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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58 views

Is the contraction of an harmonic form harmonic?

As I'm still a beginner in complex differential geometry, as soon as I tried reading an article I got stuck on what (I think) should be a minor detail. I hope someone can help me a bit. Let $M$ be a ...
1
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1answer
47 views

De Rham interpretation of $H^1(R,p,\mathbb{C})$

Let $R$ be a Riemann surface of genus $g\ge 2$ and $p\in R$ a point. This is my question: Is there a way to interpret the relative cohomology group $H^1(R,p,\mathbb{C})$ as a De Rham cohomology group ...
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11 views

Structures in Non-linear Sigma Model

I debated whether this belongs here, or on Physics SE, but all my questions correspond to the algebraic/complex geometry, not physics, so hopefully it's okay here. The non-linear sigma model ...
1
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1answer
24 views

The action of automorphisms on the Riemann sphere

If we are given that the automorphism group of the Riemann sphere is $$Aut\ \mathbb P^1=\{z\mapsto \frac{az+b}{cz+d}:ad-bc=1\}$$ Why this group does not have any proper subgroups that act without ...
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1answer
61 views

Local sections of $\mathcal{O}(1)$

Let $\mathcal{O}(-1)$ be the tautological bundle over $\mathbb{P}^n$ and $\mathcal{O}(1)$ its dual bundle, also known as the hyperplane bundle. I know that there is a bijection between $\Gamma(\mathbb{...
3
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1answer
61 views

First cohomology group on a Riemann surface with all wedge products equal to zero

Sorry for the strong edit, but I realized my question had a easier formulation: Can there be a Riemann surface $X$ with the property $\sigma\wedge \tau=0$ for every $\sigma,\tau\in H^1(X,\mathbb{C})$?...
3
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1answer
29 views

Ratio of holomorphic forms on a Riemann surface

Let $R$ be a Riemann surface of genus $g>1$ and $\omega$, $\sigma$ two holomorphic 1-forms on $R$. This means that locally we can write $\omega=fdz$ and $\sigma=gdz$ with $f$ and $g$ holomorphic. ...
0
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1answer
20 views

What is a primitive element in a Fuchsian group?

I am reading some introductory texts about Selberg's trace formula for hyperbolic surfaces and I have encountered the concept of "primitive element" $\gamma$ in a hyperbolic Fuchsian group $\Gamma \...
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0answers
34 views

Function as a combination of 1-forms on a Riemann surface

My question is quite simple, I hope it's not also stupid.. Consider $R$ a Riemann surface and $\omega_1$, $\omega_2$ two $(1,0)$-forms (i.e. holomorphic forms) and $\varphi_1$, $\varphi_2$ two $(0,1)$-...
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0answers
21 views

Describe Singular Locus of Hyperelliptic Curves?

Previously, I asked a question here: Moduli Space of Hyperelliptic Curves as Fibration? about fibering the moduli space of hyperelliptic curves $\rm{Conf}_{2n}(\mathbb{P}^{1}) \big/ \rm{Aut}(\mathbb{P}...
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1answer
59 views

Normal bundle to an exceptional sphere in a blowup along a smooth subvariety

Let $M$ be a smooth algebraic variety of dimension $m$ over $\mathbb{C}$, $S \subset M$ a smooth embedded subvariety of codimension two. Let $M_S$ denote the blowup along $S$ and $E$ denote the ...
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1answer
63 views

Moduli Space of Hyperelliptic Curves as Fibration?

Basically, I had a thought about a way to think of the moduli space of hyperelliptic curves. I'm sure it's wrong most likely, but I was hoping someone could maybe point out the flaw in my reasoning. ...
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0answers
39 views

Exciting applications of the Riemann-Roch-theorem for Riemann-surfaces

This semester I took a lecture on Riemann surfaces. The professor proved the Riemann-Roch-theorem (stated below). As an application of it, he proved elementary results, we did earlier in the course ...
4
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1answer
57 views

“Barred” Tensor Indices in Complex Manifolds

I'm having an embarrassingly hard time straightening out how to work with the "barred" indices that show up in tensors on complex manifolds. For example, the Kahler form $\omega = \frac{i}{2}g_{i \...
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26 views

Introduction to flag manifolds

What is a good self-contained introduction to the geometry of complex flag manifolds?
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40 views

Is there only one complex structure on complex plane $\mathbb{C}$? [duplicate]

There is a trivial complex structure on $\mathbb{C}$. Do we have other complex structures on complex plane $\mathbb{C}$? If not, how to prove it?
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1answer
26 views

How to determine all the complex structures on torus $T^2$?

I have known that the lattice given by the pair $(\tau_1,\tau_2)$ can determine a complex structures on torus $T^2$. But how to prove that all the complex structures of torus can be obtained in this ...
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38 views

Pushforward of canonical bundle restricted to divisor isomorphic to restriction of pushfoward of canonical bundle

Consider the branched covering $f \colon X \to \mathcal{Q}_7$ of the $7$-dimensional smooth projective quadric by a smooth connected projective variety $X$. Since we have the $6$-dimensional quadric $\...
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1answer
46 views

The associated Schubert variety of a flag of subspaces of a vector space.

Let $V$ be a vector space and $W_1 \subsetneq W_2 \subsetneq ... \subsetneq W_\ell \subsetneq V $ a flag of subspaces. The associated Schubert variety is defined as : $ \Omega ( W_{ \bullet } ) = \{ \...
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62 views

Metric transformation

I'm trying to understand the paper by Hitchin called: ''Polygons and gravitons". I'm stuck at page 471. At this point, he does some computations and obtains a metric: $$ \gamma dz d\bar{z}+\gamma^{...
13
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2answers
986 views

Conditions such that taking global sections of line bundles commutes with tensor product?

Let us work with projective algebraic varieties over $k= \mathbb{C}$. If necessary we can also assume smoothness of the varieties. Of course it is not in general true that given two line bundles $L, ...
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0answers
30 views

Alternating form from hermitian product

Let $V_\mathbb{C}$ be a complex vector space and $h$ an hermitian product on it. In particular let $\{e_1,\dots,e_n\}$ be a base of $V_\mathbb{C}$, then $h:=\sum_{i,j=1}^nh_{ij}dz_id\overline{z}_j$ ...
4
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2answers
187 views

complex submanifolds in complex euclidean space

Assume all manifolds are without boundaries. In Euclidean space $\mathbb{R}^n$, there are many submanifolds (Whitney Embedding Theorem). In complex Euclidean space $\mathbb{C}^n$, are there any ...
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33 views

dimension of $\Omega^1 \left( X \right)$ the space of holomorphic $1$-forms.

I'm reading $1$-forms on "Rick Miranda, Algebraic Curves and Riemann surfaces". According to the book's notation: Let $X$ be a compact Riemann surface of genus $g$ and $\Omega^1 \left( X \right)$ be ...
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52 views

Chern classes of a double cover

Let $X$ be a compact complex surface and let $D$ be a double cover of $X$. Let $\pi:D\to X$ be the double cover map (a 2:1) map. If $E$ is a vector bundle (rank at least 2) on $X$ with $c_1(E) = A$ ...
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1answer
60 views

Does there exist a holomorphic structure on $S^6$?

Does the six-sphere $S^6$ admits any holomorphic structure? Can someone tell me if there is any development in research of holomorphic structures on $S^6$ as we know $S^6$ has an almost complex ...
4
votes
1answer
356 views

Why the moduli space of complex structure in a compact complex manifold is of finite dimension

I admit the statement in the title might be much too unclear. I just heard from my teacher that we can form a finite dimension moduli space of all the complex structure in a compact complex manifold. ...
3
votes
2answers
368 views

Equivalent definitions of Kähler metric

Two different ways to define a Kähler metric on a complex manifold are: 1) The fundamental form $\omega = g(J\cdot,\cdot)$ is closed, ie, $d\omega=0$; 2) The complex structure $J$ is parallel with ...
3
votes
1answer
49 views

When does a potential give a Kähler metric?

Suppose that I have a complex manifold $X$ with a symplectic form $\omega$ and a continuous function $f:X\to\Bbb R$ such that $\omega=2i\partial\bar{\partial}f$. Does that imply that $X$ is Kähler? If ...
10
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1answer
87 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 $p \in M$. ...
15
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2answers
225 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 ...
0
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1answer
24 views

What's the definition of being zero to infinite order at some point?

I'm reading a paper and it says The differential form $\psi$ is zero to infinite order at all points of $F$.That is,all coefficients of $\psi$ are zero to infinite order at points of $F$.(Here $\psi$...
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0answers
18 views

Reference for complex curve theory

Recently, I started to study complex curve theory with textbook written by Clemens. The thing is, I think I need a little more references for this study. I think my background is not enough. What I ...
2
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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 ($...
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0answers
56 views

Roots of canonical line bundles that are not necessarily square roots

I understand that holomorphic square roots of the canonical line bundle of a compact Riemann surface always exist, and that there are $2^{2g}$ choices of such a root. But what about further roots? ...
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3answers
102 views

Almost complex manifolds are orientable

I want to verify the fact that every almost complex manifold is orientable. By definition, an almost complex manifold is an even-dimensional smooth manifold $M^{2n}$ with a complex structure, ...
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34 views

Examples of the primitive decomposition of a form

Let $(X,\omega)$ be a Kahler manifold of dimension $n$, let $L = \omega \wedge -$ and let $\Lambda$ denote its adjoint. There is a unique ''primitive decomposition'' of a $k$-form $u$ which looks like ...
0
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1answer
17 views

Show that $\Omega^1(X) \to \operatorname{Rh}^1(X)$ is injective.

Problem: Let $X$ be a compact Riemann surface. Show that $$\Omega^1(X) \to \operatorname{Rh}^1(X) = \frac{\ker (d : \mathcal{E}^{(1)}(X) \to \mathcal{E}^{(2)}(X))}{\operatorname{im}( d: \mathcal{E}(X) ...
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0answers
19 views

$\mathcal{M}_1$ and conformal structures on $\mathbb{T}$

I'm kind of lost trying to understand both what is usually denoted by $\mathcal{M}_1$ and the moduli space of conformal/complex structures on the 2-torus $\mathbb{T}$ (closed orientable surface of ...
1
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1answer
50 views

Exact Sequence of Line Bundles on $\mathbb{P}^{2}$

I'm considering an example in the great book "Mirror Symmetry" where they consider the exact sequence of line bundles $\mathcal{O}(-2) \to \mathcal{O}(-1) \oplus \mathcal{O}(-1) \to \mathcal{O}$, ...
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0answers
43 views

Connection of $\mathcal{O}(n)$ on a toric manifold

The holomorphic line bundle $\mathcal{O}_X(1)$ over a toric manifold $X$, admits a hermitian connection, $A^{(1)}$, whose $U(1)$ gauge transformation in a local patch of the base space is $$ A^{(1)}...
3
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0answers
26 views

Griffiths and Harris $\mu=\mathcal{H}(\mu)+dd^*G(\mu)$

Griffith and Harris state on page $116$ that for a closed form $\mu$ on a Kahler manifold of type $(p,q)$ we have $$\mu=\mathcal{H}(\mu)+dd^*G(\mu)$$ Here $$\mathcal{H}:\Omega^{p,q}(M)\to\mathcal{H}^{...
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23 views

Identification of the holomorphic tangent space with the real tangent space

I'm reading Griffiths and Harris and just want to check I'm interpreting a passage correctly. Let $M$ be a $n$ dimensional complex manifold. We define $T_p^\mathbb{R}M$ as the tangent space of the ...
4
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1answer
49 views

Finitely Many genus-g Quotients of Compact Riemann Surface

I hear there is a semi-famous theorem from my advisor, but he didn't know the name and I was unable to find it online. Does anybody know of the following? Let $S$ be a compact Riemann surface. Then ...
6
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1answer
88 views

The torus as a projective plane curve $x^3+y^3+z^3=0$

The homogeneous polynomial $F(x,y,z)=x^3+y^3+z^3$ clearly defines a smooth projective curve $X\subset\mathbb{P}^2$. It is easy to see that $\pi:X\rightarrow\mathbb{P}^1$ defined by $$\pi([x:y:z])=[x:...
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1answer
55 views

Picard rank of K3s — How can it be small?

I must be missing something. How can the Picard rank of a compact Kahler manifold drop below its Hodge number $H^{1,1}(X)$? For simplicity, let $X$ be a K3 surface. After all, we have the standard ...
3
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2answers
65 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}...
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2answers
34 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 ...
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
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0answers
39 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 ...