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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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When is the pullback of a holomorphic $(n,0)$-form by such a map a holomorphic $(n, 0)$-form?

Suppose we have two Kähler manifolds $(M, w_1, J_1)$ and $(N, w_2,J_2)$ and a diffeomorphism $f:M \rightarrow N$ such that $f$ preserves the complex structure and the symplectic form. If $\Omega$ is a holomorphic $(n,0)$-form on $N$, when can we say…
Ashley
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Prove the Jacobian of a curve of genus g is a complex torus

As stated in the title I am about to prove the Jacobian of a curve of genus g is a complex torus. Here is what I have done so far: I know the first homology group of $X$ is $H_1(X,\mathbb{Z}) \cong \mathbb{Z}^{2g}$. Let $\alpha_1, \dots, \alpha_2g$…
Federico
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first Chern class and divisor under modifications

Assume that $X$ is a Moishezon manifold, then there exists a modification $\pi:\tilde{X}\rightarrow X$, where $\tilde{X}$ is a projective algebraic manifold. Let $\tilde{w}$ be a Kahler metric on $\tilde{X}$, then we can construct a holomorphic line…
user95633
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confusion about the notion of $\bar{\theta^j}$ and $\bar{\theta_i^j}$

Let $(M,J,g)$ be an almost Hermitian manifold, and $\{e_i\}$ be $(1,0)$-vector field basis, $\{\theta^i\}$ be its dual basis. We have $$g=g_{i\bar{j}}\theta^i\otimes\bar{\theta^j}$$ If connection $D$…
gaoxinge
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Extending a d-closed (p,q) form of a fibre of complex analytic family

Let $\phi: X \to B$ be a family of complex manifolds. Fix a point $0 \in B$ and $X_0 := \phi^{-1}(0)$. For any $\alpha \in A^{p,q}(X_0) := \{C^\infty (p,q)\text{ forms on }X_0\}$ such that $d \alpha = 0$, can we extend it to $\tilde\alpha \in…
Tei Huang
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complex coordinate system

what is the actual definition of "complex coordinate system"? I am not referring for 'complex number' nor 'polar coordinate system'. These terms are overlapping in my mind and i am unable to get what "complex coordinate system" is? Please help me ,…
nayab
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Newlander-Nerenberg theorem in Voisin's book

I am reading the proof of Newlander-Nerenberg's theorem in the real analytic case, and there are some parts I don't understand, can someone help me please? 1) In the beginning, she said: " since everything is local, we may assume that X is an open…
Long
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A step in the proof of holomorphic Frobenius theorem

Let $M$ be a complex manifold and $I$ be the almost complex structure on $TM_{\mathbb{R}}$, and $E$ be a rank $k$ distribution stable under $I$. Let $\psi\colon U \to V$ be a submersion whose fibre are maximal integral manifold of $E$, or say $\ker…
fyx1123581347
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A question on Demailly's proof to the canonical isomorphism of tangent bundle of Grassmannian

Let $G_{r}(V)$ be the Grassmannian of a complex vector space $V$ consists with subspace of codimension $r$. It is well known that $$TG_{r}(V)=Hom(S,Q)$$ where $S$ is the tautological subbundle and $Q=G_{r}\times V/S$. I have some trouble with…
Jiang Tianshu
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Induced hermitic metric on the exterior algebra bundles

Let X be a complex manifold equipped with an hermitic metric $h:T_X \times T_X \rightarrow \mathbb{C}$. For me it is clear that this induces an hermitian metric on the exterior algebra bundles $\bigwedge\nolimits^k T^*_X$. My question is now, how…
dr01
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Line bundle on a complex torus whose Chern class is given.

Let $\mathbb C$ be a standard complex plane, $\Lambda=\mathbb Z+i\mathbb Z$ be a lattice in $\mathbb C$, then we can get a torus $T=\mathbb C/\Lambda$, when $T$ is equipped with a standard Kähler metric $h=dz\otimes d\bar z$, then its associated…
Tom
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A little problem in a proof of a corollary in Huybrechts' book

In proof of corollary 1.1.19 in Page 16. In the last paragraph he claims that Since $g$ is irreducible and $\frac{\partial g}{\partial z_1}$ is of degree $d-1$, there exist elements $h_1,h_2\in\mathcal{O}_{\Bbb{C}^{n-1},0}[z_1]$ and…
user867836
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Notation in the proof of the existence of Chern connection (Huybrechts's book)

My question is simple. I had read about the existence of Chern connection in the Huybrechts book Complex Geometry p. 177 but I don't remember meaning of some notation : Here the $A$ is the connection matrix with respect to the $\{e_i\}$ Q. What is…
Plantation
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ideal generated by elements of sheaf of holomorphic functions

Let $I \subset \mathcal{O}_{\mathbb{C}^2,0}$ be the ideal generated by $z_1^2-z_2^3+z_1$ and $z_1^4-2z_1z_2^3+z_1^2$. Describe $\sqrt{I}$. I know $\sqrt{I}:=\{f \in \mathcal{O}_{\mathbb{C}^2,0}: f^k \in I, \space \text{for some $k$}\}$. So how so I…
homosapien
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Question about the Kodaira embedding theorem

I'm now studying Daniel Huybrechts, Complex Geometry. But I can't understand some defitnions ; Q.1) What is the natural restriction map $H^{0}(X,L) \to L(x)$ ? $s \mapsto s(x)$? If so, why the surjectivity of the maps is equivalent to $BS(L) =…
Plantation
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