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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5
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3answers
409 views

Kähler form is harmonic

Let $M$ be a Kähler manifold with fundamental form $\omega(X,Y) = h(JX, Y)$. I am trying to show that $\omega$ is harmonic. The Kähler condition implies that $\omega$ is closed with respect to $d$, so ...
4
votes
3answers
462 views

A complex algebraic variety which is connected in the usual topology

Hartshorne wrote in his book's Appendix B that it can be easily proved that a complex algebraic variety is connected in the usual topology if and only if it is connected in Zariski topology. How can ...
7
votes
1answer
654 views

Meaning of holomorphic Euler characteristics?

I wonder what holomorphic Euler characteristic $\chi(\mathcal{O}_X)$ of a variety represents. For example, I have seen someone fix $\chi(\mathcal{O}_C)=n$ for a complex curve $C$. What does this mean ...
5
votes
1answer
123 views

Hilbert schemes of one point.

Let $X$ be a smooth complex manifold. Then the Hilbert scheme of one point is canonically isomorphic to $X$ $$ \mathrm{Hilb}^1(X)\cong X. $$ Is this still true for general scheme $X$ (not necessarily ...
0
votes
1answer
74 views

A rational curve on projective complex surfaces.

Let $X$ be a smooth projective complex surface and $C$ a smooth curve on it. How can one conclude that $(C,C)=-1$ provided that $(C,C)+(K_X,C)<0$? By adjunction formula, $(C,C)+(K_X,C)=2g(C)-2$, ...
5
votes
1answer
141 views

Algebraic classes in Hodge decomposition.

Let $X$ be a Kähler manifold. The torsion free part of the singular cohomology $H^n(X,\mathbb{C})$ has a Hodge decomposition $$ H^n(X,\mathbb{C})=\bigoplus_{p+q=n}H^{p,q}(X), $$ where $H^{p,q}(X)$ ...
1
vote
1answer
154 views

A question on elliptic fibration of K3 surfaces.

Let $X$ be a K3 surface and $L$ be a non-trivial nef line bundle with self-intersection $(L,L)=0$. Then it is known that $h^0(X,L)=2$. How can one prove that the map $\phi_{|L|}\rightarrow ...
3
votes
2answers
177 views

tangent vector on a complex manifold

Let $x_1,x_2, \ldots, x_n$ be local coordinates on a manifolds $M$. One can interpret $\frac{\partial}{\partial x_i}(p)$ as a tangent vetor to a curve with constant $x_j$ (where $j \neq i$). What is ...
1
vote
1answer
130 views

Complex Numbers geometry question

on each edge of a quadrilateral ABCD you build a square such that the points H, G, F, E are the centers of these squares (the intersection of the diagonals). I need to use complex numbers to prove ...
2
votes
1answer
62 views

Is $\Delta^{\bar{\partial}}f = \Delta f$ for $f \in C^{\infty}(M)$?

Let $M$ be a complex manifold and $\Delta^{\bar \partial} = \bar\partial^* \bar\partial + \bar\partial\bar\partial^*$ the complex laplacian. Is it true that $\Delta^{\bar\partial} f = \Delta f$ (the ...
8
votes
1answer
167 views

Almost complex structure on $\mathbb CP^2 \mathbin\# \mathbb CP^2$

Why doesn't $\mathbb CP^2 \mathbin\# \mathbb CP^2$ admit an almost complex structure?
7
votes
2answers
366 views

Why is $x^2$ +$ y^2$ = 1, where $x$ and $ y$ are complex numbers, a sphere?

I've heard $x^2 + y^2$ = 1, where $x$, $y$ are complex numbers, is supposed to be a sphere with two points removed, or also a cylinder. The problem is I've been trying to wrap my head around this for ...
2
votes
0answers
69 views

Complex structure on the product of two complex Kähler manifolds

I have the following question: Let $(M,J_{M}, \omega_{M})$ and $(N,J_{N}, \omega_{N})$ be two Kähler manifolds. I consider $M \times N$. Can I introduce on $M \times N$ a complex structure ? I think ...
2
votes
1answer
157 views

A question on a linear system on a complex surface.

For a complex surface $X$ with a line bundle $L$, the base locus Bs$|L|$ consists of $0$-dimensional and $1$-dimensional components. The fixed part of $|L|$ is the $1$-dimensional locus and of Bs$|L|$ ...
0
votes
1answer
151 views

An embedding of a projective variety.

I came across with a statement Let $X\rightarrow \mathbb{P}^n$ be a map defined by a linear system $|L|$ for some line bundle $L$ on $X$. It is embeding if $H^0(\mathbb{P}^n,O(1))\rightarrow ...
0
votes
1answer
238 views

cotangent bundle splits as a product?

Let $M$ and $N$ be two Riemannian manifolds with Riemannian metrics $g$, $h$ respectively. We consider the product $M \times N$ with metric $g \oplus h$. By the metric we get an isomorphism of bundles ...
3
votes
2answers
187 views

Lines on algebraic surfaces and duality

In the following, all varieties will be algebraic over $\mathbb{C}$. I have some general problems with concepts like the "space of lines in $\mathbb{P}^5$", "space of lines on a surface in some ...
4
votes
1answer
660 views

$n$-sheeted branched covering

Michael Artin's algebra let $f(x,y)$ be an irreducible polynomial in $\mathbb{C}[x,y]$ which has degree $ n>0$ in the variable $y$. The Riemann surface of $f(x,y)$ is an $n$-sheeted branched ...
10
votes
2answers
785 views

A question on Hermitian metric on complex manifold.

We say that a Riemannian metric $g$ on a complex manifold $(X,I)$ is Hermitian if $$ g(x,y)=g(Ix,Iy) $$ for any $x,y\in \Gamma(X,TX)$. Here we consider $X$ as a real even dimensional manifold with ...
11
votes
1answer
382 views

What is the line bundle $\mathcal{O}_{X}(k)$ intuitively?

I am always confused about how to understand the line bundle $\mathcal{O}_{X}(k)$ on a projective scheme $X=\mathrm{Proj}(\oplus_{n=0}^\infty A_{n})$. Of course this is by definition the ...
1
vote
1answer
137 views

Kähler–Einstein metric on Calabi–Yau manifold

I am reading "Complex geometry" by D. Huybrecht. On p.223 the books says that "If $c_{1}(X)=0$, e.g. if the canonical bundle $K_{X}$ is trivial, and $g$ is Kähler–Einstein metric, then ...
3
votes
1answer
178 views

Finite subgroups of $\operatorname{Aut}(\mathbb{P}^1)$

I would like to know all finite subgroups of $\operatorname{Aut}(\mathbb{P}^1)$. I am aware that any automorphism of $\mathbb{P}^1$ is given by a Möbius transformation $$ z\mapsto\frac{az+b}{cz+d} ...
1
vote
2answers
336 views

Circle in the complex plane

Show analytically (finding the centre and radius) that $z(t)=\frac{1}{(1-i)^{-1}-t}=\frac{2}{1+i-2t}$ where $z(t)\in C $, that $z(t)$ traces out a circle in the complex plane as $t$ is varied.
3
votes
1answer
135 views

Does pullback and d and dbar commute?

If $M$ is a complex manifold. we can write $d = \partial + \overline{\partial}$. Does pullback commute with either $\partial$ or $\overline{\partial}$?
6
votes
2answers
547 views

When does a line bundle have a meromorphic section?

Let $X$ be a scheme and $D$ be a Cartier divisor on $X$. Then $D$ determines a line bundle $\mathcal{O}(D)$ on $X$. Under which condition, is the converse true? That is, when does a line bundle come ...
5
votes
1answer
333 views

Existence of Complex Structures on Complex Vector Bundles

Let $E$ be a real vector bundle on a smooth manifold $X$. Let $J : E \to E$ be a vector bundle morphism (i.e. $\pi \circ J = \pi$, where $\pi : E \to X$ is the projection map) with $J^2 = ...
2
votes
1answer
311 views

Definition of tautological section.

Reading Barth, Peters: Compact complex surfaces, i stumbled across the following: Let $Y$ be an algebraic surface over $k =\mathbb{C}$, and $\mathcal{L}$ an invertible sheaf on $Y$. Denote by $p: L ...
3
votes
1answer
140 views

real part of a holomorphic function from a PDE

I have some problem that I can't figure out myself. Hope that someone can help me out. The problem is: Let $f : U \to \mathbb{R}$ be some real function on a simply-connected domain $U \subset ...
2
votes
0answers
98 views

On morphisms on varieties: 1-1 and projective implies iso?

For my thesis i have to learn about double covers of surfaces. All varieties in this question are projective over $\mathbb{C}$ if necessary. Let $\phi: V \rightarrow W$ be a double cover of surfaces ...
1
vote
1answer
65 views

Example about hyperbolicity.

$\def\abs#1{\left|#1\right|}$I would like to understand this example: Why is the following set a hyperbolic manifold? $X=\{[1:z:w]\in \mathbb{CP}_2\mid0<\abs z< 1, \abs w < ...
3
votes
0answers
234 views

Anti-holomorphic involution of $\mathbb{P}^1$

I wonder if anti-holomorphic involution of $\mathbb{P}^1$ is, up to change of coordinate, given by either $$ z\mapsto \overline{z}, \ \ \,z\mapsto -\overline{z}, \ \ \ or \ \ \ z\mapsto ...
6
votes
1answer
500 views

How should one think of non-projective compact manifolds and Moishezon manifolds?

A Moishezon manifold $M$ is a compact connected complex manifold such that the field of meromorphic functions on $M$ has transcendence degree equal to the complex dimension of $M$. There exists a ...
20
votes
2answers
846 views

When is the sheaf corresponding to a vector bundle on a smooth manifold coherent?

In algebraic and analytic geometry, vector bundles are usually interpreted as locally free sheaves of modules (over the structure sheaves). They are in particular examples of quasi-coherent sheaves. ...
0
votes
1answer
33 views

Action of $D_{2n}$ on $\mathbb{P}^1$

Let $D_{n}=\langle a,b \ | a^n=b^2=abab=e\rangle$ be a dihedral group. Assume that $b$ acts on $\mathbb{P}^1$ by $z\mapsto \overline{z}$ where $z$ is an inhomogeneous coordinate of $\mathbb{P}^1$. ...
2
votes
1answer
50 views

A map $\mathbb{C}^2\rightarrow \mathbb{C}^2$ that preserves a holomorphic 2-form.

Let $\mathbb{C}^2\rightarrow \mathbb{C}^2$ be a map. If $f$ preserves a non-zero holomorphic 2-form, is $f$ a holomorphic map?
1
vote
1answer
41 views

Holomorphic action of $C_{2}\times C_{2}$ on $\mathbb{P}^1$

I am looking for a holomorphic action of $C_{2}\times C_{2}$ on $\mathbb{P}^1$. Is it true that there is a unique effective action given by $$ a:z\rightarrow -z, \ \ \ \ b:z\mapsto 1/z $$ up to ...
2
votes
1answer
1k views

What is a Holomorphic Vector Field?

On a smooth manifold $M$, a smooth vector field is an element of $\Gamma(M, TM)$ which is the space of all smooth sections of the bundle $TM \to M$. If $M$ is a complex manifold, then we have the ...
3
votes
1answer
297 views

Quotient Riemann surfaces

Let $\mathbb{H}$ be an upper half plane (this is a Riemann surface), then $PSL(2,\mathbb{Z})$ acts on $\mathbb{H}$ and it is well-know that $$ \mathbb{H}/PSL(2,\mathbb{Z})\cong \mathbb{C} $$ is again ...
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vote
0answers
92 views

Complexification of a complex bundle.

Suppose that $E$ is a complex bundle already. This is to say that is can be viewed as a real bundle with an almost complex structure $j$. Then Taubes assert $E$ sits inside its complexification ...
10
votes
1answer
231 views

When is a $k$-form a $(p, q)$-form?

Let $X$ be a complex manifold and denote the space of all $(p, q)$-forms on $X$ by $\mathcal{E}^{p,q}(X)$. Forgetting about the complex structure, we can consider the real differential $k$-forms on ...
2
votes
1answer
298 views

Weitzenböck Identities

The Wikipedia page for Weitzenböck identities is explicitly example based. I am looking for a reference which takes a more rigorous approach (as well as a discussion of the Bochner technique). In ...
6
votes
1answer
161 views

Holomorphic structures on associated bundles

Suppose $M$ is Kähler. Let $P \to M$ be the principal $U(n)$-frame bundle of $M$. Let $(\pi,V)$ be a finite dimensional unitary representation of $U(n)$ and let $E = P \times_\pi V$ be the ...
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vote
3answers
434 views

Graphing the complex function

I'm looking for some software that my help me to graph some complex functions on unit circle. I.e. let say if I have $\ f(z)=1/(1-z)$ I want to see to give an input an image with unit circle and want ...
1
vote
1answer
190 views

Complex Space : Why unit disc?

This may sound a newbie question anyway I'm yet new to this area, In complex space for simplicity the properties of the functions on curves are sometimes considered on the unit circle on complex ...
3
votes
0answers
72 views

Complex structure of hyper-Kähler manifold

Let $X$ be a hyper-Kähler manifold with complex structure $I$ and a metric $g$. It is known that there are two other complex structures $J,K$ on $X$ that generate $S^2$ of possible complex structures ...
2
votes
1answer
234 views

Is there a geometric projection for every complex function?

I was wondering about the best way to visualize complex functions. As they're $$ {\mathbb R}^2 \rightarrow {\mathbb R}^2\ ,$$ I think best way are complex plane image/grid transforms like they used in ...
3
votes
3answers
157 views

Definition of vector bundle

Everywhere i see definition of vector bundle as triple $(E, p, B)$, $B$ and $E$ are manifold and local trivialization condition holds. For example see the definition here. . Local trivialization ...
6
votes
4answers
2k views

Number of points at which a tangent touches a curve

My teacher told me that we are mistaken coming out of school that tangent tocuhes a point at one point. According to him, a tangent is just a special type of secant where two points share the same ...
3
votes
0answers
60 views

Complex atlas and Zorn's lemma, maximal complex atlas [duplicate]

Possible Duplicate: Is Zorn's lemma required to prove the existence of a maximal atlas on a manifold? Could any one explain me in detail how to prove the following statements in ...
2
votes
0answers
399 views

Trivial canonical bundle

Let $X$ be a complex compact Kähler minimal surface with zero algebraic dimension and $H^2(X,\mathcal{O}_X) \ne 0$. We know that according to Enriques–Kodaira classification, $X$ is either a torus or ...