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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$h^{1,1}$ of blow-up of a surface

Let $S$ be a smooth projective variety of dimension $2$ over $\mathbb{C}$, consider the blow-up $\tilde{S}$ of $S$ along one point $x\in S$. How can I show that $h^{1,1}(\tilde{S})=1+h^{1,1}(S)$?
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How to write down a loop on $\mathrm{Diff}(\mathbb{S}^2)$ concretely?

If we regard $\mathbb{S^2}$ as the complex projective line $\mathbb{P}^1$, and define a loop on its Autodiffeomorphism group: $$\gamma: \mathbb{S}^1\longrightarrow\mathrm{Diff}(\mathbb{S}^2)$$ $$z\...
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On global section of a line bundle.

At Proposition 1.1.1. (1) of https://arxiv.org/pdf/0706.0494.pdf it is written that $H^0(X, L \otimes {\cal{J}}(X, ||L||)) = H^0(X, L)$ ensures that every global holomorphic section $s$ of $L$, i.e., $...
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Why are these two definitions of conjugate vector bundle the same?

Let $E$ be a complex vector bundle over some space $X$. Definition 1: Let $E'$ be the complex vector bundle with the same total space as $E$, but with conjugate complex multiplication. That is, if $\...
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Does tautological line bundle $\mathcal{O}(-1)$ over complex projective space have zero section?

In Complex Geometry written by Huybrechts, it defines the tautological line bundle $\mathcal{O}(-1)$ as followed $$ \mathcal{O}(-1):=\{(\ell,z)\in \mathbb{C}P^n \times \mathbb{C}^{n+1}: z\in \ell\} $$ ...
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What does it mean when a chern class of a vector bundle is postive(resp. negative)?

Recently i was studying line bundles on $\mathbb{C}P^1$. Here is my confusion: for any holomorphic map $f:\mathbb{C}P^1 \to M$, where $(M,E,\nabla)$ is a $r$-rank holomorphic vector bundle with a ...
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Cellular decomposition of complex manifold

In which cases a complex manifold can be decomposed in the sense of a cellular decomposition into cells which are complex local submanifolds? A positive example are the complex projective spaces. But ...
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Why use orientation-preserving diffeomorphism (instead of all diffeo's) in the construction of the moduli space of a Riemannian manifold

The question is basically in the title, but I want to make it more precise: Given an oriented Riemannian 2-manifold $\Sigma$ one can take a quotient of the set $$ \mathcal{M}_+(\Sigma)=\{~c~\mid (\...
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Calculation on Riemannian manifolds

I am learning the variational calculation of Yang Mills functional, but I can't understand 2 steps in the following calculation: Given a variation of the connection $A$ in local coordinates: $A\to A+\...
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What is an instanton? (On a complex surface or a differentiable 4-manifold )

The question is as in the title. I have browsed online (Wikipedia, etc) and while they do give me the definition, it gets a bit too much physics-y for me. Therefore I would appreciate it if someone ...
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Lie derivative in Kahler manifold

I have the following question $X$- a compact Kahler manifold and $v\in \Gamma(X,TX)$- Killing vector field. I don't understand why Lie derivatives equal to zero $L_v \omega=0$ where $\omega$ is the ...
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Purely-imaginary Harmonic function on hyperbolic space is constant

Suppose I have a function $$ f: \mathbb H^3 \to \mathbb C $$ that is harmonic and whose image can be shown to be purely imaginary. Is it possible to deduce that it is in fact constant?
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Rank of a coherent sheaf using resolution by vector bundles

The rank of a coherent sheaf is defined in terms of the Hilbert polynomial (See Huybrechts-Lehn 1.2.2 or Rank of a coherent sheaf in terms of coefficients of the Hilbert polynomial). Now let $\mathcal{...
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holomorphic line bundle over germ of complex space

I wonder if holomorphic line bundle over zero of holomorphic function(over $\mathbb{C}^n$) is trivial?(We can assume it to be a manifold if necessary) I Know there is a principle that, for Stein ...
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When determinant line bundle is holomorphically trivial

I'm learning the deformation theory of holomorphic structure over given smooth vector bundle by the book Smooth four - manifolds and complex surfaces. However, when talk about holomorphic vector ...
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Kahler manifold computation

In the following derivation, done on a Kahler manifold, where $\nabla$ is the complexification of the Riemannian connection (i.e., since we are on a Kahler manifold, this is the same thing as the ...
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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 ...
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Construction of the isomorphism $V^*\simeq H^0(V/L,\Omega^1_{V/L})$ for a complex vector space $V$ with lattice $L$

Le $V$ be a complex vector space and $L\subset V$ be a lattice in $V$. Then $T=V/L$ is by definition a complex torus with universal covering space $V$. In Beauville ch 5, it is stated that there is an ...
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Difference between stabilizer and automorphism group

Let $X$ be a smooth closed subvariety of a complex abelian variety $A$. Assume $X$ is of general type and of codimension one with $\omega_X$ ample. Often, people speak about the stabilizer $\mathrm{...
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Example of closed non-exact torsion differential form on variety

I am interested in finding a particular example. I would like to find a variety (analytic or algebraic over $\mathbb{C}$) such that the de Rahm-Sequence is not exact and the variety admits a ...
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Proving the holonomy group of Kahler manifolds is a subgroup of $U(n)$

I'm trying to prove that the holonomy group of an $n$-dimensional Kahler manifold $(M,g)$ is a subgroup of the unitary group $U(n)$. Texts usually say something like "the affine connection $\...
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Is a usual open ball in a complex algebraic variety Zariski dense?

Let $X$ be an affine variety $\operatorname{Spec} \mathbb C[x_1,\dots, x_n]/(f_1,\dots, f_m)$. Suppose the set of closed points gives a smooth complex analytic variety in $\mathbb C^n$. Pick any $p\in ...
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It's true that $\mathrm{D}_{X\times Y}(M^\bullet\boxtimes N^\bullet)=\mathrm{D}_X(M^\bullet)\boxtimes \mathrm{D}_Y(N^\bullet)$ for D-modules?

Let $X,Y$ be smooth algebraic varieties over $\mathbb{C}$, $M^\bullet\in\mathsf{D}^b_{\text{coh}}(\mathcal{D}_X)$, $N^\bullet\in\mathsf{D}^b_{\text{coh}}(\mathcal{D}_Y)$, and denote by $\mathrm{D}_X$ ...
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linearly independent coordinate functions implies nondegenerate image

I am currently working through chapter 5 Rick Miranda book on Riemann surfaces. On page 157 he makes the comment that if $f_0, \dots, f_n $ are meromorphic functions and $ \psi: X \rightarrow \mathbb{...
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Chern forms and tensor products

Let $E\to X$ be a rank $r$ holomorphic vector bundle over a Kahler manifold and let $L\to X$ be a holomorphic line bundle. The following relation between Chern classes is (well) known: $$c_2(E\otimes ...
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What is the scalar curvature of $\mathbb{CP}^n$?

I'm trying to find the scalar curvature/Ricci scalar $R$ of $\mathbb{CP}^n$ under the Fubini-Study metric, $g_{\mu \bar{\nu}}^{\textrm{FS}}$. As $\mathbb{CP}^n$ is a Kahler manifold, we know that the ...
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Complex structures on SL(2,C) [closed]

I know that only few questions are solved in the case of deformation of complex structures on non-compact manifold, but do we know what are all the complex structures on $\operatorname{SL}(2,\mathbb{C}...
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A question about Constant sectional curvature between the Riemannian and Kähler manifold

According to the corollary (3.6) on An Introduction to Differentiable Manifolds and Riemannian Geometry, Def. A Riemannian manifold $M$ is isotropic at $p \in M$ if the curvature is the same constant ...
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How the subject Several Complex Variables motivates to study Complex Manifolds?

How the subject Several Complex Variables motivates to study Complex Manifolds/ Complex Geometry? How one can study Several Complex variables? Please advise me a roadmap to learn this subject
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Adjoint of a end-valued complex differential form

Let $E\to X$ be a holomorphic Hermitian vector bundle over a complex manifold. Let $\xi\in \Omega^1(X,\operatorname{End}(E))$ be an end-valued form. We define its adjoint $\xi^*$ by the identity $$h(\...
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Stratification of a space which induces a stratification of a subspace

Let $\{X_i\}_i$ be a stratification of the (smooth, complex, algebraic) manifold $X$ and let $Y\subset X$ be a closed submanifold of $X$. Is it true that the family $\{Y\cap X_i\}_i$ defines a ...
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How can I show that set is cross-ratio?

We have $4$ points: $P_1 =[0:0:0:1], P_2=[1:0:0:-1], P_3=[1:0:0:1], P_4=[1:0:0:0]$ in $\mathbb{P}^3$. I want to show that this set of points is cross- ratio. I am trying to prove it by use equation: $$...
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Dimension of holomorphic vector fields on $\mathbb{P}^1$

Let $SL_2(\mathbb{C})$ act on $\mathbb{P}^1$ by $g([v]) = [gv]$. Let $\Gamma(T\mathbb{P}^1)$ be the space of sections on $T\mathbb{P}^1$. Find its the dimension and show it is isomorphic to the ...
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Does the residue map Res commutes with d?

Let $M$ be a complex manifold of dimension $n$, and $D$ be a smooth hypersurface. Let $\varphi$ be a $C^{\infty}$ $ k$-form on $M \backslash D$. We say that $\varphi$ has logarithmic singularities on $...
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How to define the pullback of a local section of an coherent analytic sheaf?

In some book on singularities, many singularity is defined via the ``pullback of generator" of the dualizing sheaf under some resolution. But they did not define the pullback of generator. Let $...
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Integration of complex $(p,q)$-form

In complex geometry, we have $(p,q)$form $\in$ $\wedge^{p,q}T^*X$, I wonder how to define their integration on submanifold, or top-form on all manifold. For instance, in the Riemann surface book I ...
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Compactness of the linear system when varying complex structure

Let $M$ be a compact, simply connected smooth manifold and $L\rightarrow M$ a complex line bundle over $M$. Assume there is a continuous path of Kähler structures $(g_t,I_t)\; t\in [0,1]$ on $M$ ...
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1 vote
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Relationship between Kahler volume form and Riemannian volume form

On an Kähler manifold $(M,g)$ of complex dimensional $n$, equipped with a Kähler form related from the metric as $$ \omega = \frac{i}{2} g_{\mu \bar{\nu}} dz^{\mu} \wedge d\bar{z}^{\nu},$$ we can ...
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2 votes
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Topological classification of complex surfaces

The famous Enriques–Kodaira classification classfies the minimal complex surfaces by algebraic invariants, which give tight restrictions on the topology of the underlying manifolds. What about the ...
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Holomorphic locally trivial fibrations over the puctured disk

Let $X$ be a complex manifold, and assume that I have a holomorphic locally trivial fibration $X \to \mathbb{C}^\times$. How does the complex orientation affect the possibilities for the structure of ...
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Elementary proof that a bijective morphism between smooth algebraic varieties is an isomorphism?

Let $f:W\rightarrow V$ be a bijective morphism of smooth (irreducible) algebraic varieties over the complex numbers. It is a fact that $f$ is an isomorphism. This fact is typically seen as a special ...
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How do one show that the quotient space is a projective manifold?

I want to prove the following statement. Let $\Omega$ be a bounded domain, and $\Gamma \subset \text{Aut}(\Omega)$ be the subgroup acting totally discontinuously on $\Omega$ without fixed points such ...
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Resolution of $(f=0)$ where $f(x,y,z)=x^ay+y^az+\omega z^ax$

Let $u\geq 2$ be an integer, $a=3u-1$, $\omega=e^{2\pi i /3u}$, and let $f$ be the polynomial $f(x,y,z)=x^ay+y^az+\omega z^ax$. In Example 23 of https://www.intlpress.com/site/pub/files/_fulltext/...
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Complex gauge group

We define the action of complex gauge group on a connection $d_A$ satisfies the condition of curvature: $F_A\in \Omega^{1,1}$ as follows: Suppose $d_A=\partial_A+\bar{\partial}_A$, for any $g \in \...
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Do we have a pushforward for complex vector bundles?

Let $f:X\rightarrow Y$ be a morphism of schemes, we know that the pushforward for an algebraic vector bundle i.e. a locally free sheaf are defined by the direct image of a sheaf. Do we have similar ...
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decomposition of differential from in (1,1) tpye

Suppose $X$ a complex manifold with hermitian metric $g$ and complex structure $J$ over its tangent bundle. We define a $(1,1)$ from $\omega$ as $\omega(v_1, v_2)=g(v_1,Jv_2)$. My question is, how to ...
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Simply connected smooth points of complex analyic/algebraic variety

If $A\subseteq \Omega\subseteq \mathbb{C}^n$ is a complex analytic/algebraic variety of codimension $2$ or greater, in an open subset $\Omega \subseteq \mathbb{C}^n$, then there exists a neighborhood ...
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1 vote
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Integration of forms on the Riemann sphere

Imagine you want to integrate a specific differential form around the equator of the Riemann sphere. This form is such that it is holomorphic at all points above the equator but there is a pole ...
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3 votes
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$P(a,b,c)=P(bc,ca,ab)$ weighted projective planes for pairwise coprime $a,b,c$

Let $a,b,c\geq 2$ be pairwise coprime integers. The (complex) weighted projective plane $P(a,b,c)$ is the quotient of $\Bbb C^3-\{0\}$ by the action of $\Bbb C^*=\Bbb C-\{0\}$ given by $t\cdot (x,y,z)=...
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Holomorphic forms on some Calabi Yau

Let $(X, h)$ a compact Kahlerian manifold of complex dimension $n$ and $\omega$ a holomorphic form of type $(n, 0)$ which never vanishes (i.e., $X$ is Calabi Yau). One writes $h = g + iw$ with $dw = 0$...
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