**58**

votes

**2**answers

10k views

### Why is the Hodge Conjecture so important?

The Hodge Conjecture states that every Hodge class of a non singular projective variety over $\mathbf{C}$ is a rational linear combination of cohomology classes of algebraic cycles: Even though I'm ...

**58**

votes

**5**answers

3k views

### Gaussian prime spirals

Imagine a particle in the complex plane, starting at $c_0$, a Gaussian integer,
moving initially $\pm$ in the horizontal
or vertical directions. When it hits a Gaussian prime, it turns left ...

**46**

votes

**6**answers

3k views

### Simplest examples of nonisomorphic complex algebraic varieties with isomorphic analytifications

If they are not proper, two complex algebraic varieties can be nonisomorphic yet have isomorphic analytifications. I've heard informal examples (often involving moduli spaces), but am not sure of the ...

**37**

votes

**6**answers

2k views

### What do Weierstrass points look like?

As somebody who mostly works with smooth, real manifolds, I've always been a little uncomfortable with Weierstrass points. Smooth manifolds are totally homogeneous, but in the complex category you ...

**29**

votes

**6**answers

5k views

### Is there a complex structure on the 6-sphere?

I don't know who first asked this question, but it's a question that I think many differential and complex geometers have tried to answer because it sounds so simple and fundamental. There are even a ...

**29**

votes

**5**answers

2k views

### Which nonlinear PDEs are of interest to algebraic geometers and why?

Motivation
I have recently started thinking about the interrelations among algebraic geometry and nonlinear PDEs. It is well known that the methods and ideas of algebraic geometry have lead to a ...

**29**

votes

**1**answer

912 views

### Example of a compact Kähler manifold with non-finitely generated canonical ring?

A celebrated recent theorem of Birkar-Cascini-Hacon-McKernan and Siu says that the canonical ring $R(X)=\oplus_{m\geq 0}H^0(X,mK_X)$ of any smooth algebraic variety $X$ over $\mathbb{C}$ is a finitely ...

**26**

votes

**2**answers

849 views

### Can the holomorphic image of $(\mathbb{C}^*)^n$ be open but not dense

Let $M$ be a compact complex connected [but not necessarily kähler] $n$-manifold, and suppose we have a holomorphic map $$(\mathbb{C}^*)^n \to M$$ such that the image is open. Is the image ...

**26**

votes

**2**answers

2k views

### A geometric characterization for arithmetic genus

Let $X$ be a smooth projective variety over $\mathbb{C}$. The following information is all equivalent (any of these numbers can be computed by a linear equation from any of the others):
the ...

**26**

votes

**0**answers

3k views

### A paper to the question, if the six dimensional sphere is a complex manifold

Hi,
for a few days a paper was published on arxiv.org with the title "The six dimensional sphere is a complex manifold": http://arxiv.org/PS_cache/math/pdf/0505/0505634v3.pdf
Because I am not able to ...

**24**

votes

**0**answers

1k views

### Is S^2 x S^4 a complex manifold?

As observed by Calabi a long time ago, the manifold $S^2\times S^4$ admits an almost-complex structure (obtained by embedding it in $\mathbb{R}^7$ and using the octonionic product), which however is ...

**23**

votes

**10**answers

3k views

### Algebraic Geometry versus Complex Geometry

This question is motivated by this one.
I would like to hear about results concerning complex projective varieties which
have a complex analytic proof but no known algebraic proof; or
have an ...

**22**

votes

**3**answers

780 views

### Rep Theory Consequences of Bott--Weil--Borel

I've been getting interested in the (Bott--)Borel--Weil theorem lately. As a (mainly) geometer it is very interesting to see representation appearing (from nowhere as far as I can see) in the theory ...

**22**

votes

**8**answers

3k views

### What's the difference between a real manifold and a smooth variety?

I am teaching a course on Riemann Surfaces next term, and would like a list of facts illustrating the difference between the theory of real (differentiable) manifolds and the theory non-singular ...

**22**

votes

**1**answer

1k views

### Complex vector bundles that are not holomorphic

Is there an example of a complex bundle on $CP^n$ or on a Fano variety (defined over complex numbers), that does not admit a holomorphic structure? We impose of course that the Chern classes of the ...

**21**

votes

**4**answers

3k views

### “The complex version of Nash's theorem is not true”

The title quote is from p.221 of the 2010 book,
The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions
by Shing-Tung Yau and Steve Nadis. "Nash's theorem" here ...

**21**

votes

**5**answers

2k views

### References for “modern” proof of Newlander-Nirenberg Theorem

Hi,
I'm starting to prepare a graduate topics course on Complex and Kahler manifolds for January 2011. I want to use this course as an excuse to teach the students some geometric analysis. In ...

**21**

votes

**5**answers

2k views

### Two definitions of Calabi-Yau manifolds

Why is it that the vanishing of the integral first Chern class of a compact Kahler manifold is equivalent to the canonical bundle being trivial? I can see that it implies that the canonical bundle ...

**21**

votes

**3**answers

2k views

### Which almost complex manifolds admit a complex structure?

I was reading Yau's list of problems in geometry, and one of them is to prove that any almost complex manifold of complex dimension $n \geq 3$ admits a complex structure. It's been some time since ...

**21**

votes

**2**answers

1k views

### Accumulation of algebraic subvarieties: Near one subvariety there are many others (?)

Let's work over the field $\mathbb{C}$ of complex numbers, and let $X\subset \mathbb{P}^n$ be a projective variety. Let $\tilde{X}\subset \mathbb{P}^n$ be any small open neighborhood of $X$, in the ...

**19**

votes

**3**answers

1k views

### Hsiung on the Complex Structure of $S^6$

In 1986 C. C. Hsiung published a paper "Nonexistence of a Complex Structure on the Six-Sphere" and in 1995 he even wrote a monograph "Almost Complex and Complex Structures" to further elaborate on his ...

**19**

votes

**1**answer

1k views

### Diffeomorphic Kähler manifolds with different Hodge numbers

This question made me wonder about the following:
Are there orientedly diffeomorphic Kähler manifolds with different Hodge numbers?
It seems that this would require that those manifolds are not ...

**19**

votes

**1**answer

878 views

### Do hyperKahler manifolds live in quaternionic-Kahler families?

A geometry question that I thought about more seriously a few years ago... thought it'd be a good first question for MO.
I'm aware that there are a number of Torelli type theorems now proven for ...

**19**

votes

**1**answer

372 views

### Biholomophic non-Algebraically Isomorphic Varieties

Recently, when writing a review for MathSciNet, the following question arose:
Is it true that two smooth complex varieties that are biholomorphic are algebraically isomorphic? The converse is true ...

**18**

votes

**2**answers

2k views

### Which Kahler Manifolds are also Einstein Manifolds?

Is it known which Kahler manifolds are also Einstein manifolds? For example complex projective spaces are Einstein. Are the Grassmannians Einsein? Are all flag manifolds Einstein?

**18**

votes

**2**answers

1k views

### Non-compact complex surfaces which are not Kähler

Not every complex manifold is a Kähler manifold (i.e. a manifold which can be equipped with a Kähler metric). All Riemann surfaces are Kähler, but in dimension two and above, at least for compact ...

**18**

votes

**2**answers

802 views

### Is Hodge theory somehow connected with a Galois group action Gal(C/R)?

I'm currently taking a course in Hodge theory ... and I wonder if all the splittings in $\{i,-i\}$ Eigenvalue pairs come from the Galois group action (of the extension ...

**18**

votes

**3**answers

2k views

### Exercises in Hodge Theory

I was wondering: is there a good place to find exercises in Hodge theory? Mostly computations and proving small (preferably nifty) theorems, is what I have in mind. Something roughly like the ...

**18**

votes

**1**answer

1k views

### Projective embedding of symplectic manifolds

Let $(M^{2n},\omega)$ be a symplectic manifold with an integral symplectic form $\omega$. Due to the work of M.Gromov and D.Tischler (M.Gromov "A topological technique for the construction of ...

**18**

votes

**0**answers

666 views

### Almost complex 4-manifolds with a “holomorphic” vector field

Main question. What is the class of smooth orientable 4-dimensional manifolds that admit an almost complex structure $J$ and a vector field v, that preserves $J$?
The following sub question is ...

**17**

votes

**5**answers

3k views

### The Relationship between Complex and Algebraic Geomety

I have recently begun to study algebraic geometry, coming from a differential geometry background. It seems that there is a deep link between complex manifolds and complex varieties. For example, one ...

**17**

votes

**4**answers

2k views

### Why do people think that abelian varieties are the hardest case for the Hodge conjecture?

Today, I heard that people think that if you can prove the Hodge conjecture for abelian varieties, then it should be true in general. Apparently this case is important enough (and hard enough) that ...

**17**

votes

**7**answers

2k views

### Kähler structure on cotangent bundle?

The total space of cotangent bundle of any manifold M is a symplectic manifold.
Is it true\false\unknown that for any M, $T^*M$ has Kähler structure?
Please support your claim with reference or ...

**17**

votes

**2**answers

1k views

### Complex manifolds in which the exponential map is holomorphic

Let $X$ be a complex manifold and $g$ a hermitian metric on $X$. Consider the Riemannian exponential $\exp_p: T_p X \to X$.
If $\exp_p$ is holomorphic for every $p \in X$, then $(\exp_p)^{-1}$, ...

**17**

votes

**0**answers

582 views

### Are the eigenvalues of the Laplacian of a generic Kähler metric simple?

It is a theorem of Uhlenbeck that for a generic Riemannian metric, the Laplacian acting on functions has simple eigenvalues, i.e., all the eigenspaces are 1-dimensional. (Here "generic" means the set ...

**16**

votes

**3**answers

1k views

### Holomorphic vector fields acting on Dolbeault cohomology

The question.
Let $(X, J)$ be a complex manifold and $u$ a holomorphic vector field, i.e. $L_uJ = 0$. The holomorphicity of $u$ implies that the Lie derivative $L_u$ on forms preserves the (p,q) ...

**16**

votes

**1**answer

957 views

### Accumulation of algebraic subvarieties: Near one subvariety there are many others (?), 3

Part 3 of this series of questions. In the meantime, I realized that there is some very simple question that was left open in Accumulation of algebraic subvarieties: Near one subvariety there are many ...

**16**

votes

**1**answer

747 views

### Rational or elliptic curves on Calabi-Yau threefolds

Let $X$ be a Calabi-Yau threefold. From a complex analytic point of view, it is widely believed that it should not be Kobayashi hyperbolic, that is it should always admit some non-constant entire map ...

**16**

votes

**2**answers

572 views

### Moishezon manifolds with vanishing first Chern class

Suppose $M$ is a Moishezon manifold with $c_1(M)=0$ in $H^2(M,\mathbb{R})$. Does it follow that $K_M$ is torsion in $\mathrm{Pic}(M)$?
This is true whenever $M$ is Kähler (and therefore ...

**15**

votes

**2**answers

632 views

### Projective variety with no syzygies but not isomorphic to projective space

Let $A:={\mathbb{C}}[x_0,\ldots,x_n]=\oplus_{d=0}^{\infty} {\mathbb{C}}[x_0,\ldots, x_n]_d$ be the graded complex algebra of polynomials in $n+1$ variables, graded by degree.
Suppose $L$ is a line ...

**15**

votes

**4**answers

694 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 ...

**15**

votes

**1**answer

1k views

### Stein Manifolds and Affine Varieties

When is a Stein manifold a complex affine variety? I had thought that there was a theorem saying that a variety which is Stein and has finitely generated ring of regular functions implies affine, but ...

**15**

votes

**1**answer

2k views

### GAGA and Chern classes

My question is as follows.
Do the Chern classes as defined by Grothendieck for smooth projective varieties coincide with the Chern classes as defined with the aid of invariant polynomials and ...

**15**

votes

**2**answers

814 views

### When can you reverse the orientation of a complex manifold and still get a complex manifold?

I'm told that $\overline{\mathbb{C}P^2}$, i.e. $\mathbb{C}P^2$ with reverse orientation, is not a complex manifold. But for example, $\overline{\mathbb{C}}$ is still a complex manifold and ...

**14**

votes

**5**answers

1k views

### de Rham vs Dolbeault Cohomology

For a complex manifold $M$, one can consider (A) its de Rham cohomology, or (B) its Dolbeault cohomology. I'm looking for some motivation as to why one would bother introducing Dolbeault cohomology. ...

**14**

votes

**4**answers

1k views

### Weitzenböck Identities

I asked this question at Maths Stack Exchange, but I haven't received any replies yet (I'm not sure how long I should wait before it is acceptable to ask here, assuming there is such a period of ...

**14**

votes

**4**answers

1k views

### Compact holomorphic symplectic manifolds: what's the state of the art?

This came up in a discussion I had yesterday. Since my understanding is limited,
I thought I ask here, because I know there are quite a few experts lurking about.
Recall that a holomorphic ...

**14**

votes

**1**answer

954 views

### Almost Complex Structure approach to Deformation of Compact Complex Manifolds

I don't know much about the deformation of compact complex manifolds, I've only read chapter 6 of Huybrechts' book Complex Geometry: An Introduction. There are two parts to this chapter. The second ...

**14**

votes

**3**answers

999 views

### Deformations of Kähler manifolds where Hodge decomposition fails?

This is partly inspired by answers to the question:
Question about Hodge number .
Is there a family of compact complex manifolds, where the general fibres are
Kähler, but for which $E_1$ degeneration ...

**14**

votes

**2**answers

559 views

### Does equality of Laplacians imply Kähler?

This question follows on from this one.
Let $(X, \omega)$ be a Hermitian manifold and define the Laplacians $\Delta_{\partial} = \partial\partial^* + \partial^*\partial$ and $\Delta_{\bar{\partial}} ...