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

1
vote
0answers
21 views

Compute total curvature of a curve

Let $u:[0,2\pi]\rightarrow \mathbb{C}$, $\theta\mapsto e^{i\theta}$, be the unit circle. Let $f:\mathbb{C}^*\rightarrow \mathbb{C}^*$ be a holomorphic function such that $\frac 1 2 <|df|<2$ in ...
2
votes
0answers
30 views

Question about definition of non-compact Calabi-Yau manifolds

Using the following definition: Definition $(X, J, \omega, \Omega)$ is a Calabi-Yau manifold if $g(\cdot, \cdot)= \omega(\cdot, J \cdot)$ and $\Omega$ is a nowhere vanishing holomorphic $(n,0)$-form ...
1
vote
1answer
26 views

Integral of $\omega\wedge\overline{\omega}$ on Riemann surface

Let $X$ be a Riemann surface of genus $g$ and $\omega$ a meromorphic 1-form on it. I've read that if $\omega$ has just a simple pole in $x\in X$ (and is holomorphic on $X\setminus\{x\}$) then the ...
2
votes
0answers
17 views

Coordinates for a neighborhood of a totally real submanifold

Motivation: I'm interested in which submanifolds of (half dimension) a complex manifold can be expressed locally in coordinates as the real part of the complex coordinates, and was wondering if the ...
3
votes
2answers
27 views

Holomorphic vector bundles on almost complex manifolds

Let $M$ be a real manifold with complex structure $J$, making $M$ into an almost complex manifold. I know that the complexification $T_{\textbf{C}}M = TM\otimes \textbf{C}$ of the tangent bundle $TM$ ...
0
votes
0answers
49 views

(Reference Request) Proofs for basic facts about regular functions on algebraic sets.

I am writing an assignment about algebraic and analytic sets in $\mathbb{C}^n$ and, when searching for references, came across the book Algebraic Geometry III. The book is a bit out of my depth, yet ...
3
votes
0answers
59 views

Ricci curvature of sum of metrics

Is there an estimate for Ric$(g+h)$ in terms of Ric$(g)$ and Ric$(h)$, where $g,h$ are smooth Riemannian metrics? More specifically can one say that the eigenvalues will decrease (resp. increase) if ...
2
votes
1answer
69 views

How does a complex algebraic variety know about its analytic topology?

This question has two parts. The first is a reference request regarding a result I assume is standard, and the second is a soft question asking for philosophy and intuition about an issue the first ...
8
votes
1answer
47 views

Necessary and sufficient condition for branch points on a Riemann surface.

I've been reading out of a book by V.B. Alekseev about Abel's theorem on the insolubility of the quintic, and I'm a bit troubled by its presentation on Riemann surfaces. My question is as follows: ...
1
vote
1answer
29 views

The dimension of a projectivised, complexified tangent bundle

I am looking at page 15 here, starting from the line 'Let us suppose we are given a...' The Zoll projective structure is not relevant to this question, so don't worry about that. We take, in this ...
1
vote
1answer
72 views

What's wrong with my “proof” that branch cuts are not arbitrary?

Recently, I've been thinking about contour integrals around branch cuts in the complex plane. Now clearly the choice of contour is arbitrary, so long as you don't deform past any poles or cuts, but I'...
0
votes
1answer
48 views

How can I equip the universal vector bundle over $\mathbf{G}(k,n)$ with a connection?

When constructing the grassmannians $\mathbf{G}_\mathbb{F}(k,n)$ for $\mathbb{F} = \{\mathbb{R},\mathbb{C} \}$ there is a natural vector bundle $$ \mathbf{G}_{k,n} \to \mathbf{G}_\mathbb{F}(k,n) $$ ...
0
votes
1answer
19 views

Explicit formulation of hermitian form and corresponding alternating form

I can't understand the following basic thing: Let $X$ be a complex manifold of dimension $n$ and $T_xX$ its tangent space in $x\in X$. Then on $T_xX$ we can define a hermitian form $\sum_{i=1}^ndz_id\...
0
votes
1answer
38 views

Base point free linear system

Let $X$ be a (compact) Riemann surface. Let $D$ be a divisor. In Rick Miranda's book on Riemann surfaces, on page 160, there is a bijection between Base-point-free linear systems of dimension $n$ on ...
1
vote
1answer
27 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
vote
1answer
29 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 ...
1
vote
1answer
58 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 ...
3
votes
1answer
34 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. ...
3
votes
1answer
69 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 $\sigma,\tau\in H^1(X,\mathbb{C})$?...
1
vote
1answer
26 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 \...
2
votes
2answers
114 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{...
0
votes
0answers
25 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}...
2
votes
0answers
46 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 ...
2
votes
1answer
66 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 ...
2
votes
0answers
32 views

Introduction to flag manifolds

What is a good self-contained introduction to the geometry of complex flag manifolds?
5
votes
1answer
66 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
vote
0answers
41 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?
1
vote
1answer
67 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. ...
4
votes
1answer
64 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 \...
1
vote
1answer
33 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 ...
1
vote
0answers
41 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 $\...
1
vote
0answers
33 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 $h:=\sum_{i,j=1}^nh_{ij}dz_id\overline{z}_j$ ...
0
votes
0answers
36 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 ...
1
vote
0answers
62 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 $c_1(E) = A$ ...
0
votes
1answer
26 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 $\psi$...
1
vote
0answers
20 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
votes
1answer
66 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 ...
0
votes
0answers
61 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? ...
1
vote
3answers
108 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, ...
0
votes
1answer
18 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) ...
0
votes
0answers
21 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 ...
4
votes
0answers
36 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 ...
1
vote
1answer
52 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 $\mathcal{O}(-2) \to \mathcal{O}(-1) \oplus \mathcal{O}(-1) \to \mathcal{O}$, ...
4
votes
1answer
39 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}^{...
2
votes
0answers
25 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 ...
6
votes
1answer
94 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)}...
1
vote
1answer
57 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
votes
1answer
50 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 ...
3
votes
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\...