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

3
votes
1answer
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_{(...
4
votes
0answers
42 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
356 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
46 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
55 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
21 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
32 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
78 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
23 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
79 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
224 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
26 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
25 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
vote
0answers
24 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
78 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
68 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
30 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
35 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
48 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
52 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
38 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
39 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
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 $p \in M$. ...
1
vote
0answers
17 views

Cohomology of $\mathcal{O}^*$ and projection map

Suppose $X$ is a complex manifold and $T$ a complex space (or complex manifold maybe) and let $\pi:T\times X \rightarrow T$ denote the projection. What are sufficient conditions on $X$ that make $$H^2(...
0
votes
1answer
87 views

Wells 'Differential Analysis on Complex Manifolds' page 127

How does the first equation for $Qu(x)$on this page follow from the defining equation (3.10) on the previous page. This is from the section on pseudodifferential operators in chapter 4. I'm starting ...
1
vote
2answers
70 views

Least value of $|z-w|$

On an argand diagram, sketch the locus representing complex numbers $z$ satisfying $|z+i|=1$ and the locus representing complex numbers $w$ satisfying arg$(w-2)=\frac{3}{4}\pi$ find the least value of ...
0
votes
1answer
20 views

Let $ f(w)=\frac{w(1-i)-(i-1)}{w-1} $, where $w$ is the left hand plane. What is the image of this map?

Let $$ f(w)=\frac{w(1-i)-(i-1)}{w-1} $$, where $w$ is the left hand plane. What is the image of this map? The answer should be $|z|^2<2$ if I did everything before correctly. This is a show ...
1
vote
2answers
24 views

Solving inequality in complex plane

I have to graphically represent the following subset in the complex plane being z a complex number: $A={1<|z|<2}$ However after trying to do it on WolframAlpha it says that "inequalities are ...
10
votes
1answer
89 views

How to interpret the cotangent bundle of a complex manifold?

Let $X$ be a complex manifold. I am not sure what people mean when they talk about the cotangent bundle $T^*X$ of $X$. I have two interpretations: At each point $x\in X$, $T_x^*X$ is the complex ...
0
votes
0answers
45 views

Is the complementary of a 2 codimensional set in a complex space simply connected?

It is well-known that if X is a complex manifold and $A \subset X$ an analytic set of codimension at least $2$ then $\pi_1(X)=\pi_1(X\setminus A)$. Can we generalize this equality when $X$ is a ...
2
votes
0answers
30 views

Verlinde Formula in geometric quantization?

I think I have a fair grasp on the $\rm{SU}(2)$ Verlinde Formula from the algebraic geometry perspective. I'm hoping to understand better how exactly this relates to the geometric quantization of a ...
0
votes
0answers
21 views

Relation between moduli space of flat connections and moduli space of bundles on curves

I would like to let $X$ be a genus $g$ curve, with $M_{g}(r,d)$ the moduli space of bundles on the curve such that $(r,d)=1$. Alternatively, we can pick a group $G$, consider principal $G$-bundles $...
2
votes
0answers
34 views

Dimension of Moduli Space of Bundles on Curves

I think I'm getting conflicting results for the dimension of the moduli space of rank $r$, degree $d$ stable vector bundles on a curve $X$ of genus $g$. I'm happy to look only at the nice case of $r$ ...
3
votes
1answer
58 views

Topological Degree of Map of Effective Divisors

Let $\Sigma$ be a compact Riemann surface. Is it possible to show that the map $$f:\text{Div}(\Sigma)^d_+\to \text{Div}(\Sigma)^{qd}_+$$ Given by $\sum_{i} n_ix_i\mapsto \sum_{i} qn_ix_i$, has ...
1
vote
1answer
75 views

Can we prove uniformization by solving the Yamabe problem directly?

One version of the uniformization theorem says that a simply connected complex manifold is biholomorphic to either the unit disc, $\Bbb C$, or $\Bbb{CP}^1$. The proof of this goes through potential ...
0
votes
1answer
14 views

Given two hermitian matrices of signature (2,1) there exists a Cayley transform between them?

Given a matrix $A\in M_{k\times l}(\mathbb{C})$ we define the hermitian transpose of $A$ as the matrix $A^*=\overline{A}^t\in M_{l\times k}(\mathbb{C})$. We say a matrix $H\in M_k(\mathbb{C})$ is ...
0
votes
0answers
8 views

Proper surjective holomorphic function is a topological bundle

If I have $u:M\rightarrow D\setminus 0$, $M\subseteq (\mathbb C^2,0)$ compact, $\mathbb C^2 = \mathbb C \times \mathbb C$ and $D$ the Poincare disk, $u$ is holomorphic, surjective, proper and every ...
1
vote
0answers
30 views

Unifying Transformations in Complex 3-Space

I am currently researching the vector space $\mathbb{C}^{3}$ and I was wondering if it is possible to generate a scheme of unifying the rigid transformations in $\mathbb{C}^{3}$. I know that in the ...
4
votes
0answers
47 views

Volume of a hypersurface in flat families?

Suppose that $X_t$ is the vanishing of $x^2 + y^2 + t z^2 = 0$ in $\mathbb{CP}^2$. For $t \not = 0$, this is a submanifold, and it inherits a Riemannian metric from the Fubini-Studi metric on $\mathbb{...
0
votes
0answers
38 views

Locally nilpotent vector field is complete

Suppose $X$ is a smooth affine algebraic variety over $\mathbb{C}$ and let $V$ be an algebraic vector field (i.e. an algebraic section of the tangent bundle). I am trying to show that if the $V$ is ...
3
votes
0answers
42 views

Equation of the form $\overline{\partial}_{J}f=g$ made holomorphic

In section 2.6 of the book "Holomorphic curves in symplectic geometry" by Audin and Lafontaine there is explained when one can transform a perturbed holomorphic curve in a holomorphic curve. I tried ...
0
votes
1answer
45 views

Question about the proof of $\Delta_d = 2\Delta_{\bar{\partial}} = 2\Delta_{\partial}$ in Principles of Algebraic Geometry by Griffiths and Harris.

On page $115$ of Principles of Algebraic Geometry by Griffiths and Harris, in the proof of $\Delta_d = 2\Delta_{\bar{\partial}} = 2\Delta_{\partial}$, they state that $$\sqrt{-1}(\partial\bar{\...
5
votes
0answers
48 views

Intuition behind definition of Stable Bundles?

To my understanding, given a rank $r$ and degree $d$, we can fix a $C^{\infty}$ vector bundle $\mathcal{V}$ over some curve $\Sigma$ of genus $g$. We then want to study the moduli space $M_{g}(r,d)$ ...