{C}^2$

I am stuck on the following problem: For each real $p\geq 1$ consider the the set $$D_p=\{(z,w)\in\mathbb{C}^2:|z|^{2p}+|w|^2<1\}.$$ Let $p\neq q$, then, there does not exist any biholomorphism ...
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### Group actions on product varieties by diagonal

Let $X,Y$ be two varieties, and $G$ be a group together with actions on $X$ and $Y$. Moreover, we assume the action on $X$ is free. I would like to know if the following statement is true: There is ...
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### formal andjoints in hermitian structures

Given a hermitian vectorbundle $E\to \Sigma$ on a complex plane. The hermitian structure be given by $\langle \cdot, \cdot \rangle$. Let $A \in \Omega^1 (\Sigma; End(E))$ be a complex smooth bundle ...
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### Intersection number of complex curves in a complex surface

Suppose $C_1,C_2$ are embedded complex curves in a complex surface $S$, and $C_1,C_2$ have no common component. Assuming $C_1$ and $C_2$ intersect transversally, the intersection number $C_1\cdot C_2$ ...