All Questions
17 questions with no upvoted or accepted answers
9
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0
answers
996
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Complexification of smooth manifolds
Given a $C^\infty$- smooth manifold $M$, is there a natural way to complexify it?
By this I mean finding a complex manifold $N$ and a smooth function $f:M\to N$ such that $df:T_xM\otimes \mathbb{C}\to ...
- 435
6
votes
0
answers
239
views
Complex submanifolds via Kähler reduction
Let $X$ be a Kähler manifold with an isometric $S^1$-action (which of course complexifies to a $\mathbb C^*$ action). Consider the corresponding Hamiltonian $H$ and let $X_0=H^{-1}(0)/S^1$ be the ...
- 14.3k
4
votes
0
answers
114
views
Degeneration formula and Donaldson-Floer theory
Is there a relation between the degeneration formula of GW Invariants of Jun Li and the Donaldson-Floer theory? Is there an example / discussion anywhere of/on this relation?
- 153
4
votes
0
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310
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PDE obtained while trying to construct a complex structure
Upon reading this answer to this question, the last paragraph mentions the following. "Requiring the [almost complex] structure to be integrable corresponds to a certain PDE for this map." ...
- 1,763
4
votes
0
answers
129
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Geodesic distances and Grassmannians
Let $Gr(n, V)$ be the Grassmannian parametrising $n$-dimensional subspaces of a vector space $V$.
When $V$ is an inner product space, formulae exist for calculating the geodesic distance between ...
- 61
4
votes
0
answers
129
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Coordinate-free B.Feix's construction of a hyperkähler metric
In the 2001's paper 'Hyperkähler metrics on cotangent bundles' B.Feix gives a construction of a hyperkähler metric on a neighbourhood of zero section in $T^*X$ where $X$ is a real analytic Kähler ...
- 2,305
4
votes
0
answers
157
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Analytic maps $\varphi: \mathbb C^n\to \mathbb C^n$ with degenerate differentials
Let $B^n\subset \mathbb C^n$ be a unit ball with center $p$ . Let $\varphi: B^n\to \mathbb C^n$ be a complex analytic map such that $d\varphi$ has rank at most $n-1$ at $p$. I would like to know if ...
- 14.3k
3
votes
0
answers
336
views
Understanding Calabi's conjecture proof: What is it meant by the logarithm of a differential form?
I'm reading several books and articles concerning Yau's proof of the Calabi conjecture. I want to have a deep understading of how and why such proof actually works, but most articles are aimed at ...
- 153
3
votes
0
answers
637
views
English reference for Fischer-Grauert theorem and its generalization by Schuster
From this MSE question and its answer, and from this MO question I have learned of the following remarkable theorem of Wolfgang Fischer and Hans Grauert.
Theorem. A proper holomorphic submersion with ...
- 10.5k
3
votes
0
answers
447
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Complex structures on Riemann surfaces
This is cross posted from math.SE: https://math.stackexchange.com/q/876432/9
Let $M$ be a Riemann surface and $[\alpha] \in H^{0,1}(M; T^{1,0} M) \simeq (H^0(M;K^2))^*$. Considering $\alpha$ as a ...
- 3,244
2
votes
0
answers
152
views
Period map on non-Kähler manifold
Is there a theory of period map on non-Kähler manifolds that has Hodge decomposition? Any reference is helpful. Thank you.
- 263
2
votes
0
answers
172
views
Construction of Kahler Einstein Metric of Poincare Type
I am reading Kobayashi's Kahler-Einstein metric on an open algebraic manifold. In this paper he constructs a Kahler-Einstein of Poincare type on an open manifold X' = X\D, where X is projective and D ...
- 31
2
votes
0
answers
75
views
Notation and geometry facts in a paper on the Diederich-Fornæss index
I am reading this article by Bingyuan Liu on the Diederich-Fornæss index.
I am having some problems with both the notation and the geometrical side.
1)I don't know what kind of objects $N,L$ are ...
- 779
2
votes
0
answers
59
views
Group of real analytic isometries of $g$-fold product of the Poincare upper half plane
Let $\mathfrak{h}^g$ be the cartesian product of $g$ copies of the Poincare upper half plane. We endow $\mathfrak{h}^g$ with the usual Poincare metric given in local coordinates by $ds^2=\sum_{i=1}^g ...
- 7,609
2
votes
0
answers
282
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Generalizing a result of Paul Andi Nagy
I am trying to generalizate a result of Paul Andi Nagy which says that in a almost Kähler manifold with parallel torsion we have $\langle \rho,\Phi-\Psi\rangle $ is a nonnegative number; in fact
$$
4\...
- 21
1
vote
0
answers
150
views
A vector field over a complex riemannian manifold
Let be a complex riemannian manifold $(M,g,J)$. Is the following canonical vector field studied ?
$$
X_J = \sum_{i=1}^{2n} \nabla^{LC}_{e_i}e_i +\nabla^{LC}_{Je_i}Je_i+ J[e_i,Je_i],
$$
with the $(e_i)...
- 377
1
vote
0
answers
382
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Question in the paper of Robert Bryant "Calibrated embeddings in the special Lagrangian and coassociative cases"
Hallo,
I am trying to read the paper "Calibrated embeddings in the special Lagrangian and coassociative cases" by Robert Bryant and I have a question. I hope that maybe one of you could give me some ...
- 339