Questions tagged [complex-geometry]

Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry. For elementary questions about geometry in the complex plane, use the tags (complex-numbers) and (geometry) instead.

Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry.

3187 questions
65
votes
1 answer

Does $\Bbb{CP}^{2n} \mathbin{\#} \Bbb{CP}^{2n}$ ever support an almost complex structure?

This question has now been crossposted to MathOverflow, in the hopes that it reaches a larger audience there. $\Bbb{CP}^{2n+1} \mathbin{\#} \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has an orientation-reversing diffeomorphism…
user98602
50
votes
2 answers

What exactly is a Kähler Manifold?

Please scroll down to the bold subheaded section called Exact questions if you are too bored to read through the whole thing. I am a physics undergrad, trying to carry out some research work on topological solitons. I have been trying to read a…
48
votes
1 answer

Is there a proof of Bézout's theorem via residue theory?

Let's define intersection numbers as follows. Consider a collection $f_1,\dots, f_n$ of holomorphic functions on some neighborhood of zero in $\mathbb C^N$ cutting out divisors $D_1$, all of which vanish at $0$. Define $$\omega(f_1,\dots,…
Potato
  • 37,797
  • 15
  • 117
  • 251
44
votes
2 answers

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 matter of convention. Let $\mathbb{P}^n$ be a…
Li Yutong
  • 3,715
  • 20
  • 38
36
votes
0 answers

Arithmetic and geometric genus

There are two notion of genus in algebraic geometry, namely arithmetic genus $p_a=(-1)^{\dim X}(\chi(\mathcal{O}_X)-1)$ and geometric genus $p_g=\dim H^0(X,\Omega^{\dim X})$. I keep forgetting definition of these, or being confused which is which.…
M. K.
  • 4,561
  • 25
  • 47
34
votes
1 answer

Why is the hard Lefschetz theorem "hard"?

Let $X$ be a compact Kähler manifold of complex dimension $\dim_{\mathbb C} = n$. Let $[\omega]$ be the cohomology class of a Kähler metric on $X$. Then powers of the class $[\omega]$ defines a linear morphism between cohomology groups $$ L^k : …
31
votes
2 answers

Learning Complex Geometry - Textbook Recommendation Request

I wish to learn Complex Geometry and am aware of the following books : Huybretchs, Voisin, Griffths-Harris, R O Wells, Demailly. But I am not sure which one or two to choose. I am interested in learning complex analytic & complex algberaic geometry…
31
votes
2 answers

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. If the bundle is of finite rank, then the sheaf is…
29
votes
1 answer

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 $V \to V$ which commute with $J$. One can think of $J$ as…
28
votes
1 answer

How to properly use GAGA correspondence

currently studying algebraic surfaces over the complex numbers. Before i did some algebraic geometry (I,II,start of III of Hartshorne) and a course on Riemann surfaces. Now i understood that by GAGA, a lot of results transfer from complex analytic…
27
votes
5 answers

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 treats Chern classes in algebraic geometry over…
27
votes
4 answers

Where can I learn about complex differential forms?

So I'm a 3rd year grad student in number theory/modular forms/algebraic geometry, and I've worked with differential forms from an algebraic point of view without ever knowing what they really are. I'd like to fix this. I've recently read a section…
26
votes
1 answer

Is there a complex surface into which every Riemann surface embeds?

Every Riemann surface can be embedded in some complex projective space. In fact, every Riemann surface $\Sigma$ admits an embedding $\varphi : \Sigma \to \mathbb{CP}^3$. It follows from the degree-genus formula that the same is not true if we…
26
votes
3 answers

Sine of a Complex Number

While I know that $\sin(x)=2$ has no real solution, I tried seeing if it has a complex solution. That equality is equal to $$e^{2ix}-4ie^{ix}-1=0$$ Taking a quadratic in $e^{ix}$ I got $$x=-i\log(i)-i\log(2 \pm \sqrt{3})$$ I then expressed $i$ as…
Gridley Quayle
  • 1,559
  • 1
  • 12
  • 16
25
votes
2 answers

Is there an algebraic reason why a torus can't contain a projective space?

Let $X$ be an abelian variety. As abelian varieties are projective then $X$ contains lots and lots of subvarieties. Why can't one of them be a projective space? If $X$ is defined over the complex numbers, then there is a relatively painless way to…
1
2 3
99 100