Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry.

learn more… | top users | synonyms

3
votes
1answer
52 views

Vector bundle morphisms and determinants

Let $F,E$ be two locally free sheaves of equal rank on a complex manifold $X$. Assume that we have an injection $0\to F\to E$ such that the induced map $0\to \det F\to \det E$ is an isomorphism. Is ...
0
votes
1answer
59 views

Possible subbundles on $\mathbb{CP}^1$

I have the vector bundle $E:= \mathcal{O}(1)^{\oplus k}$ on the projective line, $\mathbb{CP}^1$. I want to know what are the possible subbundles of this, and why? We know that the Birkhoff-...
3
votes
1answer
65 views

Complex Elliptic Surfaces without sections

Is there a description of smooth complex projective surfaces without sections? While working on a problem a surface $X$ showed up with the following property: it is a non-ruled surface that has an ...
0
votes
1answer
38 views

Why there is no biholomorphism between complex plane and unit disk? [closed]

Why there is no biholomorphism between complex plane and the unit disk?
0
votes
1answer
62 views

Choice of Fundamental Domain of Torus (Dehn Twists?)

So I would like to consider a lattice $\Lambda \subseteq \mathbb{C}$ generated by $(1,\tau)$ with $\tau$ in the upper-half complex plane. This lattice $\Lambda$ will remain fixed. If you choose the ...
2
votes
1answer
41 views

Polyakov action in complex coordinates

Let $\Sigma$ be a compact $2$-manifold with riemannian metric $g$ and $f:\Sigma \to \mathbf{R}^n$ given locally by $f_1(x_1,x_2),\dots,f_n(x_1,x_2)$. Define $$ S(f,g) = -\frac{1}{2\pi\alpha'}\int_{\...
0
votes
1answer
24 views

Natural map from cokernel of a monad

If I have a monad $$ U \stackrel{\alpha}{\longrightarrow} V \stackrel{\beta}{\longrightarrow}W $$ then there should be a natural map $$ \text{cokernel}(\alpha) \rightarrow W $$ but I can't think of ...
0
votes
2answers
74 views

Reference request on complex projective algebraic geometry

I am looking for a reference on complex algebraic projective geometry. Specifically, I would like to become more acquainted with notions like the dimension and the degree of a projective algebraic set,...
0
votes
0answers
18 views

Cohomology with adding an infinite point

Let $F$ be a closed subset of $\mathbb{C}$. (We assume $F \neq \emptyset, F \neq \mathbb{C},$ and $0 \notin F$.) Of course $F$ is not a closed subset of $\overline{\mathbb{C}}$ in general but it ...
1
vote
1answer
81 views

Kähler differential and higher derivations (geometric interpretation of diagonal here)

I am studing Kähler differentials and I tried to understand the geometric motivation behind these settings. What I do not understand is the role which plays the diagonal in all these theory. The ...
0
votes
1answer
36 views

On the proof of Riemann extension theorem in Huybrechts

In the proof of the generalized Riemann extension theorem in Daniel Huybrechts's Complex Geometry, which is: Proposition 1.1.7 (Riemann extension theorem) Let $f$ be a holomorphic function on an ...
6
votes
1answer
201 views

Find type of a differential form on an almost complex manifold

If $M$ is a nearly Kähler manifold (that is, an almost Hermitian manifold on which $\nabla_X(J)X=0$) we have the three-forms $$ A(X,Y,Z)=\langle\nabla_X(J)Y,Z\rangle \quad\text{and}\quad B(X,Y,Z)=\...
0
votes
0answers
26 views

Show that $|1-h \lambda| <1 $ is a disc

From the stability region of Euler, show that $|1-h \lambda| <1$ is a disc, where $\lambda$ is imaginary. I am wondering why it is a disc with center at $h \lambda = -1$
4
votes
1answer
151 views

How much algebraic geometry is there in complex geometry (for example, Demailly)?

I wonder how much of modern algebraic geometry (schemes, etc) is there in complex (algebro-analytic) geometry. What I mean is complex algebraic (analytic) geometry in the sense of Demailly, Lasarsfeld ...
4
votes
1answer
125 views

Vector bundle as locally free coherent sheaves

I am studying coherent sheaves and was looking for a geometric motivation. Hence, in wikipedia and although here is stated that it can be seen as a generalization of vector bundles, which is quite ...
6
votes
0answers
94 views

Is there an irreducible projective hypersurface such that its complement has zero Euler characteristic?

We know that, if $f=X_0X_1...X_n \in \mathbb{C}[X_0,...,X_n]$ and $Z(f)\subset \mathbb{CP}^n$, then the Euler characteristic of its complement is zero, i.e. $$ \chi(\mathbb{CP}^n\setminus Z(f))=0. $$ ...
3
votes
1answer
73 views

Which line bundle has transition function $\psi_{12}([z_1,z_2])=\frac{z_1/z_2}{|z_1/z_2|}$?

We know that the complex line bundles over $\Bbb{CP}^1$ are classified by the integers. Each is isomorphic to one of the bundles $\mathcal{O}(n)$ for $n\in\Bbb Z$, where $\mathcal{O}(-1)$ is the ...
3
votes
1answer
67 views

Analytic proof of Serre vanishing theorem

Consider the following equivalent statement of Serre vanishing theorem (replacing ampleness condition on the line bundle with postivity condition). Let $X$ be a compact complex manifold. Let $L$ be ...
6
votes
1answer
104 views

Imaginary lines and tangents

My geometry text seems to say that the tangent to a circle from an interior point in the real plane is imaginary. Further...it seems that when a double cone is intersected by a plane with an angle ...
3
votes
0answers
50 views

valid definition of complex geodesic

Let $X$ be a complex manifold, and let $Y$ be a complex submanifold of $X$. If $X$ has an hermitian structure(on its tangent bundle), we can consider the Chern connection $\nabla$ on the holomorphic ...
2
votes
0answers
74 views

Functions that maps unit circle into unit circle

This and This problems discuses the characterization of Analytic functions which maps unit circle on to itself. I would like to know the characterization of functions which map unit circle into ...
0
votes
1answer
75 views

Horrocks-Mumford bundle

Let $E$ be the Horrocks-Mumford bundle, which is a rank-2 vector bundle on $\mathbb P^4$ with $c_1(E)=5$ and $c_2(E)=10$, defined by some combinatorical construction (see Okonek, Schneider, Schindler, ...
0
votes
1answer
47 views

How do we get from $\Delta f = \rho$ to $\partial\bar{\partial}f = \text{Const.} \rho\,dz\wedge d\bar{z}$?

I asked this question about the Kähler potential on MathOverflow. Donu Arapura left a comment saying Classically, a potential satisfies $\Delta f = \rho$. In the plane, this can be rewritten as $\...
2
votes
1answer
95 views

Neron-Severi group as the image of first Chern class

Let $X$ be a smooth projective variety over $\mathbb{C}$, then the Neron-Severi group $NS(X)$ of $X$ is defined to be the Picard group of $X$ modulo algebraically equivalent relations. On the other ...
1
vote
0answers
39 views

The Levi-Civita and the Covariantly Constant Tensors in Kahler Manifold?

Please scroll down to the bold section if you are too bored to read the whole details. Aiming to explain the mathematical structure of Kahler manifolds, Freedman and Van Proeyen, in their book ...
3
votes
0answers
28 views

almost complex structure on a surface

Let $M$ be an oriented smooth surface, $GL(M) \to M$ the bundle of oriented frames of $M$. Why is the space $S(M)$ of almost complex structures on $S$ equal to smooth sections of $GL(M) \times_{GL_2^+(...
1
vote
0answers
30 views

One of Hermitian metric's properties?

We now define a Hermitian manifold is a complex manifold in which unmixed components of metric tensor vanish $g_{ij}=g_{\bar{i}\bar{j}}=0$. Is this a propert of a Hermitian manifold? Or is it an extra ...
3
votes
1answer
70 views

first chern class of holomorphic tangent bundle $T\mathbb{C}P^n$

Let $L$ be tautopological bundle of $\mathbb{C}P^n$ and $L^{-1}$ be its duality. Because $L$ is a subbundle of $\underline{\mathbb{C}}^{n+1}$, $\underline{\mathbb{C}}=L\otimes L^{-1}$ is a subbundle ...
4
votes
1answer
66 views

Algebraic surface with infinitely many exceptional curves

I am learning about the classification of Projective Algebraic Surfaces (in fact, Compact Complex Surfaces) and I am troubled with the following point. If I understood correctly every surface $X$ ...
2
votes
1answer
88 views

the $\partial\bar{\partial}$-lemma dilemma

In the question here Simplifying the Kahler form, user290605 asked a question about how is that when we take the differential of Kahler form:$$\mathcal{K}=\frac{\sqrt{-1}}{2\pi}g_{i\bar{j}}dz^i\wedge ...
0
votes
0answers
43 views

Real Quadratic Forms, Complex Quadratic forms, and the Inertia Theorem.

I am very confused about the classification of quadratic forms. Scroll to the last paragraph for my question. Here is what I know: A $\bf \text{real}$ quadratic form is obtained from any bilinear ...
1
vote
0answers
30 views

Why was this terminology of holomorphism used here?

page 6 in this notesays: A note on index convention: $i$,$j$, and $\bar{i}$, $\bar{j}$ index over the holomorphic and antiholomorphic components. I never knew there were holomorphic components,...
2
votes
1answer
60 views

Why do those terms vanish if the metric is Hermitian?

On this page, the author says: We now define a Hermitian manifold as a complex manifold where there is a preferred class of coordinate systems in which unmixed components of metric tensor vanish ($...
0
votes
0answers
48 views

Equivalence of a vector bundle being trivial on $\mathbb{P}^1$

I am looking for various statements about a vector bundle $E$ of arbitrary rank being trivial on the complex projective line, $\mathbb{P}^1$. In particular, some arguments about cohomology would be ...
1
vote
0answers
24 views

Moduli Space of elliptic fibration

Given an elliptically fibered Calabi-Yau threefold in Weierstrass form I want to compute the number of complex structure moduli of the fibration. I know how it is done for the generic Weierstrass ...
2
votes
3answers
41 views

What is $\partial \bar{\partial}$ and when is it non-zero?

I often see $\partial \bar{\partial}$ arising in complex geometry, what is it (explain in relatively simpler language)? And sometimes $\partial \bar{\partial}T$ is non-zero (for example, the "current"...
2
votes
0answers
38 views

Questions about the connected component of a relative Picard Scheme.

Let $X$ be a smooth, projective surface (i.e. $2$-dimensional connected variety) over $k=\mathbb{C}$. Denote by $\mathrm{Pic}_{X/k}$ the associated relative Picard scheme. We write $\mathrm{Pic}^0_{X/...
11
votes
1answer
154 views

Is the homology class of a compact complex submanifold non-trivial?

Let $X$ be a connected complex manifold (not necessarily compact). Let $C \subset X$ be a compact complex $k$-dimensional submanifold (for some $k>0$). Is it true, in this generality, that the ...
3
votes
1answer
58 views

When is a quasiprojective variety Kobayashi hyperbolic?

I am looking for some (simple and well-known) sufficient conditions on a quasiprojective variety to be Kobayashi hyperbolic. I realize that in this generality it may be a complicated (maybe even ...
1
vote
1answer
25 views

Proving the principle symbol is globally defined

I want to prove the principle symbol is globally defined as an element \begin{align} \Gamma(T^* M / \{0\}, \textrm{Hom}(\pi^*E, \pi^* F) ) \end{align} To more specify, let me explain the definition ...
2
votes
1answer
57 views

Riemann Surface of $w^{2}=\sqrt{1-z^{2}}$

I'm working in the problem of finding branch points and build the Riemann Surface of the following complex function: $$ w(z)=\sqrt{1-z^{2}} \,\, . $$ I'm reading lots of texts about how to do this, ...
2
votes
1answer
36 views

Doubt with an illustration of algebraic curves and Riemann surfaces

The complex equation $w - z = 0$, $z$, $w \in \mathbb{C}$, represents a complex curve (also called $1$-dimensional complex manifold). This complex curve corresponds to the complex plane $\mathbb{C}$ ...
2
votes
0answers
14 views

k-symbol differential opeartor L, and its independent of choices.

This material is in O. Well's Differential analysis on complex manifold, page 115. Let, $(x,v) \in T'(X)$ ($T^*(X)$ with deleted zero section) and $e \in E_x$, Find $g\in \epsilon(X)$ and $f\in \...
2
votes
0answers
26 views

Is there a description of variation of (mixed) Hodge Structures in terms of a Deligne operator?

A complex Mixed Hodge Structure is given by a complex vector space $V$ together with a descending filtration $W$ and two ascending filtrations $F,\bar{F}$ that satisfy the condition \begin{equation} ...
2
votes
0answers
20 views

Hermitian vector space and relation of associated operators.

Here what i want to do is prove proposition 1.1 in chapter 5, on Wells, Differential analysis on complex manifold, The propositions are follows For Hermitian vector space of complex dimension $n$. ...
6
votes
2answers
149 views

(Anti-) Holomorphic significance?

What are holomorphic and anti-holomorphic components? Why don't we call them complex components and their conjugates? What is holomorphic coordinate transformation?
2
votes
1answer
72 views

Simplifying the Kahler form

In the link here, p.4, it says that, given a fundamental 2-from $\mathcal{K}$ $$\mathcal{K}=\frac{\sqrt{-1}}{2\pi}g_{i\bar{j}}dz^i\wedge d\bar{z}^{\bar{j}},$$ a manifold is said to be Kahler if this ...
6
votes
1answer
138 views

Is there an elementary way to see that there is only one complex manifold structure on $R^2$?

Is there an elementary way to see that there is only one complex manifold structure on $\mathbb{R}^2$? (Up to biholomorphism, naturally.) Elementary in the sense of not appealing to the ...
2
votes
1answer
42 views

What does $dz^2$ mean?

I'm reading a paper ("La Formule de Verlinde" by Christoph Sorger) and at a certain point, the author switches from algebro geometric language to complex geometric language. He uses the symbol $dz^2$, ...
2
votes
0answers
78 views

Primitive cohomology, example request

$X$ is a compact Kähler manifold or smooth projective variety. is there an example that a primitive class $0\neq [\omega]$ of $H^{p+q}(X, \mathbb{C})$ is wedge product of other two primitive classes: $...