# Tagged Questions

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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### Question about a definition of non-compact Calabi-Yau manifold

The specific definition I'm referring to is the following. Definition $(X, J, \omega, \Omega)$ is a Calabi-Yau manifold if $g(\cdot, \cdot)= \omega(\cdot, J \cdot)$ and $\Omega$ is a nowhere ...
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### 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 ...
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### 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 ...
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### 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 $TM$ ...
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### (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 ...
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### 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 ...
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### Every holomorphic map between Kähler manifolds is harmonic

I was reading the Wikipedia article on harmonic maps and saw the following statement in the 'examples' section: Every holomorphic map between Kähler manifolds is harmonic. I am not that familiar ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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'...
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### 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 $$\mathbf{G}_{k,n} \to \mathbf{G}_\mathbb{F}(k,n)$$ ...
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### Cross product in complex vector spaces

When inner product is defined in complex vector space, conjugation is performed on one of the vectors. What about is the cross product of two complex 3D vectors? I suppose that one possible ...
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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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### 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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### 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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### 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 $\sigma,\tau\in H^1(X,\mathbb{C})$?...
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### 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. ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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. ...
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### 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 ...
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### 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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### 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 $p \in M$. ...
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### 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 ...
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### 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 $\psi$...