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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13 views

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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58 views

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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Map $w(z)=\frac{1}{1+z^2}$ for $0<|z|<1$ is Conformal

Question Map from punctured disc $$D^*:=\lbrace z\in\mathbb{C}:0<|z|<1\rbrace$$ to a domain $P$ $$P:=\lbrace w\in\mathbb{C}:1/2<\Re(w), w\neq 1\rbrace$$ defined as $$w:D^* \to P $$ $$w(z)=\...
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Effective divisors $D_1\leq D_2$ such that $h^0(D_1)=h^0(D_2)$, then $D_1=D_2?$

Let $X$ be a smooth algebraic surface over $\Bbb{C}$. I'll use the following notation: \begin{align*} H^0(D)&:=H^0(X,\mathcal{O}_X(D))\\ h^0(D)&:=\dim_\Bbb{C}H^0(D)\\ |D|&:=\{\text{...
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How to plot the graph in polar coordinates of $r^2< \cos(2θ)$

$z$ is a complex number, $|z^2-1|<1$. Question is, to verify whether the above set is a region? So i tried to plot the graph of the above set, to get idea of the set. First i tried converting to, ...
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$w=\frac{1}{1+z^2}$ maps the unit disc onto the plane $\Re(w)>1/2$

Define a map on the unit disc $\mathbb{D}=\{z\in \mathbb{C} \mid |z| \text{ <1 }\}$ $$w(z)=f(z)=\frac{1}{1+z^2}$$ Question Prove that $w(z)=\frac{1}{1+z^2}$ maps the unit disc $D$ onto the plane $\...
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Kähler metrics on holomorphic vector bundles

Let $(X,\omega)$ be a Kähler manifold not necessarily compact of complex dimension $n$. Let $\pi:E\to X$ be a holomorphic vector bundle of rank $r$, then $E$ can be seen as a complex manifold of ...
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Deformations of Kähler manifolds are Kähler.

Suppose $f: X \to B$ is a proper and smooth morphism of complex manifolds, and suppose the fiber $X_0 \subset X$ over $0 \in B$ is a Kähler manifold. How can I show that there is a neighbourhood $U \...
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Upper-semicontinutiy of points of intersections of deformed complex submanifolds

Suppose $Z,F, M$ are complex manifolds and $Z\overset{\eta}{\leftarrow} F \overset{\tau}{\to}M$ is an analytic family of compact complex submanifolds of $Z$. Meaning $\tau$ is a proper submersion and $...
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Existence of a global section of $L(D)$ vanishing on $D$.

Given a complex surface $X$ and $D$ a smooth curve on $X$, there is an associated line bundle $L(D)$ over $X$. Is there always a global section $s\in H^0(X,L(X))$, s.t. $s$ is just vanishing on $D$?
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Complex Analytic Subset vs. Complex Analytic Set

In my complex geometry class we have introduced two concepts of analytic sets. Let $M$ be a complex manifold. A subset $A\subset M$ is called complex analytic subset, if for each $p\in M$ there ...
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Zariski topology & Analytic topology 2 [duplicate]

Let $Z \subset \mathbb{P}_{\mathbb{C}}^n$ be an irreducible projective variety. Now we can regard the complex space $\mathbb{P}_{\mathbb{C}}^n$ (therefore also $Z$) as a topological space with two ...
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Fubini-Study on complex abelian varieties

Let $A$ be a complex abelian variety. Then the Fubini-Study metric on $\mathbb{P}^N_{\mathbb{C}}$ restricts on to $A$ by pulling back along an embedding $A\hookrightarrow \mathbb{P}^N_{\mathbb{C}}$. I'...
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Metric on the dual line bundle

Let $X$ be a compact Kähler manifold and $L$ be a holomorphic line bundle on $X$ with a Hermitian metric $h$. I am struggling to understand how one induces a canonical dual metric $h^*$ on $L^*$. Now ...
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extension of plurisubharmonic functions across complex hypersurfaces

Let $U \subset \mathbb{C}^n$ be an open set and let $f: U \to \mathbb{R}$ be a continuous function. Moreover, assume that $f$ is smooth on the complement of a complex hypersurface $Z \subset U$ and ...
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Examples/classification of algebraic symplectomorphisms

I'm curious about examples of algebraic automorphisms of complex varieties which are symplectomorphism. For instance, can we classify the algebraic symplectomorphisms of $\mathbb{P}_{\mathbb{C}}^n$ ...
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Common components, common factors and common roots of polynomials

I am currently working on understanding the proof for the Bézout’stheorem and I get confused by the properties of Polynomials of the form P(x,y,z) and their respective curves in the 2-dim complex ...
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Video lectures for Complex Geometry

Does anyone know the video lectures of a Complex Geometry course which are freely available online? I aim to understand the book lectures on K3 surfaces by Daniel Huybrechts afterward.
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114 views

Lefschetz Hyperplane Theorem's original proof

I'm trying to understand the main ideas used in the original proof by Lefschetz of his Hyperplane theorem. Here it is sketched shortly (source: Here) and I want to fill the gaps: Let $X$ be an $n$-...
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71 views

Prove that $\int_{\mathbb{C}\mathbb{P}^3}c_1(\mathbb{C}\mathbb{P}^3)^3=64$.

I want to prove that $$\int_{\mathbb{C}\mathbb{P}^3}c_1(\mathbb{C}\mathbb{P}^3)^3=64,$$ where $c_1$ is the first Chern class. I know that for projective spaces of dimension $n$ (but maybe also in ...
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80 views

An almost complex structure on the real 2-sphere $S^2$

If $R:=\mathbb{R}[x,y,z]/(x^2+y^2+z^2-1)$ and $S^2:=Spec(R)$ is the real 2-spere, a classical result of Borel and Serre says that the only real spheres with an almost complex structure is $S^2$ and $S^...
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Canonical Bundle of product manifold

Let $X$ and $Y$ be two complex manifolds. I want to know how to identify the canonical bundle $K_{X\times Y}$ in terms of $K_X$ and $K_Y$. As we know the canonical bundle $K_X:=\det(\Omega_X)$, $\...
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Existence of coframe for Hermitian metric on complex manifold?

I am reading page 28 of the 1994 version of Principles of Algebraic Geometry by Griffith. Let $M$ be a complex manifold of dimension n, Griffith defined a Hermitian metric to be a positive definite ...
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79 views

Blowup extends a regular map to $\mathbb{P}^{N+1}$

Let $(X_0,X_1,...,X_n)$ homogeneous coordinates of $\mathbb{P}^n$ and let assume that $X^r \subset \mathbb{P}^n$ is a complex variety where $x:= (1,0,...,0) \in X$ and $X$ isn't a cone with vertex $x$....
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$H_1(\mathbb C\mathbb P^n \setminus D,\mathbb Z)=\mathbb Z/\text{deg} D\mathbb Z$

Let $D$ be a hypersurface in $\mathbb C\mathbb P^n$, I saw the claim from the class that $H_1(\mathbb C\mathbb P^n \setminus D,\mathbb Z)=\mathbb Z/\text{deg} D\mathbb Z$ but couldn't see why. I ...
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Pushforward of fundamental class under rational function

Consider we have a rational function $f$ on $S^2=\mathbb{C} \cup \{\infty \}$ with algebraic degree$f=n$, where the algebraic degree is defined to be the maximum of degrees of its numerator and ...
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20 views

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 ...
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112 views

Definition of a complex space

A complex analytic subset of a complex manifold $M$ is a closed subset which is locally defined as common zeros of finitely manly analytic functions. A complex space is a second countable Hausdorff ...
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General properties of cubic hypersurfaces

Is there any literature dealing with cubic hypersurfaces in full generality (over $\mathbb{C}$)? Couldn't find any. We know everything about hyperplanes. We also know a lot of things about quadric ...
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57 views

Convert Weierstrass form to real $T^2$

A one-dimensional complex torus can be described as the quotient of $\mathbb{C}/\{m_1 \omega_1 + m_2 \omega_2 \}$, where $m_i \in \mathbb{Z}$ and the $\omega_i$ are complex numbers which form a basis ...
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Relation of Euler number of a line bundle over a Riemann surface and a section of it with finitely many isolated singularities

Suppose $M$ is a compact Riemann surface and $E\to M$ a holomorphic line bundle over $M$. Also suppose $s:M\to E$ is a section with finitely many isolated singularities (undefined points, and not ...
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A Formula of Jet Bundles in Gelfand's Book

In Gelfand's book Discriminants, resultants, and multidimensional determinants he gives a formula: $$J_{1}(L)\cong J_{1}(\mathcal{O}_{X})\otimes L$$ Here $L$ is a line bundle on some complex variety. ...
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71 views

Locally given complex structure on a 2-plane bundle over a compact manifold can be extended globally

Not any even rank real vector bundle over a smooth manifold has a complex structure, because there are even dimensional manifolds that have no almost complex structures. I am curious about the ...
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Using Quaternion Coefficients to transform a vector from one reference frame to another

I hope this question is not too trivial, and I welcome any pointers to good resources for this problem. I am not familiar with quaternions and have never had to use them before--all my learning about ...
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Understanding the Euler sequence on $\mathbb{P}^n$

I want to get an intuition for the Euler sequence, by understanding the explicit construction of maps between terms. I prefer to use this version: $$ 0 \longrightarrow \mathcal{O}_{\mathbb{P}^n} \...
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1answer
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Does the natural exact sequence of the holomorphic jet bundle spilt?

Let $X$ be a complex manifold. Let us consider the jet bundle of the trivial line bundle on $X$. We denote it as $J_{1}(\mathbb{C})$. We have the short exact sequence: $$0\rightarrow\Omega_{X}\...
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Does the Bochner-Kodaira-Nakano inequality hold w.r.t. some singular metric?

The following contents are copied from Demailly's e-book Chapter VII-(2.7)-Complex Analytic and Differential Geometry. Let $(X, \omega)$ be a compact hermitian manifold, $\operatorname{dim}_{\mathbb{...
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83 views

An identity on Kähler manifold

I am reading The Seiberg–Witten equations and applications to the topology of smooth four manifolds by John Morgan. In the calculation for a Kähler surface in page 116, he uses an identity: for a ...
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1answer
40 views

Integrate a 2-form on Kaehler manifold

Let $X$ be a Kaehler 3-fold, with associated Kaehler form $\omega$ and metric $g_{i\bar{j}}$, $$ \omega = \omega_{i\bar{j}} \, dz^{i} \wedge d\bar{z}^j = \frac{i}{2} g_{i\bar{j}} \, dz^{i} \wedge d\...
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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 $C_1\cdot C_2$ ...
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Explicit integration of Kahler form to get volume

Consider a compact Kaehler manifold $X$ of dimension $d=3$. The K"ahler form in local coordinates $z^i,\bar{z}^i$ is $$ \omega = \omega_{i \bar{j}}dz^i \wedge d\bar{z}^j, $$ with $\omega_{i \bar{...
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Complex Structure on S^6 using Stereographic projection. Why does not work?

Consider $S^6$ and do a Stereographic projection over $\mathbb{R}^6$ (https://en.wikipedia.org/wiki/Stereographic_projection). Give to $\mathbb{R}^6$ the natural complex structure, so it is $\mathbb{C}...
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Griffith and Harris Page 21: singular variety $V_s$ is contained in a subveriety of M not equal to V

$V_{s} = V - V^{*}$ where $V^{*}$ is the locus of smooth points of $V$. Proof: For $p \in V$ let k be the largest integer such that there exist k functions $f_{1},...,f_{k}$ in a neighborhood $U$ of $...
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Issues with the $\overline{ \partial}$-operator and the almost complex structure of a hermitian manifold

I'm working through "Lecture on Kahler Geometry" by Andrei Moroianu, and am stuck on Lemma 11.7 (p. 85). The lemma says: For every section $Y$ of the complex vector bundle $(TM, J)$ the $\...

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