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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
56 views

$\mathbb{C}-$linear extension of 2-forms to (1,1)-forms

I am trying to analyze a bit how we can extend a differential form to the complexification $V\otimes\mathbb{C}=V_{\mathbb{C}}$ of the vector space. Of course, you can do this for a general $k$-form, ...
領域展開's user avatar
  • 2,077
0 votes
1 answer
45 views

tensor notation in complex geometry

Suppose I have $g$, a symmetric, Hermitian metric on a complex manifold of complex dimension $n$. Can anyone please suggest how to evaluate the following tensorial quantity (assuming Einstein ...
Soumya Ganguly's user avatar
0 votes
0 answers
71 views

commutativity of $\iota^*$ and ${\bar\partial}^*$ / $\mathcal G$ on smooth differential froms

Let $(X,\omega_X)$ be a compact Kaehler manifold. Denote by $\iota:Y\rightarrow X$ the natural embedding, where $Y$ is a submanifold in $X$. Recall that we have the De Rham-Kodaira Hodge ...
Invariance's user avatar
  • 1,678
1 vote
1 answer
109 views

What is the compactification of $y=e^x$ in $\mathbb P_{\mathbb C}^2$?

Denote the curve $V(y-e^x)\subset \mathbb A_{\mathbb C}^2$ by X, and consider its closure $\bar X$ in $\mathbb P_{\mathbb C}^2$. By GAGA, we know $\bar X$ is algebraic. I think this is amazing. I want ...
Richard's user avatar
  • 1,374
0 votes
1 answer
49 views

How do I find an algebraic expression for the function $F(ξ, \bar{ξ})$ from this paper?

I am working on understanding the paper "On $C^2$-smooth Surfaces of Constant Width" by Brendan Guilfoyle and Wilhelm Klingenberg. As part of their definition of equations for a 3D surface ...
Lawton's user avatar
  • 1,675
3 votes
0 answers
80 views
+50

how does a metric change under the parametrization and topological point of views

In the past few days, I've been delving into characteristic classes for the Torus, and I've realized that my understanding of metrics isn't concise enough. Initially, I derived a metric from the ...
falamiw's user avatar
  • 844
1 vote
1 answer
57 views

$S^2$ with the two stereographic projections is a Riemannian surface.

Consider $S^2$ with the two stereographic projections obtained by removing respectively the North pole and the South pole, find the gluing function obtained through the definition of Riemannian ...
Andreadel1988's user avatar
0 votes
0 answers
48 views

Example of a Riemann surface

Let $\Gamma = \omega_1 \mathbb{Z} + \omega_2 \mathbb{Z}$ with $\omega_1, \omega_2$ independent in $\mathbb{R}$. Let $E_{\Gamma} = \mathbb{C}{/ \Gamma}$. Show that $E_{\Gamma}$ is a Riemann surface. ...
Andreadel1988's user avatar
1 vote
1 answer
72 views

A question on isolated singularity involving analytic varieties.

Suppose $V\subset \mathbb{D}^2$ ($V$ is a subset of the unit bidisc) is a set. Also, suppose that the following conditions hold: If $\hat{V}$ is the polynomial convex hull of $V$, then $\hat{V}\cap \...
Anindya Biswas's user avatar
0 votes
0 answers
32 views

Geometric explanation of Fueter-Sce-Qian theorem and similar situations

In Clifford analysis there is a fundamental theorem due to Fueter and extended by Sce and Qian that says (in modern terminology) that the given a slice regular function $f:\mathbb{R}^{m+1}\to\mathbb{R}...
Giulio Binosi's user avatar
1 vote
0 answers
200 views

Covering Space for 3-punctured sphere

I have been considering whether we can find a branched covering map $p:Y \rightarrow \mathbb{CP}^1$, having 3 critical values, such that exactly five points are mapped to those critical values. That ...
Jacob Lee's user avatar
0 votes
1 answer
115 views

a problem with complex numbers that may have a geometric solution

The statement of the problem : For a strictly positive real number r and z $\in \mathbb C$, we consider the set : $D_z(r) = \{ v \in \mathbb C \,|\, |z-v|\leq r \}$ . We also consider the following ...
Last X's user avatar
  • 171
0 votes
1 answer
26 views

linear complex structure as submanifold of general linear group

Given a finite dimensional real vector space, we can view all the complex structures on it as a subspace of ${\rm GL}(V)$. I wonder if it is a submanifold of ${\rm GL}(V)$. And moreover, given a ...
Qhejaz's user avatar
  • 106
0 votes
0 answers
22 views

Clarification of the proof of the form $\partial_{1,0}$ in Restrictions on harmonic maps of surfaces by J. Eells and J. C. Wood

I am currently reading "Restrictions on harmonic maps of surfaces" by J. Eells and J. C. Wood and have trouble understanding the proof of the lemma on p. 265. The paper states: Let $X$ and $...
chrstphfrtz's user avatar
0 votes
0 answers
38 views

Holomorphic sectional curvature of complex projective spaces

I have heard that the holomorphic sectional curvature of the complex projective space (w.r.t the Fubini Study metric) is 2. Can anyone give me a self-contained reference to this fact, please? Thanks ...
Soumya Ganguly's user avatar
1 vote
0 answers
56 views

Fixed group of right-equivalences for $f = x^3 - y^2$.

I have the polynomial $f = x^3 - y^2$, considered as a convergent power series $f \in \mathbb C\{x,y\}$. Is there a strategy to compute the group $G$ of right-equivalences (coordinate transformations) ...
red_trumpet's user avatar
  • 8,302
0 votes
1 answer
42 views

De rham cohomology over complex manifolds

I am studying sheaf cohomology of complex manifolds and, while reading some proof about Dolbeault cohomology, I realized that there is a $\bar{\partial}$-Poincaré Lemma which gives us the local ...
user720386's user avatar
1 vote
1 answer
69 views

Why is the complex Lie group $(\mathbb C^*)^n$ called "Complex Torus"

While studying complex Lie groups theory, and more generally complex geometry, I've found two different objects which are called "complex tori". Consider the multiplicative group $\mathbb C^...
Federico T.'s user avatar
-1 votes
0 answers
22 views

Do points enclosed by circle on upper half Riemann sphere form a closed or compact set in the Extended Complex Plane?

Consider the Riemann sphere and a circle draw as shown below: https://i.stack.imgur.com/u6nA1.png The green points indicate the set in the upper hemisphere enclosed by the line. Is this an open/closed/...
Dani Lisle's user avatar
2 votes
1 answer
73 views

When is a nef line bundle big

Suppose $M^n$ is a smooth projective variety. A line bundle $L$ on $M$ is nef (numerically effective) if on any complete curve $C$ in $M$, $L$ has positive degree, i.e. $$ L\cdot C=\int_{C}R_h\geq 0. $...
eulershi's user avatar
  • 541
0 votes
0 answers
64 views

The ample cone of complex Hirzebruch surface [closed]

How one can discribe the ample cone of a Hirzebruch surface? It is identified with the projective bundle over $P^1$ . Its picard group is generated by two elements: its fibre and the zero section.
Penguin deprime's user avatar
0 votes
1 answer
40 views

Fundamental form $\omega=\sum_{i\leq m}v^*_i\wedge (Jv_i)^* $ with a complex structure $J$

Let $V$ be a $\mathbb{C}-$ vector space, $J$ an almost complex structure on $V$ and take a real orthonormal basis $\langle v_1,Jv_1,\ldots,v_n,Jv_n\rangle $ with a scalar product $\langle,\rangle = \...
領域展開's user avatar
  • 2,077
1 vote
0 answers
33 views

Why the local mass-ratio in a complex submanifold of the Euclidean space is nonzero? [closed]

Okay, I'm not very comfortable with either complex geometry or minimal surfaces, so bear with me. I've encountered the following theorem, but I can't find a proof: Suppose $V\subset \mathbb{C}^n$ is a ...
unsure's user avatar
  • 31
0 votes
0 answers
44 views

Exterior algebra decomposition natural projections $ Π^{p,q}:\bigwedge {V}^*_{\mathbb{C}} \to \bigwedge^{p,q}V$ (Huybrechts book)

Let $\bigwedge^{p,q}V:=\bigwedge^{p}V^{1,0}\wedge \bigwedge^{p}V^{0,1}$ and $\bigwedge {V}^*_{\mathbb{C}}=\bigoplus_k \bigwedge^{k} V_{\mathbb{C}} $. Then one defines the natural projection $$ Π^{p,q}...
領域展開's user avatar
  • 2,077
0 votes
0 answers
35 views

Sylvester orthonormal frame for a possibly degenerate Hermitian differential form

Let $M$ be a complex manifold and $\Omega$ a Hermitian differential $(1,1)$-form on $M$. Is it true that for every $x \in M$ there exists an open neighborhood $x \in U \subseteq M$, a local ...
Riccardo Pengo's user avatar
0 votes
0 answers
35 views

Isomorphism between Tangent Sheaf and Cotangent Sheaf

I'm studying K3 surfaces and I often encountered the fact that the tangent sheaf $\mathcal{T}$ is isomorphic to the sheaf of differentials $\Omega$. Why is this true? I guess this follows from the ...
WindUpBird's user avatar
1 vote
2 answers
103 views

Tangent space of $\mathbb{P}(V)$

I encountered this problem studying the local period map and I'm wondering how to solve it. I would like to prove that, given $V$ a complex vector space and $W \subseteq V$ a one-dimensional subspace, ...
WindUpBird's user avatar
0 votes
0 answers
49 views

There are No Non-trivial Rational Function Solutions to $n=4$ Fermat's Curve

Pardon me for the homework-like question, I'm just trying to better understand how we go about using infinite descent. Problem Statement Suppose we have rational functions $f,g \in \mathbb C[t]$. We ...
M Rozzzz's user avatar
0 votes
0 answers
26 views

Fundamental group of the complement of a quadric cone with two tangent planes

Let $X=\mathbb{C}^3$, $Q=V(x^2-yz)$, $H_1=V(y)$, $H_2=V(z)$. The hyperplanes $H_i$ are tangent to $Q$ along a line. Let $U$ be the complement of $Q\cup H_1\cup H_2$. Is the fundamental group $\pi_1(U,\...
Sergey Guminov's user avatar
4 votes
1 answer
118 views

Second homology group of Kähler surfaces

Let $M$ be a closed Kähler surface, i.e., Kähler manifold of complex dimension $2$. Is it true that for any element $\alpha \in H_2(M,\mathbb Z)$, there exists a nonsingular holomorphic curve $\Sigma$ ...
Summer's user avatar
  • 6,833
2 votes
0 answers
31 views

Why is the stabilizer of a holomorphic complex Lie group action a complex Lie subgroup?

Suppose $G$ is a complex Lie group and $M$ is a complex manifold. Suppose we have an action of $G$ on $M$ which is a holomorphic map $G \times M \rightarrow M$. I have seen the claim that it is easy ...
rosecabbage's user avatar
  • 1,621
1 vote
1 answer
121 views

Example 2.13 in Wells "Differential Analysis on Complex Manifolds" Conclusion

I'm currently working through Raymond O. Wells' "Differential Analysis on Complex Manifolds" and I'm confused by example 2.13 in chapter 1. In this example he is computing the global ...
geometric_20's user avatar
0 votes
1 answer
78 views

Understanding the claim that holomorphic divisors contained in a divisor class are in bijection with the projectivisation of the space of sections

Let $V$ be a complex manifold. I am trying to approach the subject of divisors through complex geometry since I am not familiar enough with algebraic geometry to come that way. A divisor $D$ on $V$ is ...
rosecabbage's user avatar
  • 1,621
4 votes
1 answer
143 views

$\partial_{z}$ and $\partial_{\bar{z}}$: what are these vector fields from a geometrical point of view?

In complex analysis, we are taught that instead of coordinates $x$, $y$ on the complex plane, one can use $z$, $\bar{z}$, then, for instance, the Cauchy-Riemann conditions become $\frac{\partial }{\...
Daigaku no Baku's user avatar
2 votes
0 answers
41 views

Geometric notion of modality

I'm reading Singularity Theory (one of the authors is Arnold). I am a bit confused on the concept of modality. Is there an easy geometric description of it? The book has a relation between codimension,...
quantum's user avatar
  • 1,645
0 votes
1 answer
76 views

Germ as subset of $\mathbb{C}^n$

I am reading "Complex Geometry: an introduction", by D. Huybrechts. In the definition, 1.1.22 says that a germ $X\subset \mathbb{C}^n$ in $0$ is called analytic if $X$ and $Z(f_1,f_2,\ldots,...
領域展開's user avatar
  • 2,077
2 votes
1 answer
105 views

The Hermitian condition is equivalent to $\nabla_{JX}J=J(\nabla_XJ)$, right?

I saw the property $\nabla_{JX}J=J(\nabla_XJ)$ on a Hermitian manifold in a paper, where $X$ is a vector field, $J$ is complex structure and $\nabla $ is the Levi-Civita connection. This seems to be a ...
Geom Zari's user avatar
  • 183
1 vote
1 answer
80 views

Hyperkähler structure of Quaternion

I think there was an issue with the question I asked earlier. I want to prove that the quaternion is a hyperkahler manifold. I know that there is a natural metric $\rho$ on that given by $\rho(a,a)=a·...
ymm's user avatar
  • 61
1 vote
0 answers
32 views

On the definition of exhaustion functions

I'll first express my confusion:why can the exhaustion function of a domain in $\mathbb{C}^n$ be bounded? More precisely: In p. 45 of the book Partial differential equations in several complex ...
msecauchy's user avatar
2 votes
0 answers
73 views

Complex geodesic coordinate, local ramified map, and the conic metric

Let $X$ be a compact Kaehler manifold of dimension $n$, and let $Y=\sum_{i\in I} Y_i$ be a snc divisor. In other words, one can find a finite trivializing cover $\left\{V_k;z_k^1,\ldots,z_k^n\right\}$ ...
Invariance's user avatar
  • 1,678
2 votes
0 answers
118 views

"Universal" line bundle over the Picard variety $\operatorname{Pic}^0(X)$.

$\DeclareMathOperator{\Pic}{Pic}$This question is inspired by my attempt to answer another question. At the end of my answer, there is a missing step, which I don't know how to fill. Let $X$ be a ...
red_trumpet's user avatar
  • 8,302
0 votes
1 answer
48 views

Cohomology group induced by $d^*$

It is well known that for a smooth manifold $M$, the de Rham cohomology group is defined by $$H_{dR}^k(M):=\frac{A^k(M)\cap \ker d}{A^k(M)\cap \text{im }d}.$$ Similarly, if we assume that $M$ being a ...
Tom's user avatar
  • 671
3 votes
2 answers
118 views

Prerequisites to begin Beauville's "Complex Algebraic Surfaces"

I'm going to read through the first three chapters of Arnaud Beauville's Complex Algebraic Surfaces. My background in algebraic geometry only consists of Gathmann's algebraic geometry notes (the part ...
Ezio Greggio's user avatar
2 votes
1 answer
112 views

Deformation and algebraic equivalence relation

I found the algebraic equivalence relation is closely related to deformation, the most intuitive description that I found is in Griffiths' Topics in Transcendental Algebraic Geometric Chapter 1 which ...
yi li's user avatar
  • 4,746
4 votes
1 answer
213 views

Polynomial identity : $|P(z)|^2=|Q(z)|^2-|R(z)|^2$ for $z \in \mathbb{D}$

Let us denote by $\mathbb{D}=\{z : |z| <1\}$ and $\mathbb{T}=\{z: |z|=1\}$. Suppose that $P, Q$ and $R$ are polynomials that satisfy the following: $|Q(z)| \geq |R(z)| $ for all $z \in \overline{\...
Curious's user avatar
  • 833
0 votes
1 answer
75 views

Bogomolov-inequality reference of proof

Where can I find a written proof of the bogomolov inequality using complex analysis (aka the existence of a ricci flat metric on semistable bundles)?
user135743's user avatar
4 votes
1 answer
100 views

Is non-constant holomorphic map $f:\mathbb P^n\to \mathbb P^n$ surjective?

Let $\mathbb P^n$ denote the $n$-dimensional complex projective space. From Forster's book Lectures on Riemann surfaces p.11, Theorem 2.7, we know any non-constant holomorphic map $f:\mathbb P^1\to \...
Tom's user avatar
  • 671
1 vote
1 answer
50 views

Prove if there exists a point on an elliptic curve over the complex numbers satisfying the condition re(x) = 0

Consider the elliptic curve over complex numbers defined by $y^2=x^3+b$, where $re(b)$ and $im(b)$ are both integer-valued. Is there a way to prove whether or not there exists a point on this curve ...
questionasker's user avatar
1 vote
0 answers
28 views

trace map of jacobians of Riemann surface

I'am reading 'A first course in modular form'. On page 221 Given two compact riemann surface and holomophic map $h$ : $X \to Y$. Suppose $h$ is a surjection of finite degree $d$, that $h$ is locally $...
abcdetale's user avatar
  • 140
1 vote
0 answers
35 views

Lefschetz operator on bundle-valued forms

For a holomorphic vector bundle $V \rightarrow X$ endowed with a Hermitian structure, one may define the corresponding Dolbeault-like operators $\bar{\partial}_V: \Omega^{p,q}(V) \rightarrow \Omega^{p,...
Eweler's user avatar
  • 681

1
2 3 4 5
73