Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

7
votes
0answers
44 views

Can a complex manifold that is not a Calabi-Yau manifold be homeomorphic to a Calabi-Yau manifold?

This is a kind of a follow up to this question, which actually already had an answer here, in which it is asserted that Hodge numbers in general are not topological invariants. Could it be so extreme ...
3
votes
1answer
55 views

Are Hodge numbers topological invariants for manifolds that admit a Kähler structure?

I know that all fibers in a analytic fibration (proper, holomorpic) are homeomorphic, and if the fibers are Kählerian manifolds, then they have equal Hodge numbers. Could it happen however that a ...
0
votes
0answers
18 views

The Fubini-Study metric $ \omega_{ \mathrm{FS} } $ on the projective space $ \mathbb{P}^1 $

I have a small question about the Fubini-Study metric $ \omega_{ \mathrm{FS} } $ on the projective space $ \mathbb{P}^1 $ appearing in page : $ 119 $ of Daniel Huybrechts's book intiteled : Complex ...
1
vote
1answer
47 views

Connectedness of complex sphere

Let $X_{n}$ be a set $$ X_{n} = \{(x_{1}, \dots, x_{n})\in \mathbb{C}^{n}\,:\, x_{1}^{2}+ \cdots + x_{n}^{2} = 1\}. $$ For $n\geq 2$. Then $X_{n}$ is connected. In the case of $\mathbb{R}$, it is ...
1
vote
1answer
36 views

Blow down map induced injection on homology?

Let $\tilde X$ be the blow up of $X\subset \mathbb{CP^{n+1}}$ along some closed subscheme $C\subset X$, $f : \tilde X\to X$ be the blow down map, and $f_*:H_k(\tilde X)\to H_k(X)$ be the induced map ...
0
votes
0answers
22 views

Riemann surface Logarithm

I would like to show, that the Logarithm gives a Riemann surface. Therefore I define the set $M:=\{(z,w)\in D \vert \ f(z,w)=0 \}$ with $D:=ℂ^*\times ℂ$ and $f(z,w):=e^w-z$. Afterwards I will use ...
2
votes
1answer
42 views

$\Bbb{CP}^1$ how many charts does it have?

When we define $\Bbb{CP}^1$ as a complex $1$-manifold, we give it two charts $(U_0,\gamma_0)$ and $(U_1,\gamma_1)$. We also say it has a complex structure $\Sigma$, which is an equivalence class of ...
0
votes
0answers
18 views

Question on Kahler geometry: Kahler form and $\mathcal{O}_X(1)$

Given a projective surface (2 complex dimensions) $X$ we can equip it with the line bundle $\mathcal{O}_X(1)$. Let us fix this line bundle. Recall that all projective surfaces are Kahler surfaces. ...
0
votes
1answer
20 views

If $p\circ f$ has a pole at $z_0$ then so does $f$, where $p$ is a polynomial

Let $f : U$ \ ${{z_0}} \to C$ be a holomorphic function, where $U$ is an open disk and let $p$ be a non-constant polynomial. Show that the singularity of $f$ at $z_0$ is a pole of $f$ iff it is a pole ...
1
vote
0answers
52 views

Decompositon of Tangent Bundles

Let $X\subset \mathbb P^n$ be a compact manifold over $\mathbb C$, $Y\subset X$ be a submanifold. We know there is a smooth decomposition $$T_X|_Y=T_Y \oplus N_{Y/X}$$ My question is: is there ...
2
votes
1answer
18 views

Holomorphic tangent vector field determines a infinitesimal holomorphic equivalence?

Let $X$ be a complex compact manifold. I want to understand the holomorphic tangent vector field on $X$. I know a smooth vector field can define an infinitesimal diffeomorphism of $X$ (more precisely, ...
3
votes
2answers
47 views

Book to learn the use of complex number to solve geometric problem

I want to learn to use complex number to solve geometric problems, Specially to solve olympiad questions. There are a couple of books in the market and i am confused which one should i buy. Here is ...
0
votes
0answers
20 views

Integrals of the number of $z$ and $\bar{z}$ factors are not equal on $\mathbb{C}^n$

I want to conclude that $$\sum_{i,j,k,l} \int_{\mathbb{C}^n} e^{-|z|^2} R_{i\bar{j}k\bar{l}} z^i \bar{z}^j z^k \bar{z}^l dV = \sum_{i=1}^n \int_{\mathbb{C}^n} e^{-|z|^2} R_{i\bar{i}i\bar{i}} |z^i|^4 ...
8
votes
1answer
189 views

Compactness of a set in the complex plane.

Consider $$A=\Big\{(z_1,z_2) \in \Bbb{C}^2 : z_1^2+z_2^2=1\Big\}$$ Is $A$ compact? My try: If $A$ is compact, then it is closed and bounded But it is not bounded! Since $(n,\sqrt{1-n^2});n\in \...
1
vote
0answers
21 views

How can I justify that the compact manifold in string theory must be orientable in one dimension

I am preparing a presentation on Calabi-Yau manifolds where I do not want to use advanced mathematics, but I would like to show that under some minimal assumptions in one (complex) dimension, the ...
0
votes
0answers
17 views

Modified homotopy and relation with intersection theory.

Denote by $\Gamma$ a hypersurface in $\mathbb{C}^2$, i.e. the zero locus of a polynomial of two complex variables. Denote by $X$ the complement of $\Gamma$ in $\mathbb{C}^2$. I am trying to define a ...
2
votes
1answer
32 views

Size and shape moduli of Calabi-Yau manifolds

Let's say that Calabi-Yau manifolds are compact Kähler manifolds with a trivial canonical bundle. We know that $H^2(X,\mathcal T_X) = H^1(X,\Omega^2_X)$ is the tangent space to the deformation functor ...
4
votes
1answer
56 views

Does every possible Kähler metric on a projective variety arise from the Fubini-Study metric for some embedding?

Every projective variety inherits a Kähler structure from a projective embedding, by restriction of the Fubini-Study metric. They will generally admit many Kähler structures though. I was wondering if ...
1
vote
1answer
34 views

How to properly deduce the Holomorphic Implicit Function Theorem from the Smooth Real Implicit Function Theorem?

I have seen at several places, incl. some notes and books, the following inference of the Holomorphic Implicit Function Theorem from the Smooth Real Function Theorem, but I believe this proof to be ...
3
votes
0answers
78 views

How can it be that the Chern class fully determines a line bundle, but having Chern class zero doesn't imply a line bundle is trivial?

It is well-known that the Chern class of a line bundle in $H^2(M,\mathbb Z)$ fully determines the bundle up to isomorphism. However, in this wikipedia entry on Calabi-Yau manifolds it is stated that ...
1
vote
0answers
46 views

Monodromy of the family of hypersurfaces on moduli space

Let $\bar{\mathfrak X}\to \mathbb P^N$ be the universal family of hypersurfaces in $\mathbb P^{n+1}$ of degree $d$ and $\mathfrak X \to U$ ($U\subset \mathbb P^N$) be the sub-family of smooth ...
0
votes
1answer
42 views

3D lattice of a tetrahedron. What is it called?

I recently stumbled on this image and have been looking for a name for it: pyramid image http://tetraktys.de/bilder/buckminster-tetraeder.gif It’s not a Seirpinski pyramid because it doesn’t become ...
8
votes
1answer
84 views

Holomorphic Lefschetz formula and basic linear algebra.

There is a well-known way to prove the fact that the field of complex numbers is algebraically closed using Lefschetz fixed point theorem. Let me recall the idea: The existence of a root for any ...
1
vote
0answers
33 views

Hodge star computation on a Riemann surface

Let $M$ be a Riemann surface, the Hodge star is defined as $ \alpha \wedge *\beta = \langle \alpha, \overline{\beta} \rangle vol$ where $\langle \cdot , \cdot \rangle$ is the Hermitian product on the ...
0
votes
1answer
29 views

Showing that this map descends to the quotient in an injective way

Let $f : \mathbb{S}^3 \to \mathbb{S}^2$ be the map $$ f(z_1,z_2) = (2z_1 \overline{z_2}, \vert z_1 \vert^2 - \vert z_2 \vert^2), $$ where we regard $\mathbb{S}^3 \subset \mathbb{C}^2$ and $\mathbb{...
0
votes
0answers
17 views

Orbit space of an action over Simplicial complex

I'm studying the action of a group $G$ on a simplicial complex. I'm looking for the orbit space, or a fundamental domain for the action. There is a manner to define the orbit space like simplicial ...
1
vote
0answers
58 views

Deligne's theorem on the Leray spectral sequence and weights

Motivation : If $f : X \to Y$ is a smooth projective map between algebraic varieties, then there is a theorem by Deligne which says that the Leray spectral sequence degenerates at $E_2$. The proof I ...
5
votes
0answers
106 views

Direct image of tangent bundle under projection map

Let $\pi: Y \to X$ be a smooth projection map, where $Y$ and $X$ are complex manifolds and the fibres of the map have constant dimension: dim$\left(\pi^{-1}(x)\right)=d ~~~ \forall x \in X$. The ...
4
votes
0answers
57 views

A compact complex manifold admits an ample line bundle if and only if it is projective

Given a holomorphic line bundle $L$ on a complex manifold $X$, a point $x\in X$ is called a base point of $L$ if $s(x)=0$ for all $s\in H^0(X,L)$ (the space of global holomorphic sections of $L$). The ...
0
votes
1answer
21 views

Variety isomorphic to Riemann sphere

I have read that the complex solutions to $y^2=p(x)$ where $p(x)$ has degree 1 or 2 (and distinct roots) is topologically and complex analytically isomorphic to the Riemann sphere $\mathbb{P}^1$. How ...
1
vote
0answers
37 views

What is the curvature form $\Omega$ associated with the Levi-Civita connection for the complexified $n$-sphere with respect to the standard metric?

What is the curvature form $\Omega$ associated with the Levi-Civita connection $\nabla$ for the complexified $n$-sphere $(S^n)^{\mathbb{C}}$ with respect to the standard metric, i.e. what is $\Omega=d\...
0
votes
0answers
13 views

Chevalley's theorem for real Lie groups

Let $G$ be a complex Lie group, then by Chevalley's theorem; The commutator $G'$ is an algebraic group and it acts algebraically and its orbits are closed in their algebraic closures. Is there any ...
1
vote
1answer
63 views

Some questions about Hermitian metric on complex manifold

Suppose $g$ is a Riemannian metric on a complex manifold $X$ compatible with the almost complex structure. $g$ can be extended to $TX\otimes\mathbb{C}$, which is a Hermitian inner product: $<\...
1
vote
1answer
72 views

Why the conditions $w(0)=0$ and $w(2)=\infty$ map the region $|z-1|<1$ onto the region $\Re w>0$?

I want to find a linear fractional transformation which maps the region $D$ of the $z$-plane onto the region $G$ of the $w$-plane, where $D=\{z;|z-1|<1\},~G=\{w;\Re w>0\}$ This is an exercise ...
3
votes
1answer
99 views

Definition of $dz_i\otimes d\bar{z_j}(\frac{\partial}{\partial z_m},c\frac{\partial}{\partial z_n})$

Is $dz_i\otimes d\bar{z_j}(\frac{\partial}{\partial z_m},c\frac{\partial}{\partial z_n}):=dz_i(\frac{\partial}{ \partial z_m})d\bar{z_j}(\bar{c}\frac{\partial}{\partial \bar{z_n}})$? It seems that by ...
-1
votes
2answers
28 views

Finding the number of points on unit circle satisfying a criteria. [closed]

Find the number of numbers$(z)$(Or the number of solution for $z$) on the unit circle such that :- $z^{6!}-z^{5!}$ is a real number.
2
votes
1answer
30 views

From compatible Riemannian metric to Hermitian metric

By this notes p.42 It gives a hermitian metric by a compatible Riemannian metric $g$, and from p.23, it extends $g$ to $T_{\mathbb{C}}M$ complex bilinearly. I wonder if we extend $g$ via the ...
0
votes
2answers
64 views

Why is Hermitian inner product in the form of $h=\sum h_{ij}z_i\otimes\bar{z_j}$?

The following are from O'Wells' book p.156-157. Let $E$ be a complex vector space of complex dimension $n$. Let $E'$ be the real dual space to the underlying real vector space of $E$, and let $F = E'\...
0
votes
1answer
39 views

Why is $g(JZ,JW)=-g(Z,W)$?

Let $M$ be a complex manifold, with Riemannian metric $g$ and complex structure $J$. If $g$ satisfies $$g(JX, JY ) = g(X, Y ),$$ for any two vector fields $X$ and $Y$. I am reading a proof from here, ...
6
votes
1answer
69 views

Why do the first Chern classes of these line bundles span the Dolbeault cohomology group $H^{1,1}(X;\mathbb{R})$?

Forgive me for what is probably a simple question, I am new to this field. I am studying the Hirzebruch surfaces and their higher dimensional analogues $M_{n,k}$, defined to be the projective line ...
3
votes
3answers
95 views

Algebraic trick to map $|z|<2$

Suppose that we want to find the image of the region $|z|<1$ under the mapping $w=\frac z{z+1}$. Since $z=\frac{-w}{w-1}$ we should have $|\frac w{w-1}|=|\frac{u(u-1)+v^2-iv}{(u-1)^2+v^2}|<1$ or ...
2
votes
0answers
29 views

Germ at any point determines analytic function

If I remember by Riemann surfaces course correctly, then the following should be true: Let $X$ be a Riemann surface, $U\subset X$ be open, and $x\in U$. Then the map $\Gamma(U,\mathcal{O}_X)\to \...
0
votes
1answer
49 views

Definition of blowing-up along a complex submanifold (Huybrechts)

I am trying to understand Huybrecht's definition of the blow-up of a complex manifold $X$ along a submanifold $Y$ - if you don't have a hard copy to hand, I have found an electronic version here (see ...
1
vote
1answer
39 views

Does every almost complex manifold admit an almost-Kähler structure?

I couldn't find a conclusive answer to this question online. Here is my reasoning. Let $M$ be an almost complex manifold. Then, from what I understand, we can define almost complex structure $J$ on $...
3
votes
2answers
97 views

Example of nef and big, not ample

What would be a common, simple example of a nef and big divisor that is not ample? Are there any common, less simple examples? Are there any common strategies for finding examples?
2
votes
0answers
27 views

Flow in the direction of a complex vector

I'm currently doing some reading on the Witt algebra, and I'm trying to understand the meaning behind $\frac{d}{dz}$. In the book I'm reading (Conformal field theory by Martin Schottenloher), $\frac{...
1
vote
1answer
50 views

Is it true that every real manifold can be embedded into complex manifold?

I think this is true by the following way: let $M$ be a $n$-dimensional real manifold. Consider the diagonal embedding $\Delta:M\to M\times M$, $x\mapsto (x, x)$. Using the smooth structure of $M$, we ...
3
votes
1answer
31 views

Holomorphic line bundles over $\mathbb{CP}^n$ and the Hirzebruch surfaces

In Huybrechts' text Complex Geometry, I am told that any holomorphic line bundle over $\mathbb{CP}^n$ is of the form $\mathcal{O}(k)$ for some $k\in\mathbb{Z}$, where \begin{equation} \mathcal{O}(k)=\...
1
vote
1answer
54 views

Specific formula of Hodge star operator $\bar{*}$

Hodge star operator is an operator: $\bar{*}:\epsilon^{p,q}=\Gamma(X,\bigwedge^p\Omega^1_X\otimes\overline{\bigwedge^p\Omega^1_X})\to\epsilon^{n-q,n-p}$ with the relation $$\alpha\wedge\bar{*}({\beta})...
1
vote
1answer
73 views

Hodge star operator $\bar{*}$ and volume form

Hodge star operator is an operator: $\bar{*}:\epsilon^{p,q}=\Gamma(X,\bigwedge^p\Omega^1_X\otimes\overline{\bigwedge^p\Omega^1_X})\to\epsilon^{n-q,n-p}$ with the relation $$\alpha\wedge\bar{*}({\beta})...