Unanswered Questions
1,216 questions with no upvoted or accepted answers
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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 ...
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50
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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 ...
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25
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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....
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22
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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 $\...
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21
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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(...
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21
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903
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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 ...
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19
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637
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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 ...
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18
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886
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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 ...
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17
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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 ...
CommunityBot
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16
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0
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537
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Gabriel's theorem for complex analytic spaces
Let $X,Y$ be noetherian schemes over $\mathbb{C}$.
Then, it is known that
$$
\text{Coh}(X) \simeq \text{Coh}(Y) \Rightarrow X \simeq Y,
$$
by P. Gabriel(1962).
Are there some results in the case of ...
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16
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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 ...
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16
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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 ...
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15
votes
0
answers
248
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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 ...
- 66k
15
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answers
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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 ...
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14
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776
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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 ...
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