0
votes
0answers
16 views

Show that $\delta =1$

I have a problem: Let $M \subset \Bbb C^2$ be a real analytic hypersurface: $$M=\left \{(z,w) \in \Bbb C^2 \colon \text{Im}\ w=|zw|^2+|z|^8+\frac{15}{7}|z|^2\text{Re}\ z^6 \right \}. \tag 1$$ ...
0
votes
0answers
14 views

Show that $M$ is not equivalent to $O_k$.

Assume that a model hypersurfaces is described by $$O_k=\{(z,w)\in \Bbb C^2 \mid v=|z|^k\}\tag 1$$ and a real analytic hypersurface: $$M=\left \{(z,w)\in \Bbb C^2 \mid ...
2
votes
0answers
32 views

Compute $[\Lambda,\ \bar{\Lambda}]$

I have a problem: We denote by $[X,\ Y]$ the commutator of $X$ and $Y$ defined by $$[X,\ Y]f(p)=X(Yf)(p)-Y(Xf)(p), \tag{1}$$ for any smooth function $f$ defined on a hypersurface $M$. ...
1
vote
1answer
46 views

A Submanifold $M$ of $\Bbb C^N$

I have a Proposition in my book, and I write here: For every $p \in M$, with $M$ be a hypersurface in $\Bbb C^N$ the following hold. \begin{align*} \mathcal V_p &= \left \{ X \in \Bbb C ...
1
vote
1answer
28 views

On Stein manifolds and constant functions

Stein manifolds are defined here: http://en.wikipedia.org/wiki/Stein_manifold#Definition Obviously, M is Stein implies that there is a non-constant holomorphic function defined in it. Is the converse ...
0
votes
1answer
28 views

Dual isogenies of complex tori in Birkenhake-Lange

Let $f: X\to Y$ be an isogeny of complex tori, of degree $n$. On page 13 of Complex Abelian Varieties, Birkenhake-Lange show that there is a dual isogeny $g: Y\to X$. Basically, they show ...
0
votes
2answers
81 views

Homotopy invariance of line integral on manifolds

Consider a 1-form: $\omega\in\Gamma(\mathrm{T}^*M)$ and two differentiable curves: $\gamma,\tilde{\gamma}:[a,b]\to M:\gamma(a)=\tilde{\gamma}(a),\gamma(b)=\tilde{\gamma}(b)$ together with a ...
1
vote
0answers
55 views

Constructing Riemann surfaces

At the risk of asking a question that has been already answered, I have been trying to figure out how to construct the Riemann surface of slightly more complicated examples, but after reading examples ...
1
vote
1answer
39 views

Canonical isomorphism between complexified tangent space of submanifold fixed by antiholomorphic involution and tangent space of complex manifold

I haven't really studied complex manifolds and I am at a bit of a loss in regards how to approach this problem: Let $M$ be an $n$-dimensional complex manifold, and let $\phi:M\rightarrow M$ be an ...
7
votes
1answer
237 views

Möbius transformation in the complex plane.

Assume that $U$ be a line in the complex plane. And assume a Möbius transformation $\phi $ sends $ U $ again to a line. How can I classify all such $\phi$? I want to write my ideas. But, I ...
2
votes
1answer
130 views

Construct a map from unit disk to upper half-plane

I want to construct this map in high-dimensional case. Let $D=\{x \in \mathbb{R}^n:|x|^2<1\}$,and $H=\{u\in\mathbb{R}^n:u^n>0\}$. Well, it is quite clear when $n=2$, but I find it is hard for me ...
6
votes
1answer
130 views

Is there any relationship between Cauchy-Riemann equations and vector fields on manifolds?

Well, suppose we have $f : \mathbb{C} \to \mathbb{C}$ analytic, then if $f = u + iv$ the functions $u,v : \mathbb{C} \to \mathbb{R}$ satisfy the Cauchy-Riemann equations: $D_1u=D_2v$ and $D_2u=-D_1v$. ...
1
vote
0answers
40 views

Spherical Grid Identifcation

I'm trying to see how the lower half of these grids look like when I make the following identification onto the unit sphere: CLICK HERE TO SEE IMAGE In notation, how would one represent this ...
4
votes
1answer
109 views

Prove that the circle $S^1$ is not the boundary of any compact manifold with boundary in $\mathbb R^2-{(0,0)}$

Suppose it were, then define a 1-form $w:=\frac{1}{x^2+y^2}(-y\,\mathrm dx+x\,\mathrm dy)$. Firstly , I try to evaluate $\int_{S^1}w$ by two ways . Firstly, let $F\colon[0,2 \pi]\to S^1$ defined by ...
2
votes
0answers
57 views

Torus biholomorphic to smooth cubic curve?

I am trying to understand that all compact genus 1 Riemann surfaces are biholomorphic to a smooth cubic curve. ( assuming that $\dim H^{1,0} = \dim H^{0,1} = \frac{1}{2} \dim H^1$ ) I think I ...
2
votes
0answers
171 views

Flow of a complex vector field?

Suppose I have a vector field X over a m-dimensional analytical manifold $M\subset \mathbb{C}^n$; how can I define the flow of $X$? Is it done in the same way as for the real case, but instead of ...
1
vote
1answer
193 views

Tangent bundle on a complex manifold

When defining the tangent bundle of a $n$-dimensional manifold $M$ whose coordinate-change mappings are holomorphic, do I need to specify where M is immersed? That is, I must assume that ...
1
vote
1answer
252 views

any two simply connected open set in the plane R^2 are diffeomorphic

Prove that any two simply connected open set in the plane R^2 are diffeomorphic. I know that in the complex plane any simply connected open set is diffeomorphic to either complex plane or open unit ...
4
votes
1answer
326 views

Local normalization of algebraic curves

I am currently reading about the normalization theorem: Suppose $C$ is an irreducible plane algebraic curve, and let S be the set of singular points. Then there exists a compact Riemann surface $\hat ...
3
votes
1answer
223 views

The Implicit Function Theorem for complex polynomials

I'm looking for a reference that proves implicit function theorem for polynomials in two variables over the complex numbers via the real version. Such a theorem is needed, for example, in the theory ...
2
votes
0answers
308 views

Preparing for reading Penrose's “Road to Reality”

I am reading Road to Reality by Roger Penrose and I although I know about calculus, complex analysis, differential equations I do not know about manifolds, Riemann surfaces and so on. Which books can ...