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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3
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2answers
68 views

Take tautological bundle over $\mathbb{C}P^n$ less the zero section. Is this homeomorphic to $\mathbb{C}^{n+1}-{0}$?

It appears that (non zero vector in $\mathbb{C}^{n+1}$)$\mapsto$(line containing the vector, vector in this line) is a homeomorphism from $\mathbb{C}^{n+1}$ to the tautological bundle over $\mathbb{C}...
2
votes
1answer
49 views

Non-tangent lines to lines in $\mathbb{P}^3(\mathbb{C})$.

Let $\pi:\mathbb{C}^4\setminus\{0\}\to\mathbb{P}^3(\mathbb{C})$ be the quotient map. Let $Q\subset\mathbb{P}^3(\mathbb{C})$ be a smooth quadric, let $q$ be the quadratic form such that $Q=\pi[\{v\in\...
0
votes
0answers
40 views

Picard number of Kahler manifold

Let $(M,\omega)$ be a Kahler manifold. How can we define simply the Picard number for the special case where $M$ is also projective? Wikipedia defines it as the rank of the Neron-Severi group. In ...
2
votes
0answers
57 views

adjoint functor of inverse image functor

$f: U\hookrightarrow X$ an open immersion of two complex manifolds. $f^{-1}$ is inverse image functor, in usual sense, from category of sheaves of abelian groups $\mathcal{Ab}(X)$ over $X$ to category ...
0
votes
1answer
25 views

Application of Rouche theorem in order to find the roots of a polynomial in each quadrant.

I want to solve the following : (i) Show that $z^4+2z^2-z+1$ has exactly one root per plane quadrant. My idea to prove (i) is by using Rouche theorem, by considering 4 cuts of the complex plane ...
0
votes
0answers
29 views

Euler characteristic of branch cover of punctured Riemann surface

Let $\Sigma_1$, $\Sigma_2$ be two closed Riemann surfaces, $\pi: \Sigma_1 \to \Sigma_2$ is degree $m$ branched cover of $\Sigma_2$, then we have formula about their Euler number: $$\chi(\Sigma_1)= m\...
1
vote
2answers
35 views

Local defining functions on real hypersurfaces.

I'm currently reading Real Submanifolds in Complex Space and their Mappings by Baouedi, Ebenfelt and Rothschild. I'm currently stumped by what the author's claim to be an easy check (really blowing ...
1
vote
1answer
18 views

Indicate on an Argand Diagram the region of the complex plane in which $ 0 \leq \arg (z+1) \leq \frac{2\pi}{3} $

Question: Indicate on an Argand Diagram the region of the complex plane in which $$ 0 \leq \arg (z+1) \leq \frac{2\pi}{3} $$ I've tried this Consider $$ 0 \leq \arg (z+1) \leq \frac{2\pi}{3} $$...
0
votes
1answer
93 views

Prove the Jacobian of a curve of genus g is a complex torus

As stated in the title I am about to prove the Jacobian of a curve of genus g is a complex torus. Here is what I have done so far: I know the first homology group of $X$ is $H_1(X,\mathbb{Z}) \cong \...
1
vote
1answer
28 views

why $h^{2,0}$ ought to be positive for non-algebraic manifolds?

In their paper ``Nonalgebraic hyperkahler manifolds'' Campana, Oguiso and Peternell mention in Theorem 2.3 that if $Y$ is a smooth, Kähler, non-algebraic base of a fibration $f: X \dashrightarrow Y$ ...
1
vote
0answers
37 views

A counterexample of Riemann mapping theorem in high dimension

There is an exercise(1.1.16) in Huybrechts: the polidisc $B_{(1,1)}(0)\subset\mathbb C^2$ and the unit disc $D$ in $\mathbb C^2$ can not be biholomorphic. The hint is to compare the automorphisms of ...
0
votes
1answer
63 views

Derivative of a section of a vector bundle

Let $X$ be a complex algebraic variety and let $E \to X$ be a vector bundle over $X$, with sheaf of sections $\mathcal{E}$. If $s$ is a local section of $\mathcal{E}$, what is the derivative $ds$ ...
1
vote
1answer
32 views

Functions over a $C$ vector space with geometric importance. (How to find the basis?)

Searching through our suggested exercises of linear and abstract algebra for solving, I found the following exercise. The reason I am posting this, is that because we haven't went through complex ...
0
votes
1answer
29 views

Why is $\int_{X} d(\alpha \wedge *\bar{\beta})$ zero on a compact hermitian manifold?

I am reading the book Complex Geometry - An Introduction by Huybrechts. In proving Lemma 3.2.3 that $\partial$ and $\partial^*$ are formal adjoints to each other, he mention that the following ...
1
vote
0answers
27 views

PDEs on higher genus Riemann surfaces, e.g. Klein Curve

I'm trying to solve a PDE on compact Riemann surfaces of genus g > 1. Since these can be obtained as quotients of the upper half plane $\mathbb{H}_2$ by some Fuchsian group $\Gamma$, I suppose it's ...
0
votes
1answer
36 views

Subspace of infinite dimensional complex projective space generated by compact set

This question is similar to this one, but with the infinite dimensional complex space instead of the complex separable Hilbert space. My question is: if $S\subseteq \mathbb C P^\infty $ is a compact ...
4
votes
1answer
34 views

Does a hyperelliptic Riemann surface $S$ with $\# Aut(S)=2$ exist?

If a Riemann surface $S$ has genus $g\geq 2$, its automotphisms group is finite. I was wondering if there exists a hyperelliptic Riemann surface $S$ with $\# Aut(S)=2$. In other words, I was wondering ...
1
vote
1answer
71 views

Linear algebra for modern differential geometry( and other types of modern geometry, like analytic, complex and algebraic)

I wish to study real and complex analysis(for example, Pugh "Real Mathematical Analysis" and Cartan "Elementary theory of analytic functions of one and several complex variables") and modern ...
8
votes
2answers
123 views

Does the Lie derivative commute with $\partial$?

It is well-known that on a smooth manifold $M$, the Lie derivative commutes with the exterior derivative, i.e. $${\cal L}_Xd\alpha=d{\cal L}_X\alpha$$ for any vector field $X$ and differential form $\...
3
votes
0answers
32 views

Proving subgroup of $Aut(\Bbb C^2)$ that fixes a specific curve is isomorphic to $\Bbb Z^6 \times \Bbb Z^3 $

So, I have the curve $C = V(y^3 - x^6 + y^6) \subset \Bbb C^2$. I want to prove that, if $G= \{ \varphi = (f_1,f_2) \in Aut(\Bbb C^2):\varphi(C) = C,$ $ deg(f_i) = 1 \}$, then $G \simeq \Bbb Z^6 \...
3
votes
1answer
53 views

What is the horizontal space of trivial hermitian line bundle?

Suppose $L=M\times\Bbb C$ is a trivial holomorphic line bundle on a complex manifold $M$, and suppose there is a Hermitian fibre metric $h$ on $L$. Question: What is the horizontal space of $T_{(...
4
votes
0answers
45 views

Relationship between invariant and harmonic forms

Let $G$ be a compact real Lie group which is also a complex manifold (note that I do not want $G$ to be a complex Lie group). Let $\Omega^{p , q}(G)$ denote the space of smooth $(p , q)$-forms on $G$, ...
8
votes
3answers
362 views

Why only consider Dolbeault cohomology?

On a complex manifold we have the differential operators $$\partial:A^{p,q}\to A^{p+1,q}$$ $$\bar\partial:A^{p,q}\to A^{p,q+1}$$ which both square to zero. Hence one can define cohomology groups $$H^{...
0
votes
1answer
48 views

Page 276 of Principles of Algebraic Geometry by Griffiths and Harris; wrong parameter count?

The following is taken from page $276$ of Principles of Algebraic Geometry by Griffiths and Harris: Now let $S$ be a Riemann surface of genus $g\ge 3$. By our last result, if $S$ has any ...
4
votes
0answers
59 views

Mirror Symmetry of Elliptic Curve

I'm a little bit unsure about the mirror symmetry statement for elliptic curves; specifically, how the flipping of the Kähler and complex moduli works. Perhaps I should say at the outset, the reason ...
1
vote
0answers
22 views

Why is compactness needed for proving that Kahler forms are open

I am studying complex geometry from Huybrechts' Complex Geometry - An Introduction. In Corollary 3.1.8 he proves that: The set of all Kahler forms on a compact complex manifold $X$ is an open ...
2
votes
1answer
37 views

Embedding Complex Tori in Projective Space

When we talk about projectively embedding complex tori $\mathbb{C}^{g}/\Lambda$ (i.e in Lefshetz Embedding Theorem), what exactly do we mean by an embedding. Is it in the differential geometry sense ...
1
vote
1answer
83 views

Chern class of ideal sheaf

Let $X$ be a smooth projective surface. Let $Z$ be a dimensional $0$ subscheme of length $l$. Suppose $I_Z$ is the ideal sheaf of $Z$. Then it claimed that $c_1(I_Z) = 0$ and $c_2(I_Z) = l$. (1)Why ...
0
votes
0answers
26 views

Oka's coherence theorem and Cartan's theorem A and B

So reading up on some articles I've found that whilst attempting to characterize Oka's coherence theorem in sheaf-theoretic language, Cartan was ultimately led to formulating his theorem A and B in ...
6
votes
0answers
84 views

When exactly is a compact complex manifold algebraic?

It is well known that a necessary and sufficient condition for a compact Kähler manifold $\mathcal{X}$ to be a projective algebraic variety is that it admit a positive holomorphic line bundle $L \...
15
votes
2answers
228 views

Complex manifold with subvarieties but no submanifolds

Note, I have now asked this question on MathOverflow. There are examples of compact complex manifolds with no positive-dimensional compact complex submanifolds. For example, generic tori of ...
2
votes
0answers
28 views

Given an analytic continuation along $\gamma$ such that $R(t)\equiv \infty$ for some $t$, then $R(s)\equiv \infty$ for each $s\in [0,1]$

Definition: A function element is a pair $(f,U)$ where $U$ is a region and $f$ is an anaytic function on $U$. For a given function element $(f,U)$ define the germ of $f$ at $a$ to be the ...
0
votes
0answers
22 views

first Chern class and divisor under modifications

Assume that $X$ is a Moishezon manifold, then there exists a modification $\pi:\tilde{X}\rightarrow X$, where $\tilde{X}$ is a projective algebraic manifold. Let $\tilde{w}$ be a Kahler metric on $\...
1
vote
0answers
29 views

Hyperelliptic Curves and Period Integrals

I've been thinking about the period integrals on a hyperelliptic curve of the form $y^{2}=F(x)$, where $F(x)$ is a polynomial of degree-$2n$ with complex coefficients. Of course, a hyperelliptic ...
1
vote
0answers
23 views

Confusion surrounding the Koszul-Malgrange theorem

On a recent project, I ran into something which seemed related to the Koszul-Malgrange theorem. According to nlab, the original statement of the theorem is Theorem 1. (Koszul-Malgrange theorem) ...
1
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0answers
27 views

Hilbert scheme of quasi-projective variety

Suppose $X$ is a projective scheme over an algebraically closed field $k$, denote its Hilbert scheme with Hilbert polynomial $p$ by $\text{Hilb}^p_X$, then from section 1.1 of Nakajima's book, ...
0
votes
0answers
16 views

Prove L(T(γ))=L(γ) where L(γ) is the hyperbolic length

The first proof $\text{Im}(w)=(w-w^*)/2i$ I believe I have correct, but I need help on the second proof. $L(T(γ))=L(γ)$ I need to know if this is the same concept as, independent of choice of the ...
0
votes
0answers
23 views

Symmetric Product of a Projective scheme

Following the question, Theon Alexander (http://math.stackexchange.com/users/165460/theon-alexander), Reference-Request: Symmetric Product Schemes, URL (version: 2014-08-10): http://math....
1
vote
0answers
38 views

Hilbert scheme of $n$ points on a smooth curve

If $C$ is a smooth curve over a field $k$, then from lots of references, e.g. Janos Kollar, Rational Curves on Algebraic Varieties, exercise 1.4.1, that the Hilbert scheme of $n$ points is \begin{...
4
votes
1answer
80 views

The cylinder does not embed into $\Bbb C^n$

The cylinder $\Bbb R\times S^1$ can be viewed as a complex manifold with a flat metric by viewing it has the quotient $\Bbb R\times\Bbb R/\Bbb Z$, where $\Bbb R\times\Bbb R=\Bbb C$. (In fact it makes ...
4
votes
0answers
70 views

Embedding of Kähler manifolds into $\Bbb C^n$

Consider $\Bbb C^n$ with its standard hermitian product. This space produces many example of Kähler manifolds simply by taking a smooth affine variety $X\subseteq\Bbb C^n$ with the induced metric. Now ...
0
votes
0answers
31 views

Asymptotics of Harmonic Functions

I am looking for some information (answers, references etc.) on existence of solutions to \begin{equation} \Delta u = f \end{equation} on the Euclidean unit ball $B \subset \mathbb{C}^n$ with ...
10
votes
0answers
79 views

Is $X$ an algebraic subset? Analytic subset?

Suppose that $X$ is a subset of $\mathbb{C}^n$, and that every (complex) hyperplane section of $X$ is an algebraic subset (respectively analytic subset) of complex dimension at least one (or empty). ...
0
votes
1answer
37 views

Locally finitely generated sheaf

Here is the image from the book So I don't quite understand the proof of lemma 3.9. Namely, I don't see why there exists $H_{jk}$ such that the formula is true on $U'$. I was wondering if someone ...
2
votes
0answers
49 views

Ring of germs of holomorphic functions at $0\in \mathbb{C}$

So I've been reading the book and they used a induction proof where they just state that for the base case the ring of germs of holomorphic functions on $\mathbb{C}$ is Noetherian. I looked at other ...
1
vote
1answer
53 views

Obstructions to putting a complex structure on a real vector bundle (other than, obviously, dimension)

A complex vector bundle is usually described as one with structure group $GL(n,\mathbb{C})$. If I take a real $2n$ when is it not the underlying real bundle of some complex bundle?
3
votes
0answers
39 views

Example of a complex manifold with certain curvature properties?

Are there (nice) examples of a complex manifold such that the sectional curvatures through all the complex planes are non-positive but the sectional curvatures through the real planes are mixed?
3
votes
1answer
31 views

Diffeomorphism from Riemannian metric to Hermitian metric on a complex manifold.

It is known that any complex manifold, $M$ admits a Hermitian metric, i.e., a Riemannian metric, $g$, which satisfies \begin{equation} g_p(J_pX,J_pY)=g_p(X,Y) \end{equation} at each point $p\in M$, ...
8
votes
0answers
40 views

Geometric interpretation of the determinant of a complex matrix

A complex $n$-dimensional vector space $V$ can be thought of as a real $2n$-dimensional vector space equipped with a map $J:V \to V$ with $J^2 = -I$. Complex-linear maps are then linear maps $V \to V$ ...
10
votes
1answer
88 views

Is $T(M)$ necessarily equivalent to the normal bundle of $M$ in $\textbf{R}^{2n}$, if $M$ is totally real?

Consider real submanifolds, $M^n \subset \textbf{C}^n$. ($\textbf{C}^n \equiv (\textbf{R}^{2n}, J)$, where $J(x, y) = (-y, x)$). $M^n$ is totally real if $J(T_pM) \cap T_pM = 0$, for all $p \in M$. ...