Questions tagged [complex-geometry]

Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.

1,130 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
50 votes
0 answers
12k views

Atiyah's paper on complex structures on $S^6$

M. Atiyah has posted a preprint on arXiv on the non-existence of complex structure on the sphere $S^6$. https://arxiv.org/abs/1610.09366 It relies on the topological $K$-theory $KR$ and in ...
David C's user avatar
  • 9,792
48 votes
0 answers
17k views

What is the current understanding regarding complex structures on the 6-sphere?

In October 2016, Atiyah famously posted a preprint to the arXiv, "The Non-Existent Complex 6-Sphere" containing a very brief proof $S^6$ admits no complex structure, which I immediately read and ...
jdc's user avatar
  • 2,984
25 votes
0 answers
2k views

Is there a proof of Hodge theory using condensed mathematics?

As is well known, many results in complex geometry "feel" algebraic (and often have statements which are "completely algebraic") but only have "transcendental" proofs (i....
Gabriel's user avatar
  • 933
21 votes
0 answers
1k views

Cartan–Oka vanishing in one variable without $\overline{\partial}$?

This is a literature question, about possible proofs of some very basic results in complex analysis. Some key facts about holomorphic functions are proved via reduction to smooth functions, using $\...
Peter Scholze's user avatar
21 votes
0 answers
857 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 ...
Joel Fine's user avatar
  • 6,177
20 votes
0 answers
1k views

Reference request: deforming a G-local system to a variation of Hodge structure

Let $X$ be a smooth connected quasiprojective variety over $\mathbb{C}$ and let $G$ be a complex reductive group. Let $$\iota: G\to GL_N$$ be a representation and let $$\rho: \pi_1(X(\mathbb{C}))\to G(...
Daniel Litt's user avatar
  • 22.1k
19 votes
0 answers
602 views

Coarse moduli spaces of stacks for which every atlas is a scheme

Let $X = [P/G]$ be a smooth finite type separated DM-stack over $\mathbb C$ given as the quotient of a smooth quasi-projective scheme $P$ by the action of a smooth (finite type separated) reductive ...
Ariyan Javanpeykar's user avatar
18 votes
0 answers
868 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 ...
Dmitri Panov's user avatar
  • 28.7k
17 votes
0 answers
747 views

What are hyperkähler metrics used for?

It seems that a lot of effort has been devoted to endow holomorphic-symplectic manifolds with hyperkähler metrics. It started with Calabi [4] with $T^*\mathbb{CP}^n$. Other examples include coadjoint ...
user129123's user avatar
16 votes
0 answers
414 views

Can non-reduced fibers appear over a subset of codimension $\geq 2$?

I already asked this on math.stackexchange.com, but didn't receive an answer. Suppose $f: X \to Y$ is a (possibly proper) morphism of complex manifolds (resp. smooth varieties) such that all fibers of ...
red_trumpet's user avatar
  • 1,061
15 votes
0 answers
212 views

Geometry of Affine Kac-Moody Algebras

I recently asked this question on phys.SE and it was suggested to me to ask it here. One can reconstruct the unitary irreducible representations of compact Lie groups very beautifully in geometric ...
user356359's user avatar
15 votes
0 answers
1k views

Topological description of a blow up of a manifold along a submanifold

There's a very nice topological description of blow ups of complex manifolds at a point as connected sum with projective space. The following is an attmept to understand whether there's a higher ...
Saal Hardali's user avatar
  • 7,549
15 votes
0 answers
2k views

Kodaira-Spencer maps and deformation theory

This post concerns the following question: Can we black-box the analysis of PDE's which arises in the construction of Kuranishi families for complex analytic structures? The deformation theory of ...
Andy Sanders's user avatar
  • 2,890
14 votes
0 answers
739 views

Gromov's quick "proof" of Lefchetz Hyperplane Theorem

I'd say I'm fairly comfortable with standard proofs of the Lefschetz Hyperplane theorem (e.g. lefschetz pencils, morse theory, etc.). However, in the first chapter of Gromov's Partial Differential ...
user111650's user avatar
14 votes
0 answers
697 views

Is Hironaka's example the only known deformation of Kähler manifolds with non-Kähler central fibre?

A well-known example in the deformation theory of compact complex manifolds is the one given by Hironaka in his 1962 paper An Example of a Non-Kählerian Complex-Analytic Deformation of Kählerian ...
Michael Albanese's user avatar
13 votes
0 answers
370 views

Does the existence of an almost complex structure solely depend on the topology of the manifold?

To be precise, let $M$ and $N$ be two 2n-dimensional smooth, closed manifolds that are homeomorphic. If $M$ admits an almost complex structure, can we deduce that $N$ also admits an almost complex ...
Chicken feed's user avatar
13 votes
0 answers
317 views

Exotic smooth structures on Fano manifolds

If two Fano projective manifolds are homeomorphic are they diffeomorphic? There are examples with one manifold being Fano and the other of general type (Barlow surfaces). Moreover, the number of ...
user avatar
13 votes
0 answers
320 views

Varieties isomorphic $\mathrm{mod}\:p$ are diffeomorphic

If two smooth proper varieties over $\mathbb{Q}$ have isomorphic smooth reductions modulo some prime (for some choice of integral models) are they diffeomorphic after tensoring with $\mathbb{C}$?
user avatar
13 votes
0 answers
707 views

Kähler-Ricci flow approach for Beauville-Bogomolov type decomposition?

Is there any Kähler Ricci flow method for solving structure theorems in Algebraic geometry In fact If $X$ be a Calabi-Yau manifold then we can descend the Kähler Ricci flow to its finite etale ...
user avatar
13 votes
0 answers
1k views

Fundamental group of the complement homogeneous variety in $\mathbb{C}P^{n-1}$

Let $f,g:\mathbb{C}^n\to \mathbb{C}$ are two irreducible homogeneous polynomials. If there is a homeomorphism $h:\mathbb{C}^n\to \mathbb{C}^n$ such that $h(X)=Y$ and $h(0)=0$, where $X=f^{-1}(0)$ and $...
userX10's user avatar
  • 131
13 votes
0 answers
375 views

Complex manifold which is algebraic away from codimension \ge 2

If $X$ is a complex manifold and $Z$ is a closed subset of codimension $\ge 2$ such that $X-Z$ has an algebraic structure, then is $X$ algebraic; i.e. a scheme? If not is $X$ an algebraic space? ...
solbap's user avatar
  • 3,938
12 votes
0 answers
494 views

Does Lefschetz-type theorems imply ampleness?

Let $X$ be a smooth $n$-dimensional complex projective variety and $D \subset X$ a smooth (effective) divisor. Consider the following properties: $D$ is ample. (Positivity) For any $k$-dimensional ...
 V. Rogov's user avatar
  • 1,115
12 votes
0 answers
149 views

Holomorphic natural bundles and operators

I am wondering up to what extent the classical theory of (smooth) natural bundles and natural operations extends to the holomorphic setting. After a quick thought, I've gone through the standard ...
Quidam_t's user avatar
  • 121
12 votes
0 answers
1k views

How much algebraic geometry do I need to study complex geometry?

As one can deduce from the questions I have asked on MO, I'm interested in complex geometry. I am aware that there are many facets to the field, some of which I am more comfortable with than others. ...
Michael Albanese's user avatar
11 votes
0 answers
321 views

A purely algebraic argument for existence of a section of a smooth projective morphism to the projective line

If I am reading this post correctly, any smooth projective $\mathbb{C}$-morphism of schemes $X\rightarrow \mathbb{P}^1$ admits a section. I am afraid of the topological argument presented there. Is ...
user avatar
11 votes
0 answers
199 views

Holomorphically convex manifolds and Bergman complete manifolds

Suppose $X$ is a complex manifold which admits the Bergman metric (for definitions, see for instance Kobayashi's book "Hyperbolic Complex Spaces"). Suppose moreover that the Bergman metric of $X$ is ...
diverietti's user avatar
  • 7,852
11 votes
1 answer
814 views

Cohomological bounds for scalar curvature of an extremal Kähler metric

There is an interesting trick used in Chen-LeBrun-Weber's paper on the extremal Kähler metrics of $\mathbb{CP}^2\#2\overline{\mathbb{CP}^2}$, and I would like to know whether it can be (has been?) ...
macbeth's user avatar
  • 3,182
11 votes
0 answers
294 views

Computing $h^1$ of dual of graph of central fibre of the degeneration of Kaehler-Einstein manifolds

Consider a Kaehler degeneration $\mathcal X\to \Delta$ of smooth manifolds: Here $\Delta$ is the unit disc, $\pi$ a proper flat map, smooth over $\Delta^∗=\Delta−\{0\}$. The general fibres are $X_t=\...
Dima's user avatar
  • 405
11 votes
0 answers
2k views

Dolbeault cohomology of complex tori.

Let $T=\mathbb C^n/\Lambda$ a complex torus. It is completely elementary to prove that the de Rham cohomology of $T$ in degree $q$ is isomorphic to the $q$-th exterior power of the dual of $\mathbb C^...
diverietti's user avatar
  • 7,852
10 votes
0 answers
193 views

Holomorphic versus algebraic $\mathbb C^*$-actions

I believe that the following is true: Statement. A holomorphic $\mathbb C^*$-action on a complex projective manifold is algebraic if and only if it has a fixed point. Where can I find a proof of ...
aglearner's user avatar
  • 14k
10 votes
0 answers
335 views

Local meaning of the Pfaffian of the curvature

The Ricci and scalar curvatures have very nice pointwise interpretations (using the local expression for the volume form for example). So, (at least Ricci) having special metrics (like Einstein) can ...
Vamsi's user avatar
  • 3,323
10 votes
0 answers
333 views

Analytic space not embeddable in any complex manifold

I am looking for an example of a compact complex analytic space, reduced and irreducible, which does not admit any holomorphic embedding into any (smooth) complex manifold (possibly non-compact). I ...
YangMills's user avatar
  • 6,636
10 votes
0 answers
6k views

Atiyah's paper "Non-existent complex 6-sphere"

I'm trying to understand the main idea of Atiyah's proof (https://arxiv.org/abs/1610.09366). Although there were discussions on MO year ago I couldn't find answers to my questions. Consider the ...
Max Borovkov's user avatar
10 votes
0 answers
597 views

Uniqueness of singular Hermitian-Einstein metric along Yau-Donaldson flow?

The following question is related to Singular Yang-Mills theory The Hermitian metric $H$ along the fibers of the holomorphic vector bundle $E$ over a K\"ahler manifold $(M,\omega)$ is Hermitian-...
user avatar
10 votes
0 answers
303 views

Examples of quasi-negative but not negative holomorphic sectional curvature

Let $(X,\omega)$ be a compact Kähler manifold and call $\operatorname{HSC}_{\omega}(x,[v])$ the holomorphic sectional curvature of the Chern connection of $\omega$ at the point $x\in X$ in the ...
diverietti's user avatar
  • 7,852
10 votes
0 answers
2k views

How much of modern algebraic geometry is there in modern complex(algebraic, analytic, differential) geometry?

Good day to you, people of mathoverflow. I'll get to the point. I wonder how much of modern abstract algebraic geometry is there in modern complex geometry? What do I mean by complex geometry? ...
Seidarkman's user avatar
10 votes
0 answers
765 views

de Rham cohomology group and Dolbeault cohomology group on compact complex analytic spaces

My question is: On a compact complex analytic space,since Hodge Theorem becomes invalid,is it true that the de Rham cohomology group $H^p_{DR}(M)$ and the Dolbeault cohomology group $H^{p,q}_{\bar{\...
whitacre's user avatar
  • 101
10 votes
0 answers
439 views

Homology classes represented by $J$-holomorphic curves

Let $\Sigma$ be a compact Riemann surface with complex structure $j$. Let $(M,J)$ be an almost complex manifold. A map $u: \Sigma \rightarrow M$ is called $J$-holomorphic if $$ du \circ j = J \circ du....
Hwang's user avatar
  • 1,388
10 votes
0 answers
511 views

About the Bloch conjecture on entire curves

The Bloch conjecture states the following: Bloch's conjecture. Let $X$ be a compact complex Kähler variety such that the irregularity $q = h^0(X,\Omega^1_X)$ is larger than the dimension $n = \dim X$....
diverietti's user avatar
  • 7,852
9 votes
0 answers
253 views

Fundamental group of the complement to a plane curve with unramified normalization

Suppose that $C\subset\mathbb P^2$ is an irreducible projective curve over $\mathbb C$ such that the normalization morphism $\bar C\to C$ is unramified (i.e., the induced morphism $\bar C\to\mathbb P^...
Serge Lvovski's user avatar
9 votes
0 answers
373 views

Kähler metric on the Hilbert scheme of points on a surface

Question. Let $S$ be a non-singular complex projective surface and let $S^{[n]}$ be its Hilbert scheme of $n$ points. Is there a natural way to associate to a Kähler metric $\omega$ on $S$ a Kähler ...
Jost Schultze's user avatar
9 votes
0 answers
824 views

Is there a Calabi-Yau threefold with $h^{1,1}=1$ and $h^{1,2}=0$?

Is there a Calabi-Yau threefold with $h^{1,1}=1$ and $h^{1,2}=0$?
rj7k8's user avatar
  • 716
9 votes
0 answers
273 views

Hermitian sectional curvature

Let $N$ be a Riemannian manifold, denote $R$ its purely covariant Riemann curvature tensor with sign convention so that the sectional curvature is $K(X,Y) = R(X,Y,X,Y)$ for an orthonormal pair. ...
seub's user avatar
  • 1,337
9 votes
0 answers
367 views

Is it possible to glue together complex manifolds?

In the case of Riemannian manifolds, there are ways to take two manifolds and glue them together to get a new Riemannian manifold. For example, taking connected sums in local regions where the two ...
Kim's user avatar
  • 4,034
9 votes
0 answers
196 views

Batyrev's theorem in non-algebraic case

Let $X$ and $Y$ be two bimeromorphic closed Kaehler manifolds with trivial real $c_1$. Is it true that $b_n(X)=b_n(Y)$ for $n\geq 0$?
geometer's user avatar
  • 713
9 votes
0 answers
318 views

Diffeomorphism type of Ricci-flat four manifolds

Let $(M,g)$ be an irreducible compact and simply connected Ricci-flat Riemannian four-manifold. My first questions are as follows: A) Is there a classification of the possible homeomorphism types of ...
Bilateral's user avatar
  • 3,064
9 votes
0 answers
260 views

Riemann-Hilbert correspondence over nodal curve

There is a micro-local version of Riemann-Hilbert correspondence over nodal curve $X$ given by Roman Bezrukavnikov Mikhail Kapranov in the paper: Micro-local sheaves and quiver varieties The de Rham ...
Feng Hao's user avatar
  • 1,071
9 votes
0 answers
290 views

Coarse moduli space of compact polarized Fano Kaehler-Einstein manifolds

Let $\mathcal X\to \mathcal S$, be a family of polarized Kaehler manifolds with $\omega_s= Ric(\omega_s)$(i.e., fibers are Fano Kahler-Einstein manifolds). Then $dim Aut(X_s)=Const$.? Is there any ...
Dima's user avatar
  • 405
9 votes
0 answers
419 views

Demailly condition in Analytical algebraic geometry

Grauert-Riemenschneider conjecture: If a compact complex manifold $X$ possesses a smooth Hermitian line bundle which is semi-positive everywhere and positive on an open dense set, then $X$ is ...
user avatar
9 votes
0 answers
145 views

A characterization of Moishezon manifolds via sections of $L^k$ with $k\to \infty$

Let $X$ be a smooth compact complex manifold of dimension $n$. Suppose $L$ is a line bundle on $X$ such that $dim(H^0(X,L^k))>c\cdot k^n$ for $c>0$ and $k>>0$. Question. Is it true that $...
aglearner's user avatar
  • 14k

1
2 3 4 5
23