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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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
107 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
209 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
162 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
111 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
66 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
270 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
269 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
119 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
131 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
119 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
111 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
60 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
88 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
116 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
47 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
86 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
272 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
87 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
124 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
49 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 ...
2
votes
1answer
62 views

Positive curvature on holomorphic vector bundles

There must be a mistake in my understanding the definition of positivity for the curvature. Let me summarize: Let $ (L,\nabla,h) \rightarrow M $ be a hermitian hol line bundle with Chern connection. ...
5
votes
1answer
161 views

Which is the correct universal line bundle: the tautological bundle or its dual?

With topological line bundles over $\mathbb{C}$, one learns that every line bundle is a pullback of the universal line bundle, which is the tautological line bundle over $\mathbb{C}P^\infty.$ In ...
4
votes
2answers
64 views

The zero set of $z_0^2+z_1^2-1$ in $\mathbb{C}^2$.

Recently, I read the notes "Vector bundles on Riemann surfaces" by Sabin Cautis (http://www-bcf.usc.edu/~cautis/classes/notes-bundles.pdf). On the sixth page of these notes, there is a statement ...
7
votes
1answer
295 views

Projective closure in the Zariski and Euclidean topologies

In Smith's An Invitation to Algebraic Geometry, following the definition of the projective closure of an affine variety, it was remarked that "the closure may be computed in either the Zariski ...
1
vote
1answer
60 views

Degree of morphism of quotient of upper half-plane

Recall that SL$_2(\mathbf R)$ acts on the complex upper half-plane $\mathbf H$. Let $\Gamma$ be a finite index subgroup of SL$_2(\mathbf Z)$. Then there is the quotient $Y_\Gamma = \Gamma \backslash ...
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vote
0answers
94 views

A funny condition for ampleness on a curve

Let $E$ be a locally free sheaf on a complete nonsingular curve $C$ over $\mathbb C$. Suppose that, for all points $P$ in $C$, the locally free sheaf $$ E\otimes \mathcal{O}_C(P)$$ is an ample ...
8
votes
1answer
107 views

Varieties with the property that the cotangent bundle restricted to a complete nonsingular curve is free

Let $X$ be a $d$-dimensional smooth projective connected variety with cotangent sheaf $\Omega^1_X$ over $\mathbb C$. Suppose that for any nonsingular complete curve $C$ and non-constant morphism ...
8
votes
1answer
162 views

Are vector bundles on $\mathbb{P}_{\mathbb{C}}^n$ of any rank completely classified? (main interest $n=3$)

It is very well known that the group of line bundles on $\mathbb{P}_{\mathbb{C}}^n$ is exactly $\mathbb{Z}$. Are bundles of higher rank classified as well? If so, could anyone provide a nice ...
9
votes
5answers
506 views

Reference request: Chern classes in algebraic geometry

I have encountered Chern classes numerous times, but so far i have been able to work my way around them. However, the time has come to actually learn what they mean. I am looking for a reference that ...