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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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 ...
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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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25 views

Introduction to flag manifolds

What is a good self-contained introduction to the geometry of complex flag manifolds?
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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 ...
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37 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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57 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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51 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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24 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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37 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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61 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^{...
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29 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 $h:=\sum_{i,j=1}^nh_{ij}dz_id\overline{z}_j$ ...
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32 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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47 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
23 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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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 ...
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57 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 ...
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54 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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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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16 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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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 ...
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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 ...
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46 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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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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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 ...
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42 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)}...
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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 ...
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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 ...
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63 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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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\...
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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 ...
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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 ...
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24 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 ...
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22 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\...
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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 ...
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17 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 $$ 0 \leq \arg (z+1) \leq \frac{2\pi}{3} $$...
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90 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 \...
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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 $f: X \dashrightarrow Y$ ...
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33 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 ...
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61 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 $ds$ ...
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1answer
31 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 ...
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28 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 ...
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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 ...
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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 ...
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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 ...
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66 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 ...
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121 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 $\...
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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 \...
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52 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_{(...
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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 $G$, ...