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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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Cohomology vanishing on projective manifold, want to show that a line bundle $L$ is ample

I have some questions regarding the proof of the following theorem. Let $X$ be a projective manifold and $L$ a line bundle on $X$. Then $L$ is ample if and only if for all coherent sheaves $\mathcal{F}$ on $X$ there is an $m_0$ such that for all $m…
Slash_
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Is the subspace $F^pH^k(X_t,\mathbb C)$ varies smoothly in $H^k(X,\mathbb C)$ when Frölicher spectral sequence of $X$ degenerates at $E_1$?

Let $\pi:\mathcal X\to B$ be a holomorphic family of compact complex manifolds with $X_t:=\pi^{-1}(t),t\in B, X_0=X$. It is known that if the central fiber $X_0$ is Kähler, then the Hodge filtration $F^pH^k(X_t,\mathbb C):=\frac{F^pA^k(X_t)\cap \ker…
Tom
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Kodaira dimension of submanifolds

I encountered the following problem in Complex Geometry-an introduction of Dianel Huybrechts: Show that any submanifold of a complex torus has nonnegative Kodaira dimension. Since $\operatorname{kod}(T^n)=0$, I think it's sufficient to show that…
eulershi
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Understanding a proof of positivity of some relevant line bundles (Complex Geometry)

I'm reading the Daniel Huybrechts's Complex Geometry, p.249, Lemma 5.3.2. It is used to prove the Kodaira embedding theorem. Accepting the Lemma as true, I somewhat understand the Kodaira embedding theorem. And I want to understand the Lemma…
Plantation
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Complex vector fields on $2n$-dimensional smooth manifolds: Worked out example.

I am really struggling with the notion of complex vector field on a $2n$-dimensional smooth manifold and I am hoping to work out a down-to-earth example. I am very confused so the questions might be a bit wacky. Let $X=\mathbb{R}^2$ with global…
user7090
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Complex structure is parallel with respect to Chern connection

Let $(M,J)$ be a complex manifold with Hermitian metric $h$, and let $\nabla$ be the Chern connection on $TM^{\mathbb{C}}$, then $\nabla h = 0$ and $\nabla^{0,1}=\bar{\partial}$. I want to show that $\nabla J = 0$. For any $X,Z\in TM^{\mathbb{C}}$,…
string
  • 743
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K3-surface is not the blow-up of any other smooth complex surface?

Good evening, I'm stuck in the following exercise in Huybrechts, Complex Geometry, chapter 2, page 103. Let $X$ be a K3 surface, i.e. X is a compact complex surface with $K_X \cong \mathcal{O}_X$ and $h^1(X,\mathcal{O}_X)=0.$ Show that X is not the…
Đức Anh
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Affine Hypersurface is a complex manifold of dimension $n - 1$

Let $f: \mathbb{C^n} \rightarrow \mathbb{C}$ be a holomorphic function and such that $0 \in \mathbb{C}$ is a regular value. Then the result to be proven is that the set of zeroes $X := Z(f)$ is in fact a complex manifold of dimension $n-1$. I do…
Soby
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Integration on Complex Manifolds

Let $M$ be a complex manifold. Then there are three different notions of a tangent space on $M$; The real tangent space locally generated (on some coordinate patch) over smooth real functions locally on a coordinate patch by $\partial_{x},…
Yuugi
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What is the meaning of $\bar{\nabla}$?

In Griffiths and Harris's 'Principles of Algebraic Geometry', the authors use the symbol $\bar{\nabla}$ to prove the Weitzenbock identity. But they never show the definition of the symbol. What is the meaning of this symbol and where can I find a…
MiGang
  • 311
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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 of $\underline{\mathbb{C}}^{n+1}\otimes L^{-1}$.…
gaoxinge
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Definition of a Subtorus

Let $V$ be a finite vector space over $\mathbb{C}$ and consider a lattice $L$ of $V$ i.e a discrete subgroup of $V$ of maximal rank. Consider the torus $T=V/L$. Definition: Let $S\subset T$ be a subset of $T$. We say that $S$ is a subtorus of $T$ if…
Bartolomeo
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Giving Holomorphic structure to Complex line bundles on projective space and torus

I am doing questions from [Huybrechts, Complex Geometry, An Introduction] page 143 questions 3.3.7 and 3.3.8. Basically the questions ask: For question 7, for any complex line bundles on the projective space $\mathbb{P}^{n}$, we may endow it with a…
enoughsaid05
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Rank of a Holomorphism

Let $f \colon M \rightarrow N$ be a holomorphism of complex manifolds. Let $p \in M$. Let $(U,\phi)$ and $(V,\psi)$ be coordinate charts on $M,N$, respectively, satisfying $U \ni p$ and $V \ni f(p)$. I don't know what you call it, but as $f$ is…
Open Season
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holomorphic map between compact Riemann surfaces

Why nonconstant holomorphic map between compact riemann surfaces is surjection? I don't understand: if open-closed connected subset X of connected space Y, then X is all Y??
Math
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