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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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 ...
19
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2answers
651 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. ...
17
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1answer
1k views

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 ...
15
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2answers
1k views

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 ...
15
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1answer
415 views

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 ...
15
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1answer
355 views

Is there any holomorphic version of the tubular neighborhood theorem?

This question arised when I was studying Beauville's book 'Complex Algebraic Surfaces'. Castelnuovo's theorem says that a smooth rational curve $E$ on an algebraic surface $S$ is an exceptional ...
14
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1answer
632 views

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, ...
13
votes
4answers
563 views

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 ...
12
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2answers
259 views

Complex manifolds and Hermitian metrics

I've been trying to learn some complex geometry, and was getting confused in thinking about Hermitian metrics. In this post, I've written up my current understanding, in hopes that someone can look ...
12
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1answer
210 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 ...
12
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1answer
210 views

Proof of holomorphic Lefschetz fixed point formula using currents in Griffiths and Harris

I am trying to understand the proof of the Holomorphic Lefschetz fixed point formula on page 426 in Griffiths and Harris. However, I find their use of currents extremely confusing. They seem to go ...
12
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1answer
233 views

Can one use Atiyah-Singer to prove that the Chern-Weil definition of Chern classes are $\mathbb{Z}$-cohomology classes?

In Chern-Weil theory, we choose an arbitrary connection $\nabla$ on a complex vector bundle $E\rightarrow X$, obtain its curvature $F_\nabla$, and then we get Chern classes of $E$ from the curvature ...
11
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2answers
324 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 ...
11
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4answers
2k views

Equation of the complex locus: $|z-1|=2|z +1|$

This question requires finding the Cartesian equation for the locus: $|z-1| = 2|z+1|$ that is, where the modulus of $z -1$ is twice the modulus of $z+1$ I've solved this problem algebraically ...
11
votes
1answer
336 views

Dolbeault Cohomology is invariant under homeomorphisms

If $X$ and $Y$ are two complex manifolds, which are homeomorphic but not necessarily diffeomorphic, must their Dolbeault cohomology groups be isomorphic? Here the Dolbeault cohomology groups ...
11
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2answers
571 views

Conditions such that taking global sections of line bundles commutes with tensor product?

Let us work with projective algebraic varieties over $k= \mathbb{C}$. If necessary we can also assume smoothness of the varieties. Of course it is not in general true that given two line bundles $L, ...
11
votes
1answer
335 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 ...
10
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1answer
271 views

Torsion Chern class?

Can somebody give an example of a complex manifold whose first Chern class is a torsion class? In general it seems that Chern classes may have torsion part as well as free part. However when using ...
10
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4answers
164 views

holomorphic exoticness

A topological manifold is an exotic copy of another smooth manifold if it is homeomorphic to it, but not diffeomorphic (and when you switch diffeomorphic by homotopic, you get a fake copy, following ...
10
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1answer
200 views

Formula for decomposing a form into $(p,q)$ forms

Let $L: \mathbb{C}^n \to \mathbb{C}$ be a real linear map. In other words, $L(a\vec{v}_1+b\vec{v_2}) = aL(\vec{v}_1)+bL(\vec{v}_2)$ for all $a,b \in \mathbb{R}$. Then $L$ decomposes uniquely into a ...
9
votes
5answers
651 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 ...
9
votes
2answers
485 views

Riemann surfaces are algebraic

Via a very ample divisor one can embed a Riemann surface holomorphically into some $\mathbb{P}^n$. Now, we can then project the Riemann surface to $\mathbb{P}^3$, and we can even go until ...
9
votes
1answer
208 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 ...
9
votes
2answers
280 views

Fibres in algebraic geometry: multiplicity

Currently I am studying varieties over $\mathbb{C}$ and I know some scheme theory. My professor mentioned the other day that given a morphism of varieties over an alg. closed field $k$: $f: X ...
9
votes
3answers
715 views

Where can I read about elliptic operators on manifolds?

In Voisin's beautiful book on Hodge theory she gives a proof that every cohomology class can be represented by a harmonic form by referring to the the following theorem on elliptic operators: Let $P ...
9
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2answers
606 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 ...
9
votes
1answer
235 views

Hodge theory for toric varieties

Say we are given a complex smooth projective toric variety $X$. How can one read off hodge theoretic information from combinatorial data? For example I would like to extract dimensions of the various ...
9
votes
1answer
114 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 ...
9
votes
0answers
692 views

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 ...
8
votes
1answer
81 views

Zariski vs analytic cohomology of $\mathcal O_X^\times$

Let $X/\mathbf C$ be a smooth proper variety. Is it true that $H^1(X, \mathcal O_X^\times) = H^1(X^{an}, \mathcal O_{X^{an}}^\times)?$ GAGA doesn't apply, because $\mathcal O_X^\times$ is not coherent ...
8
votes
2answers
274 views

Relation between algebraic geometry over a field of characteristic $0$ and that over $\mathbb{C}$

Let $K$ be an algebraically closed field of characteristic $0$. Let $P$ be a proposition on a non-singular projective variety over $K$ which is stated in the language of algebraic geometry. Suppose ...
8
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1answer
150 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?
8
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2answers
234 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 ...
8
votes
2answers
240 views

connections on coherent sheaves

Let $X$ be a smooth variety over $\mathbb{C}$. If $\mathscr{F}$ is a coherent sheaf on $X$ with connection, does it follow that $\mathscr{F}$ is locally free? I can't think of any counterexamples. ...
8
votes
2answers
330 views

If $0$, $z_1$, $z_2$ and $z_3$ are concyclic, then $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear

If the complex numbers $0$, $z_1$, $z_2$ and $z_3$ are concyclic, prove that $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear. I really can't seem to get anywhere on this problem, ...
8
votes
2answers
125 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 $p \equiv 3 \pmod 4$ ...
8
votes
1answer
196 views

How to prove $(0,1)$ form is not $\overline\partial$-exact

On a complex manifold, if we are dealing with the $d$ operator, there's a pretty easy way of showing some form is not $d$-exact, simply by integrating in a closed loop. If you can find a loop that is ...
8
votes
1answer
262 views

Are these definitions of intersection multiplicity equivalent?

I am pretty sure the answer is yes. I normally work over $\mathbb{C}$ so i will do so here as well, to prevent myself from making silly mistakes. In projective space, one has Serre's famous ...
8
votes
1answer
169 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 ...
8
votes
1answer
617 views

The Chern class of a Kähler manifold

This question refers mainly to this mathoverflow question and its answer. Statement 1 The well known result of Aubin and Yau states that if $X$ is a compact Kähler manifold with negative first Chern ...
8
votes
1answer
125 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
0answers
202 views

Interesting applications of Kähler Identities

The Kähler identities give commutation relations between the Lefschetz operator ($\alpha \mapsto \omega \wedge \alpha$), the differential operators $\partial, \bar{\partial}$ and its adjoints ...
7
votes
1answer
548 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 ...
7
votes
2answers
179 views

Adjunction for varieties with higher codimension

For $X \subseteq \mathbb{P}^n$ a smooth hypersurface, the canonical divisor $K_X$ can be computed as $$ K_X = (K_{\mathbb{P}^n} + X)|_X. $$ Is there a similar formula where $X$ is of higher ...
7
votes
2answers
745 views

Why isn't $\log(-1)=i\pi$?

Reading http://people.math.gatech.edu/~cain/winter99/ch3.pdf, $\log(z)$ is defined as $=\ln|z|+i\arg(z)$. Looking on the Wessel plane, isn't $\arg(-1)=\pi$ (more generally $\pi \pm 2 \pi n$)? And ...
7
votes
2answers
316 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 ...
7
votes
1answer
226 views

Holomorphic mappings, Plücker's formula (Mistake in Rick Miranda's book?)

In Rick Miranda's book "Algebraic Curves and Riemann Surfaces", in order to prove Plücker's formula for smooth projective plane curves, he first defines a projective plane curve by the formula ...
7
votes
2answers
188 views

Are endomorphisms of degree one always automorphisms

Let $B$ be a smooth projective connected variety over $\mathbb C$. Let $\sigma:B\to B$ be an endomorphism of degree one. Do I understand correctly that $\sigma$ is an automorphism? I believe this ...
7
votes
2answers
958 views

Almost complex structures on spheres

It is fairly well-known that the only spheres which admit almost complex structures are $S^2$ and $S^6$. By embedding $S^6$ in the imaginary octonions, we obtain a non-integrable almost complex ...
7
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1answer
333 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 ...