Tagged Questions

4
votes
1answer
32 views

Complexified tangent vector, complex manifold

Consider a complex submanifold $M$ of a complex ambient vector space $X$. Suppose you have a base point $p \in M$ and a $C^1$ arc $\gamma(t)$ passing through $p$ and staying in $M$, with tangent ...
1
vote
0answers
40 views

Is there a relation between Super Riemannian manifolds and Kahler manifolds?

(This question has a physics motivation). Could we establish any kind of relationship between Super Riemannian manifolds (super Riemannian structures on super manifolds) and Kahler manifolds, or at ...
1
vote
1answer
23 views

Positive curvature on holomorphic vector bundles

There must be a mistake in my understanding the definition of positivity for the curvature. Let me summarize: Let $ (L,\nabla,h) \rightarrow M $ be a hermitian hol line bundle with Chern connection. ...
4
votes
1answer
73 views

Alternative Almost Complex Structures

Let $V$ be a real vector space. An almost complex structure on $V$ is a map $J : V \to V$ such that $J^2 = -\mathrm{id}_V$. An almost complex structure gives $V$ the structure of a complex vector ...
4
votes
0answers
55 views

Complex differential geometric form of the Grothendieck–Hirzebruch–Riemann–Roch theorem

From the wikipedia article, it seems that there should be a differential geometric form of the Grothendieck-Riemann-Roch theorem with schemes replaced by complex manifolds and quasi-coherent sheaves ...
9
votes
2answers
173 views

What exactly is a Kähler Manifold?

Please scroll down to the bold subheaded section called Exact questions if you are too bored to read through the whole thing. I am a physics undergrad, trying to carry out some research work on ...
1
vote
0answers
44 views

Normal Bundle of Twistor lines

I am reading the paper "hyperkaehler metrics and supersymmetry" by Hitchin etc.. Here is the link: ...
2
votes
1answer
67 views

question about conformal map

Thank you for let me ask question I am really enjoy with this website. It is great website I have question about geometry for expert geometry what is the definition of conformal map and the condition? ...
3
votes
0answers
39 views

$SU(n)$-invariant subring of $\Lambda^{*}\mathbb{R}^{2n}$

I have the following question: Let $R \subset \Lambda^{*}\mathbb{R}^{2n}$ be the sub-ring of forms which are preserved by $SU(n)$. How can one show that this subring is generated by $\Omega_{0}$ and ...
4
votes
1answer
106 views

Where can I learn about complex differential forms?

So I'm a 3rd year grad student in number theory/modular forms/algebraic geometry, and I've worked with differential forms from an algebraic point of view without ever knowing what they really are. I'd ...
1
vote
0answers
47 views

Complex projective manifolds and holomorphic mappings

Let $X\subset\mathbb{P}^2$ be a complex manifold defined by a homogeneous polynomial of degree $d>3$. Let $$\phi:\mathbb{P}^1\rightarrow X$$ be a holomorphic map. Show that $\phi$ is constant. ...
2
votes
1answer
51 views

Independence of coordinate charts in the definition of the order of a pole of a meromorphic 1-form on a Riemann surface

I am currently reading Forster. Let $X$ be a Riemann suface, and $Y$ an open subset of $X$. Assume we have a meromorphic 1-form $\omega \in \Omega(Y \setminus \{a\})$ with a pole at $a \in Y$. One ...
3
votes
0answers
88 views

Is $\bar{\partial}_E + \bar{\partial}_E^*$ a Dirac operator?

I have previously asked about Weitzenböck identities and received some great answers on MathOverflow. One question which has arisen from the post is the following: Let $E$ be a hermitian ...
1
vote
1answer
34 views

Differential action on a complex manifold

Let $M$ be a complex manifold of dimension $n$. Furthermore assume that we have a action of a Lie-Group $G$ on $M$ i.e. $G \times M \rightarrow M$, which is differential, meaning that for every $g \in ...
7
votes
2answers
294 views

Almost complex structures on spheres

It is fairly well-known that the only spheres which admit almost complex structures are $S^2$ and $S^6$. By embedding $S^6$ in the imaginary octonions, we obtain a non-integrable almost complex ...
0
votes
0answers
44 views

Codimension one foliation

let $f:M \rightarrow \mathbb{R}$ be a nowhere vanishing function defined on a Riemannian manifold $(M,g)$. consider the distribution $\ker(df)$. This is clearly involutive and thus defines a ...
3
votes
1answer
78 views

Kähler form is harmonic

Let $M$ be a Kähler manifold with fundamental form $\omega(X,Y) = h(JX, Y)$. I am trying to show that $\omega$ is harmonic. The Kahler condition implies that $\omega$ is closed with respect to $d$, so ...
1
vote
0answers
52 views

tangent vector on a complex manifold

Let $x_1,x_2, \ldots, x_n$ be local coordinates on a manifolds $M$. One can interpret $\frac{\partial}{\partial x_i}(p)$ as a tangent vetor to a curve with constant $x_j$ (where $j \neq i$). What is ...
6
votes
1answer
83 views

Almost complex structure on $\mathbb CP^2 \mathbin\# \mathbb CP^2$

Why doesn't $\mathbb CP^2 \mathbin\# \mathbb CP^2$ admit an almost complex structure?
2
votes
0answers
37 views

Complex structure on the product of two complex Kaehler manifolds

I have the following question: Let $(M,J_{M}, \omega_{M})$ and $(N,J_{N}, \omega_{N})$ be two Kaehler manifolds. I consider $M \times N$. Can I introduce on $M \times N$ a complex structure ? I think ...
0
votes
1answer
103 views

cotangent bundle splits as a product?

Let $M$ and $N$ be two Riemannian manifolds with Riemannian metrics $g$, $h$ respectively. We consider the product $M \times N$ with metric $g \oplus h$. By the metric we get an isomorphism of bundles ...
1
vote
1answer
61 views

Kaehler-Einstein metric on Calabi-Yau manifold

I am reading "Complex geometry" by D. Huybrecht. On p.223 the books says that "If $c_{1}(X)=0$, e.g. if the canonical bundle $K_{X}$ is trivial, and $g$ is Kaehler-Einstein metric, then Ric$(X,g)=0$. ...
3
votes
1answer
72 views

Does pullback and d and dbar commute?

If $M$ is a complex manifold. we can write $d = \partial + \overline{\partial}$. Does pullback commute with either $\partial$ or $\overline{\partial}$?
4
votes
1answer
117 views

Existence of Complex Structures on Complex Vector Bundles

Let $E$ be a real vector bundle on a smooth manifold $X$. Let $J : E \to E$ be a vector bundle morphism (i.e. $\pi \circ J = \pi$, where $\pi : E \to X$ is the projection map) with $J^2 = ...
3
votes
1answer
92 views

real part of a holomorphic function from a PDE

I have some problem that I can't figure out myself. Hope that someone can help me out. The problem is: Let $f : U \to \mathbb{R}$ be some real function on a simply-connected domain $U \subset ...
16
votes
2answers
254 views

When is the sheaf corresponding to a vector bundle on a smooth manifold coherent?

In algebraic and analytic geometry, vector bundles are usually interpreted as locally free sheaves of modules (over the structure sheaves). They are in particular examples of quasi-coherent sheaves. ...
7
votes
1answer
102 views

When is a $k$-form a $(p, q)$-form?

Let $X$ be a complex manifold and denote the space of all $(p, q)$-forms on $X$ by $\mathcal{E}^{p,q}(X)$. Forgetting about the complex structure, we can consider the real differential $k$-forms on ...
6
votes
1answer
102 views

Holomorphic structures on associated bundles

Suppose $M$ is Kähler. Let $P \to M$ be the principal $U(n)$-frame bundle of $M$. Let $(\pi,V)$ be a finite dimensional unitary representation of $U(n)$ and let $E = P \times_\pi V$ be the ...
3
votes
3answers
126 views

Definition of vector bundle

Everywhere i see definition of vector bundle as triple $(E, p, B)$, $B$ and $E$ are manifold and local trivialization condition holds. For example see the definition here. . Local trivialization ...
5
votes
0answers
174 views

understanding this differential operator on a tensor product

I am currently trying to read the T. Kotake's paper "An Analytical Proof of the Classical Riemann Roch Theorem", in which he defines a differential operator which acts on smooth sections of a tensored ...
1
vote
0answers
42 views

why do i need convexity of the set of positive definite hermitian matrices for hermitian structure on vector bundle?

Consider a Riemann surface $\Sigma$ and a holomorphic vector bundle $\mathcal{E} \to \Sigma$ of rank $N$. I just came across the remark that in order to endow $\mathcal{E}$ with a Hermitian metric it ...
1
vote
1answer
102 views

When does the vanishing wedge product of two forms require one form to be zero?

Let $\alpha$ and $\beta$ be two complex $(1,1)$ forms defined as: $\alpha = \alpha_{ij} dx^i \wedge d\bar x^j$ $\beta= \beta_{ij} dx^i \wedge d\bar x^j$ Let's say, I know the following: 1) $\alpha ...
5
votes
1answer
130 views

How to prove (0,1) form is not $\overline\partial$-exact

On a complex manifold, if we are dealing with the $d$ operator, there's a pretty easy way of showing some form is not $d$-exact, simply by integrating in a closed loop. If you can find a loop that is ...
1
vote
1answer
108 views

Wedge product $d(u\, dz)= \bar{\partial}u \wedge dz$.

How to show that if $u \in C_0^\infty(\mathbb{C})$ then $d(u\, dz)= \bar{\partial}u \wedge dz$. Obrigado.
4
votes
2answers
293 views

what does following matrix says geometrically

Let $M\subset \mathbb C^2$ be a hypersurface defined by $F(z,w)=0$. Then for some point $p\in M$, I've $$\text{ rank of }\left( \begin{array}{ccc} 0 &\frac{\partial F}{\partial z} ...
3
votes
0answers
79 views

Length of $\frac{\partial }{\partial z}$ in Kaehler geometry.

I am taking a Kaehler geometry course this semester. The book we use is Tian's Canonical Metrics in Kaehler Geometry. I got a little confused about the calculation there in. For example, ...
0
votes
0answers
136 views

volume form on a Riemann Surface

Suppose I have a Riemann Surface (i.e. an oriented manifold) and I have an integral that uses the volume density $|dx|$ instead of the volume form $dx = dx_1dx_2$ (here the $(x_1,x_2)$ are local real ...
3
votes
2answers
216 views

Some references for potential theory and complex differential geometry

I am looking for references on two distinct (though related) topics. Potential theory : I read some time ago the book of Ransford (Potential Theory in the complex plane). It was great (intuitive ...
5
votes
1answer
105 views

Deformation of the Kähler structure on $CP^n$

Using the homogeneous coordinate on $CP^n$, we consider the open set $U_0:=\{[1, \ldots, z_n]\}$. Then the standard Kähler form of $CP^n$ can be written as $$ ...
0
votes
2answers
131 views

Section of a holomorphic vector bundle

I am stuck on this problem. Let $E$ be a holomorphic vector bundle on a complex manifold $M$, and $N$ a complex submanifold of codimension at least two. Prove that every section of $E$ on $M\backslash ...
2
votes
1answer
100 views

A question about complex manifolds

Let $(M,J_{M})$ be a almost complex manifold and $(N,J_{N})$ be a complex manifold. I want to prove that $F^{*}(\mathcal{O}_{N})\subset\mathcal{O}_{M}$ implies that $F:M\rightarrow N$ is almost ...
2
votes
0answers
201 views

Differential forms and a chain rule

Let $U$ be a Riemann surface and let $z:U\longrightarrow B(0,1)$ be a diffeomorphism, where $B(0,1)$ is the open unit disc in $\mathbf{C}$. So $z$ is a coordinate around $P=z^{-1}(0)$. Let $Q\in U$ ...
3
votes
2answers
262 views

Calabi-Yau Manifolds

In short, I'm hoping for some reading recommendations. I'm starting to do some work with Calabi-Yau manifolds, though my prerequisites are fairly minimal in differential geometry. I've taken a ...
1
vote
0answers
84 views

Complex torus and integral of forms

Start with a complex torus $X$ of dimension $n$ and a basis $e_1,...,e_{2n}$ for its first integral cohomology. Let $f_1,...f_{2n}$ be the dual basis vectors, so they are in $H^{1}(X,\mathbb Z)^*$. ...
6
votes
3answers
316 views

Where can I read about elliptic operators on manifolds?

In Voisin's beautiful book on Hodge theory she gives a proof that every cohomology class can be represented by a harmonic form by referring to the the following theorem on elliptic operators: Let $P ...
0
votes
1answer
63 views

Differentiable connections

Let $E$ a vector bundle on a differentiable manifold and $D: E\rightarrow E\otimes \Omega^1$ an homomorphism, with $\Omega^1$ differential 1-forms. If I take the map $D\wedge D$ which is the target ...
2
votes
1answer
134 views

is the differential of the distance function holomorphic?

i have the following problem. Let $M$ be a complex n-dim manifold and $X \subset M$ be a n-dim real analytic submanifold. Consider $d_{X}(z)$ be the squared distance from $z \in M$ to $X$. For $z$ ...
5
votes
1answer
152 views

Nonintegrable almost complex structures

The Newlander-Nirenberg theorem states that any Integrable Almost Complex manifold is a complex manifold. I am looking for natural examples of complex structures that are not integrable.
12
votes
1answer
178 views

Can one use Atiyah-Singer to prove that the Chern-Weil definition of Chern classes are $\mathbb{Z}$-cohomology classes?

In Chern-Weil theory, we choose an arbitrary connection $\nabla$ on a complex vector bundle $E\rightarrow X$, obtain its curvature $F_\nabla$, and then we get Chern classes of $E$ from the curvature ...
2
votes
1answer
333 views

Symplectic form on a complex manifold

I am a little muddled and am hoping I can get some clarification about forms in a complex manifold. Since I am only concerned with local issues, consider $M = \mathbb C^n$ as a complex manifold. So ...