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

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Fundamental cycle $[Y]\in H_{2n-2}(X_{\mathbb{R}},\mathbb{Z})$ of irreducible analytic hypersurface on a complex manifold X.

I am studying about complex manifolds and I am trying to understand the following statement. Let $Y\subset X$ be an irreducible analytic hypersurface, where $X$ is an n-dimensional complex compact ...
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72 views

What is the canonical bundle of a smooth divisor?

I am currently trying to learn some complex geometry, using mainly the book by Huybrechts. There is one thing confusing me, however: For example on Wikipedia people are talking about the canonical ...
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Is the determinant bundle the pullback of the $\mathcal O(1)$ on $\mathbb P^n$ under the Plucker embedding?

Let $V$ be a $n$-dimensional complex vector space and consider the Grassmannian of complex $k$-planes $Gr(k,V)$. The Plucker embedding is an embedding $p:Gr(k,V) \to \mathbb P^M$ where $M = ...
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Smoothness and algebraicity

My question is the following : given a complex algebraic affine variety $M \subset \mathbb{C}^n$, let us assume that it has moreover a smooth differential structure as a submaifold of ...
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96 views

How do we wedge the complex differentials $\mathrm{d}z^i$ and $\mathrm{d}\bar z^{\bar j}$?

By the standard definition of the wedge product as an alternated tensor product, I would think we have $$\tag{1}\mathrm{d}z^i\wedge\mathrm{d}\bar z^{\bar j}=\mathrm{d}z^i\otimes\mathrm{d}\bar z^{\bar ...
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What are some good sources to learn about real analytic manifolds?

Asking this question as someone with a graduate student level understanding of smooth differential/Riemannian geometry (May be a bit more that Riemannian Geometry by Do Carmo). I am trying to upgrade ...
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1answer
40 views

Naive question about almost complex structures and hermitian metrics

I'm sorry if I have another naive question, but I want to understand correctly almost complex structures and hermitian metrics. Let's have $M$ complex manifold with almost complex structure $J$ and ...
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1answer
92 views

Why is the study of cones important in algebraic geometry?

It was quite difficult writing this question...I hope you can understand the goal of that. I want to ask it in order to clarify my ideas. I'm studying Hyperkahler geometry (principally on works by ...
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30 views

is that function must be constant under the following conditions

I'm talking about complex function $f$ is analytic function on a region $D$ that include the point $z=0$. for every $n\in N$ such $\frac{1}{n}$ is in $D$ , the function follows this condition ...
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1answer
100 views

Interpretation of the Weierstrass Preparation Theorem

I am reading Griffiths and Harris' Principles of Algebraic Geometry but I am having trouble making sense of a statement following the Weierstrass Preparation Theorem (p.9 in my edition). The ...
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19 views

Newlander-Nerenberg theorem in Voisin's book

I am reading the proof of Newlander-Nerenberg's theorem in the real analytic case, and there are some parts I don't understand, can someone help me please? 1) In the beginning, she said: " since ...
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Examples of complex manifolds satisfying the Kodaira-Spencer condition for unobstructed deformations

The theorem of existence proved by Kodaira and Spencer shows that a complex manifold $M$ for which the second cohomology group $H^2(M, \Theta T^{1,0} M)$ is trivial, $\Theta T^{1,0} M$ being the sheaf ...
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44 views

the laplacian in real coordinates and complex coordinates

In the notes by Szekely it says on page 35 that since $\nabla_k\frac{\partial}{\partial \overline z_l}=0$ it follows in holomorphic coordinates that the Laplacian is $\Delta f=g^{j\overline ...
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1answer
57 views

Does a left-invariant vector field on a complex Lie group preserve holomorphic functions?

Let $G$ be a (finite-dimensional) complex Lie group, and suppose $f : G \to \mathbb{C}$ is holomorphic. Let $X$ be a left-invariant vector field on $G$. Must $Xf$ be holomorphic? I think I have a ...
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1answer
36 views

Coordinate on the boundary of Riemann surface

Let $\Sigma$ be a Riemann surface with boundary. Question: Is there canonical way to parameterise the boundary components up to shift? By shift I mean change of coordinate $\phi$ to $\phi + c$. ...
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24 views

why we need stereographic projection in complex analysis?

Was this the first time we came to know that we need to extend the complex plane?why we used unit sphere there?
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40 views

Sections of inverse image sheaf of sheaf of sections of vector bundle

Let $\eta \colon Y \to Z$ be holomorphic map between complex manifolds, $E$ holomorphic vector bundle over $Z$, $\eta ^\ast E$ its pullback over $Y$, ${\cal O}_E$ sheaf of sections of $E$, and ...
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48 views

Schwarz theorem in complex analysis.

Is there a version of the Schwarz theorem $ \partial_x \partial_y = \partial_y \partial_x $ in the theory of complex functions of several variables and complex analysis ? It would be nice that you ...
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111 views

defining the canonical divisor

I am just starting to learn some basic algebraic geometry, but I am very confused with some definitions. So I hope I am not asking something that is completely trivial. Suppose $X$ is a normal ...
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25 views

computing the metric on a complex manifold

Let $M^n$ be a complex manifold with coordinates $z_1,\dots, z_n$ where $z_k=x_k+\sqrt{-1}y_k$. Let $$\frac{\partial}{\partial z_k}=\frac{\partial}{\partial x_k}-\sqrt{-1}\frac{\partial}{\partial y_i} ...
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1answer
91 views

{line bundles} $\neq$ {divisor line bundles}

Let $X$ be a compact complex manifold, and with the following sheaves $\mathscr O$, the sheaf of holomorphic function, $\mathscr O^*$, the sheaf of nonvanishing holomorphic function $\mathscr K^*$ ...
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What is the poinacre dual of the projectivization of a line bundle inside the projectivization of the sum of two line bundles?

Let $S:= \mathbb{CP}^1 \times \mathbb{CP}^1$. Let $TS \approx L_1 \oplus L_2$, where $L_1$ and $L_2$ are the pullback of $T\mathbb{CP}^1$ wrt to the respective projection maps. Note that ...
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40 views

Can a product of a Stein manifold and a compact manifold be again Stein?

A Stein manifold is a manifold which is holomorphically separable and convex. It is well known that a product of two holomorphically convex (resp. Stein) manifolds is again holomorphically convex ...
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“Pulling back a function from a neighborhood of 0 in $\mathbb{C}^2$ to $\mathbb{U}$ is analogous to computing the derivative of that function.”

I'm reading on Riemann Surfaces and after defining $$\mathbb{U} \equiv \{(z,\ell)\in\mathbb{C}^2\times\mathbb{P}_1;z\in\ell\},$$ and $B\ell_0:\mathbb{U}\to\mathbb{C}^2$, the author mentions that, ...
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Trying to understand a theorem of Chow

What is an example of a compact analytic subspace of the complex manifold $\mathbf{P}_{\mathbf{C}}^n$, the projective $n$-space over $\mathbf{C}$? Are they classified?
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90 views

Existence of a holomorphic connection implies existence of a flat connection

Let $E$ be a holomorphic vector bundle on a complex manifold $X$, and let $D: E \to E \otimes \Omega_X$ be a holomorphic connection on $E$ i.e. $$ D(fs) = s\otimes \partial(f)+ fD(s), $$ for any local ...
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76 views

Embedding a $deg \geq 4$ curve in $\mathbb{P}^2$

Reading the paper by Kollar "The structure of algebraic threefolds", among some examples he does talking about intrinsic and extrinsic geometry over $\mathbb{C}$, he mentions that "an irreducible ...
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1answer
30 views

Sketch $ z\in \mathbb{C}:0 < arg(z-(1+i)) < \frac\pi3 $

Sketch the following $$ z\in \mathbb{C}:0 < arg(z-(1+i)) < \frac\pi3 $$ I have considered this geometrically and ended up thinking that the complex numbers $z$ must satisfy $$0 < ...
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1answer
97 views

Why do we require that a complex manifold has the structure of a real manifold?

I am taking a course in complex manifolds, heavily influenced by Huybrechts' book "Complex Geometry", and in it we define a complex manifold $X$ to be a smooth, real manifold $M$ together with an ...
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1answer
154 views

Definition of complex conjugate in complex vector space

I am starting reading about Hodge theory and while reading the definition of abstract Hodge structure a very basic question came to my mind... What is the definition of the conjugate of a subspace of ...
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88 views

the top chern class of the holomorphic tangent bundle is the euler class

Is the following true? Let X be a complex manifold of complex dimension d and let V denote its holomorphic tangent bundle (ie it's $T^{1,0} \subset T \otimes_R C$, where T is the tangent bundle of ...
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1answer
100 views

Find all holomorphic diffeomorphisms $f:\mathbb{CP}^1\to\mathbb{CP}^1$

The complex projective line $\mathbb{CP}^1$ is the complex manifold defined by the quotient of $\mathbb{C}^2-\{(0,0)\}$ by the relation $z\sim w$ if $z=\lambda w$ for $\lambda\in\mathbb{C}-\{0\}$. I ...
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164 views

Exercise from Geometry of algebraic curves by Arbarello, Cornalba, Griffiths, Harris

Let $\pi:C^{'} \rightarrow C$ an unramified double cover of a complex Riemann surface $C$ of genus $g$. With the symbol $Nm_{\pi}$ we mean the norm application that takes a meromorphic function on ...
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125 views

Intuitive Aproach to Dolbeault Cohomology

I would like to understand an intuitive approach to the definitions of Dolbeault Cohomology (using $\partial$ and $\bar{\partial}$) similar to the one given here. All suggestions are welcome.
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1answer
62 views

$(p,q)$ part of a complex differential form in terms of the complex structure $J$?

Say $M$ is a complex manifold, viewed as real $C^{\infty}$ manifold with an integrable almost complex structure $J$. Let $\omega$ be a complex $r$-form on $M$. Is there a way to express the $(p,q)$ ...
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75 views

Integrate $\int_C{\tan{z}\ dz}; C: y=x^2$ (complex numbers)

Integrate $$\int_C{\tan{z}\ dz}$$ $C$ is the parabola arc $y=x^2$ that connects the points $z=0$ and $z=1+i$. This is what I've done so far: I know that $\tan{z}=\dfrac{\sin{z}}{\cos{z}}$ And ...
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47 views

Comparing vector bundle degrees coming from different embeddings into projective space

This question is a follow-up to this recent question of mine: Comparing notions of degree of vector bundle In that question, the definition of the degree of a vector bundle is discussed — in ...
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Cohomology vanishing on projective manifold, want to show that a line bundle $L$ is ample

I have some questions regarding the proof of the following theorem. Let $X$ be a projective manifold and $L$ a line bundle on $X$. Then $L$ is ample if and only if for all coherent sheaves ...
3
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1answer
204 views

What is $\mathbb{P}^{\infty}$?

Can we look at a complex projective space $\mathbb{P}^{\infty}$? I am curious to know what would it be. What is the right intuition to think about it? I know $\mathbb{P}^{n}$ is a space of ...
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56 views

In the real spherical harmonics, where does the sqrt(2) factor come from?

The real spherical harmonics can be written in terms of the complex spherical harmonics: $$ Y_{\ell m} = \begin{cases} \displaystyle \sqrt{2} \, (-1)^m \, \operatorname{Im}[{Y_\ell^{|m|}}] & ...
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1answer
75 views

Help needed to understand statements about torus

I am having trouble understanding two statements: Let $A$ be an algebraic curve in $\mathbb{P}^2$ over $\mathbb{C}.$ Consider its normalization $$\pi: \hat{A} \to A.$$ If genus $g(\hat{A})=1,$ ...
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Fano-ness of moduli space of stable vector bundles when determinant line bundle is *not* fixed…

According to Drezet-Narasimhan, Invent. Math. 97 (1989), no. 1, 53--94, the moduli space $\mathbb M$ of slope-stable holomorphic vector bundles with fixed rank $r$ and fixed determinant line bundle ...
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1answer
77 views

Harnack's curve theorem for curves in $\textit{complex}$ projective plane?

The wikipedia page gives the statement for algebraic curves in real projective plane. Is the statement also true in $\textit{complex}$ projective plane? If not, is there a similar statement about ...
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How to use complex numbers in geometry

Let $H$ be the orthocenter of an acute-angled triangle $ABC$. The circle $\Gamma_{A}$ centered at the midpoint of $BC$ and passing through $H$ intersects the sideline $BC$ at points $A_{1}$ and ...
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850 views

Sine of a Complex Number

While I know that $\sin(x)=2$ has no real solution, I tried seeing if it has a complex solution. That equality is equal to $$e^{2ix}-4ie^{ix}-1=0$$ Taking a quadratic in $e^{ix}$ I got ...
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2answers
91 views

Elements of the zero-th Čech cohomology group versus global holomorphic sections

Something that is confusing (well, to me) has come up in the course of asking other questions. Let $\pi:V\to X$ be a holomorphic vector bundle of rank $r$ on a complex manifold $X$, such that $V$ is ...
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1answer
94 views

Complex structure on the Jacobian of a Riemann surface

Let $X$ be a fixed smooth, connected, compact Riemann surface of genus $g$. The Jacobian variety $\mbox{Jac}(X)$, which parametrises isomorphism classes of holomorphic degree $0$ line bundles on $X$, ...
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1answer
92 views

Isomorphisms (and non-isomorphisms) of holomorphic degree $1$ line bundles on $\mathbb{CP}^1$ and elliptic curves

I have two highly-coupled questions concerning holomorphic line bundles, and so I will go ahead and ask them together. The first concerns line bundles on $\mathbb{CP}^1$ and the other concerns line ...
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2answers
203 views

What is the meaning of normalization of varieties in complex geometry?

There is a question already asked here about this. But I know almost nothing of algebraic geometry, nothing fancy to understand the answer. So I would highly appreciate an elementary explanation to my ...
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48 views

Constructing a complex structure on $S^2$

By definition of complex manifold, a complex manifold is a manifold with holomorphic charts $U \to D^2 \subseteq \mathbb C$. I want to define a complex structure on $S^2$. Can you tell me if ...