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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3
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121 views

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 ...
4
votes
1answer
134 views

Why holomorphic injection on $C^n$must be biholomorphic?

This result is certainly right in the 1-dim'l case. But I don't know how to show the general case by induction. Can anyone tell me the detail please?
11
votes
2answers
246 views

Is tautological bundle $\mathcal{O}(1)$ or $\mathcal{O}(-1)$?

I always confused by whether tautological bundle is $\mathcal{O}(1)$ or $\mathcal{O}(-1)$, and definitions from different sources tangled in my brain. However, I thought this might not be simply a ...
0
votes
1answer
39 views

Meaning of CR-Automorphism

What is the meaning of the CR-Automorphism and CR-Manifold? I tried to find the definition from the web. Is it Continuous Real ....? Thanks.
2
votes
1answer
102 views

Are there many almost complex structures on a (complex) manifold?

I guess one can have many almost complex structures on a manifold, can someone give me an example? How about when the manifold is complex? is the almost complex structure induced by the complex ...
2
votes
0answers
81 views

Miranda book Proposition 3.10 pag. 40

Let $F$ and $G$ be two holomorphic maps between Riemann surfaces $X$ and $Y$. If $F=G$ on a subset $S$ of $X$ with a limit point in $X$, then $F=G$. Choose two charts $(U_\alpha,\phi_\alpha)$ and ...
2
votes
0answers
43 views

geometry of Automorphism groups

It is proved that Aut group of a compact complex manifold is a lie transformation group. How do we show that a automorphism group of a tubular geometry is a lie group?
1
vote
1answer
87 views

Does the tangent bundle of an almost complex manifold split into the direct sum of $n$ two-dimensional $J$-invariant subbundles?

Is it true that for a manifold (of dimension $2n$) with an almost complex structure $J$, the tangent bundle splits as sum of $n$ $2$-dimensional invariant sub bundles? At least is it true that for a ...
5
votes
1answer
108 views

Can I check smoothness after a base-change

Let $X\to S$ be a flat morphism of noetherian schemes. I know that I can check smoothness on the geometric fibers to see whether $X\to S$ is smooth. Let $T\to S$ be a surjective morphism. Under what ...
3
votes
1answer
121 views

Deformation of family of divisors

Given a family of complex compact surfaces $X \to B$ over a smooth curve $B$ and a one-dimensional family of effective Cartier divisors $\{D_t\}$ in the central fiber $X_0$. For simplicity, let's just ...
4
votes
0answers
108 views

Fundamental group of a space under a group action

The short version: Why does a Borcea-Voisin threefold has trivial fundamental group? Well, what is a Borcea-Voisin threefold? These are named after Borcea and Voisin, who introduced a method of ...
2
votes
1answer
114 views

Isomorphism between Picard group and a Sheaf cohomology group

I would like to know how to prove that : $ \mathrm{Pic} ( X ) \simeq H^1 ( X , \mathcal{O}_{X}^* ) $. I specially want to know how to prove that $ \mathrm{Pic} ( X ) \to H^1 ( X , \mathcal{O}_{X}^* ) ...
7
votes
2answers
211 views

Equations for double etale covers of the hyperelliptic curve $y^2 = x^5+1$

Let $X$ be the (smooth projective model) of the hyperelliptic curve $y^2=x^5+1$ over $\mathbf C$. Can we "easily" write down equations for all double unramified covers of $X$? Topologically, these ...
6
votes
1answer
164 views

Questions on Chern characters.

Let $X$ be a complex manifold and $\mathcal{O}_p$ a skyscraper sheaf. How can one compute the Chern character $ch(\mathcal{O}_p)$? For vecoto bundles, we have $ch(V)\cup ch(W)=ch(V\otimes W)$, but ...
9
votes
1answer
112 views

When do local isomorphisms extend to global isomorphisms

Let $U$ be a smooth quasi-projective variety over $\mathbf C$, and let $V\subset U$ be a dense open subvariety. Let $X\to U$ and $Y\to U$ be smooth projective morphisms such that their restrictions ...
1
vote
1answer
86 views

A question on Aut$(N)$ and Aut$(N/G)$

Let $N$ be a complex manifold and $G$ is a finite group freely acting on $N$. Define another complex manifold $M$ as $M=N/G$. I would like to study Aut$(M)$, the (holomorphic) automorphism group of ...
4
votes
0answers
67 views

Negative self intersection and section of the conormal sheaf for a singular complex curve

Let $M$ be a complex $2$-dimensional manifold and $C$ be a compact complex curve in $M$ (possibly singular). Let us suppose that there exists a holomorphic function $f\in\mathcal{O}(M)$ such that ...
4
votes
0answers
43 views

Properties of divisors when moving from char 0 to char p.

Consider a smooth projective variety $X$ over $\mathbb{C}$ such that $X$ has models over $\mathbb{Z}[1/N]$ and $X_p=X_{\mathbb{Z}[1/N]}\times \text{Spec}(\mathbb{F}_p)$ is also a smooth projective ...
5
votes
0answers
125 views

Finding two curves with intersection included in a specified neighborhood

Let $V$ be a complex projective surface and $U$ an analytic neighborhood of $x\in V$. How can I prove that there are two smooth curves $C_1$ and $C_2$ s.t. $C_1\cap C_2\subset U$ ?
2
votes
2answers
273 views

How to compute the number of branch points and ramification indices of the quoteint covers?

Take a ramified Galois cover $f:X\rightarrow Z$ of Riemann surfaces over $\mathbb{C}$ with Galois group $G$. If $H$ is a non-tirival normal subgroup of the Galois group $G$, this cover factors as ...
6
votes
0answers
60 views

Automorphism on a kahlerian variety which acts as -id on $H^2(X,\mathbb{Z})$

I've read this proposition: "there is no automorphism (biholomorphic map) of a Kahlerian complex manifold $X$ which acts on $H^2(X,\mathbb{Z})$ as $-$identity" so I tried to prove this statement and ...
0
votes
1answer
276 views

The sum of the residues of a meromorphic function on a Riemann surface

How can one see that the sum of the residues of a meromorphic function on a Riemann surface $ \Sigma_g$ of positive genus is always zero? This is not true for the Riemann sphere $\mathbb{CP}^1$.
8
votes
2answers
122 views

Reference for relation between class number of $\Bbb Q[\sqrt{-p}]$ and partial quotients of $\sqrt p$

So in Ireland and Rosen's, "Classical Introduction to Modern Number Theory", they mention the following incredible fact at the end of Chapter 13, section 1. Suppose $p \neq 3$ and $p \equiv 3 \pmod 4$ ...
0
votes
1answer
53 views

complex vector fields - hard d vs. soft d?

I believe this is a computation I have done before, but now I can't write the symbols to convince myself: What is the connection between the "hard" complex differential operator d/dz and the "soft" ...
2
votes
0answers
64 views

If the intersection of the preimage of a generic point of a flat morphism with an open set is dense, does it imply the open set is dense?

Let $f: E \rightarrow M $ be a flat morphism of varieties (over $\mathbb{C}$) and $E^{\prime}$ an open subset of $E$. Assume that both $E$ and $E^{\prime}$ have pure dimension $2n$, where $n$ ...
1
vote
1answer
132 views

What is the definition of a flat morphism?

When we say that a morphism $f: E \rightarrow M $ between two algebraic varieties (over $\mathbb{C}$) is a flat morphism, what does it mean? Does it mean that that the "dimension" of every fiber ...
2
votes
0answers
72 views

Can every complex space be covered by a finite number of Stein spaces?

Can every complex space be covered by a finite number of Stein spaces?
1
vote
1answer
75 views

(Complex) Projective Space

I followed a course in projective geometry and I'm not sure about 2 things: If I have 6 lines in projective space (IP³) with commun secant, why are the 6 corresponding tensors linearly dependent? ...
-2
votes
1answer
120 views

If a subspace is simply connected, then the space itself is simply connected

Let $X$ be a "nice" connected topological space. Let $U\subset X$ be a non-empty subspace. Suppose that $U$ is simply connected. Is $X$ simply connected? In my application, I'm actually thinking ...
3
votes
0answers
73 views

How should I imagine the cup product on a topological space/manifold/variety

Let $X$ be a "space" with a vector bundle $E$. Let $n$ be the dimension of $X$. Then there is a map $$H^1(X,E)\otimes \ldots\otimes H^1(X,E)\to H^n(X,\Lambda^n E).$$ This map is given by taken the ...
2
votes
1answer
112 views

hermitean structure vs hermitean metric

Define a hermitean structure $H$ on a complex linear space $V$ as $H: V\times V \to \mathbb{C}$ s.t. i. $H(u,v)$ is $\mathbb{C}$-linear in $u$ for every $v \in V,$ ii. $H(u,v)=\overline{H(v,u)}$ ...
1
vote
1answer
98 views

Rank of Jacobian at a singularity

Is the following proposition true? Proposition: Suppose $\mathbb{C}\{x_1,\ldots,x_{d_1}\}/(f_1,\ldots,f_{k_1}) \cong \mathbb{C}\{y_1,\ldots,y_{d_2}\}/(g_1,\ldots,g_{k_2})$ are isomorphic complex ...
1
vote
1answer
40 views

A question on a map to a complex curve and fundamental groups.

Let $X$ be a simply connected complex manifold. How can one show that there exists no holomorphic map to a curve $C$ of genus $g\ge 1$? This shows up in a paper I am reading now. I thought this can ...
5
votes
0answers
61 views

Branch locus along a smooth curve

Let $f : S \to X$ be a dominant morphism of smooth complex surfaces. Let $C \subset S$ be a smooth curve such that $df$ is of rank $1$ along $C$ and that in a neighborhood of $C$, $df_x$ is an ...
2
votes
0answers
89 views

Product of two Kähler manifolds

Let $M$ and $N$ be Kähler manifolds. I know that the cartesian product $M\times N$ is a kahler manifold. I was wondering which form has the Kähler form on $M\times N$. Here's what I thought: Let ...
4
votes
1answer
93 views

How many types of surface singularities multiplicity two exist?

All varieties are over $\mathbb{C}$. Let $S$ be a reduced algebraic surface in $\mathbb{P}^3$ with a singular point $p$ of multiplicity two. The question is local so we reduce to $S \subset ...
7
votes
0answers
119 views

Why do algebraic varieties contain curves passing through two given points

Let $X$ be an algebraic variety over the complex numbers. My definition of an algebraic variety is a finite type separated $\mathbf C$-scheme. Someone told me that such varieties have the following ...
3
votes
0answers
48 views

Maps between total spaces of holomorphic vector bundles

I am wondering what is possible and what is not possible regarding maps between the total spaces of holomorphic vector bundles. Let me outline a situation that is a bit more concrete, to help focus ...
4
votes
1answer
96 views

Complexified tangent vector, complex manifold

Consider a complex submanifold $M$ of a complex ambient vector space $X$. Suppose you have a base point $p \in M$ and a $C^1$ arc $\gamma(t)$ passing through $p$ and staying in $M$, with tangent ...
4
votes
1answer
76 views

Non-vanishing 2-form on quartic surface.

Let $S\subset \mathbb P^3$ be a quartic surface defined by a homogeneous degree 4 polynomial $F\in k[x_0,x_1,x_2,x_3]$. $S$ is a K3 surface, so it has a unique non-vanishing $(2,0)$-form $\omega$ up ...
4
votes
3answers
87 views

Good source to learn about surface singularities?

I am looking for something that treats singularities on algebraic surfaces and curves over $\mathbb{C}$, starting from the very basics but not stopping there. I checked out Miles Reid his lectures on ...
12
votes
1answer
194 views

Introduction to the trace formula for people outside number theory

I am looking for references on the trace formula, by which I mean the Selberg trace formula and its successor the Arthur-Selberg trace formula. I am aware that there are "standard references" on the ...
5
votes
1answer
279 views

Why is Griffiths Transversality part of the definition of a variation of Hodge structures?

If $X \to S$ is a family of compact Kahler manifolds, then parallel transport with respect to the Gauss-Manin connection on the relative cohomology bundle does not respect the Hodge filtration, e.g. a ...
1
vote
1answer
88 views

Complex 3-D Euclidean space - inner product

1st question: Lets say we have a 3-D complex euclidean space. How do we geometrically draw this space? if 3-D real Euclidean space is represented by these base vectors: 2nd question: Is there a ...
5
votes
0answers
130 views

Is there a relation between Super Riemannian manifolds and Kähler manifolds?

(This question has a physics motivation). Could we establish any kind of relationship between Super Riemannian manifolds (super Riemannian structures on super manifolds) and Kähler manifolds, or at ...
2
votes
0answers
43 views

Algorithm for checking that a singularity is an ordinary double point

Let a complete intersection $d$-fold singularity be cut out by $k$ polynomials $\{f_1,\ldots,f_k\}$ in $\mathbb{C}^{k+d}$. Is there a relatively simple algorithm to check whether this is analytically ...
0
votes
1answer
42 views

Polynomials in the complex ring of 2 variables

Given $I = \left<x^2+y^2-1, x^2-y+1, xy-1\right>$ show that this generates $\mathbb{C}[x,y]$. I have tried pages and pages of writing a linear combination of these such that the combination is ...
3
votes
2answers
67 views

Help with unknown notation

In Ahlfors' Complex Analysis, page 19 it says (in relation with the Riemann sphere): "writing $z=x+iy$, we can verify that: $$x:y:-1=x_1:x_2:x_3-1, $$ and this means that the points ...
2
votes
0answers
50 views

Are complex submanifolds necessarily closed?

In the excellent book From Holomorphic Functions to Complex Manifolds by Klaus Fritzsche and Hans Grauert, if I follow the definition and properties of analytic subsets and the definition of a complex ...
2
votes
0answers
49 views

Riemann mapping between arbitrary triangles

Question---Is there nice formula for Riemann mapping between arbitrary triangles with vertices (a_1,a_2,a_3) and (b_1,b_2,b_3)? Comment---I look for the conformal equivalence of interiors promised ...