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.
4,026 questions
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Is $\mathbb{CP}^2-\mathbb{RP}^2$ a Stein manifold?
The space $\mathbb{CP}^2-\mathbb{RP}^2$ looks plausibly Stein to me, because it's a complex manifold with the homotopy type of $S^2$, so it has the correct homological dimension. Is it actually, ...
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Clarification on a passage of Birkenhake-Lange about the connecting homomorphism in group cohomology
I am studying from "C. Birkenhake, H. Lange - Complex abelian varieties" and I was having trouble understanding a certain passage on chapter 2.1.
Let $X=V/\Lambda$ be a complex torus.
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What is an example of a variety that is also a manifold with corners?
In the realm of Borel-Moore and intersection homology one deals abstractly with topological pseudo-manifolds; I guess because they impose conditions on how badly intersections of cycles deviate from ...
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Simplifying a proof that flat connection has flat sections
$\newcommand{\Omegait}[0]{\mathit{\Omega}} \newcommand{\qty}[1]{\left\{#1\right\}} \DeclareMathOperator{\GL}{\mathrm{GL}}$ To fix notation, let $\nabla:\mathcal E\to \Omega^1_X\otimes_{\mathcal O_X} \...
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Statements about dimension of singularity loci in families of polynomials
I am working with some homogeneous polynomials $f\in \mathbb C[x_1, \ldots, x_n]$ belonging to families $\mathcal F$ over $\mathbb C^k$, i.e. I am considering
$$f=\sum a_Ix^I, \qquad\qquad a_I\in \...
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Alternative description of $Pic(\mathbb{F}_1)$
Considering the Hirzebruch surface $\mathbb{F}_1=\mathbb{P}_{\mathbb{P}^1}(\mathcal{O}\oplus\mathcal{O}(1))$, we can view it as a blowup of $\mathbb{P}^2$ at a point $p$ and form a basis of $Pic(\...
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Conditions for a map of germs of complex spaces to be a morphism
Let $(X,x), (Y,y), (Z,z)$ be germs of complex spaces and let $f\colon(X,x)\to (Z,z), g\colon (Y,y)\to (Z,z)$ be morphisms of germs of complex spaces.
Suppose there exists a map $h\colon(X,x)\to (Y,y)$ ...
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Does every abelian variety contain an elliptic curve?
Let $X$ be an abelian variety. In this question, that will mean that $X$ is a complex torus $\mathbb C^n / \Lambda$, where $\Lambda \subset \mathbb C^n$ is a lattice, and that $X$ also a projective ...
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Equivalence of Kahler forms on a cubic curve
The vanishing of the homogeneous polynomial $f(z_{1},z_{2},z_{3})=z_{1}^{3}+z_{2}^{3}+z_{3}^{3}+3\lambda z_{1}z_{2}z_{3}$ in $\mathbb{CP}^{2}$ defines an elliptic curve, $X$. I am interested in two ...
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Cycle class map of deRham cohomology
Let $X$ be a smooth complex projective variety. For a closed subvariety $Z$ of codimension $p$, the fundamental class $[Z]\in H^{p,p}(X)$ is defined by the condition $\int_X\alpha\wedge[Z]=\int_Z\...
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Different approach to proving a Riemann surface theorem
Suppose $c_{1}, \ldots, c_{n}$ are holomorphic functions on the disk
$$ D(R) = \left\{ z \in \mathbb{C}: \left\vert z \right\vert < R \right\}, R > 0 $$
Suppose $w_{0} \in \mathbb{C}$ is a ...
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Can one generalize the Kobayasbi-Hitchin Correspondence to handle more general stability conditions like Bridgeland stability conditions?
I’m definitely going to have to edit this and I’m typing via phone — can one generalize the Kobayasbi-Hitchin correspondence to handle more general notions of stability?
Even better, is there a way to ...
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Clarification on normal bundle to an embedding
Background.
In section 27.5 of the Mirror Symmetry book, the following claim is made:
The normal bundle of any embedding $\mathbb{P}^1 \subseteq X$ (where $X$ is a Calabi–Yau threefold) is a rank $2$,...
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Signs of de Rham cycle class maps
I have the same post in mathoverflow.
For a general smooth complex variety $X$, I know two ways to define de Rham cycle classes with supports for subvarieties $Y\subseteq X$.
The theory of currents.
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Topological / Geometric Proof that $F_2$ embeds in $\text{SO}_3$. [duplicate]
A critical step in one of the common proofs for Banach-Tarski is to show that the free group $F_2$ on two generators embeds in the special orthogonal group $\text{SO}(3)$. Most of the proofs I have ...
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