25
votes
Accepted
Deformations of Calabi-Yau manifolds
The answer in general is no. Nakamura has constructed here (pp.90, 96-99, solvmanifolds of type III-(3b)) an example of a compact complex (non-Kähler) manifold $M$ with $TM$ holomorphically ...
- 6,871
15
votes
Accepted
Are the symmetric spaces $\operatorname{SU}(n)/{\operatorname{SO}(n)}$ always nontrivial in the bordism rings for $n>2$?
There is a fibration $SU(n) \overset p\to SU(n)/SO(n) \overset j\to BSO(n)$, where the $j$ is the classifying map of $p$, viewed as (the projection of) a principal $SO(n)$-bundle. The Stiefel–Whitney ...
- 2,995
12
votes
Deformations of Calabi-Yau manifolds
The answer is yes (in char. 0). Indeed, it suffices to show that for an infinitesimal deformation $\mathcal{X}$ of $X$ over an artinian algebra $A$, the cohomology $H^0(\mathcal{X}, \omega_{\mathcal{X}...
- 16.1k
12
votes
Accepted
Hodge Numbers and Leray Spectral Sequence
I don't think I defined the Hodge numbers in this way. Rather, the argument in Section 1 shows that the Hodge numbers agree with the dimensions of
the terms in the $E_2$ page of the Leray spectral ...
- 985
11
votes
Accepted
Calabi-Yau manifolds and knot theory
$X$ is certainly Calabi-Yau in the algebraic sense: the canonical line bundle is trivialized by the holomorphic volume form
$\Omega=\frac{dx}{x} \wedge \frac{dy}{y} \wedge \frac{du}{u}$
(this comes ...
- 6,920
10
votes
Accepted
$K_X+B \equiv 0$ implies $K_X + B \sim_\mathbb{Q} 0$?
For lc pair or slc pair, it is true. This is Gongyo’s result. See [J. ALGEBRAIC GEOMETRY 22 (2013) 549–564].
BTW, the relative version is also true, which is not a trivial generalization of the ...
- 1,164
8
votes
What is the geometrical meaning of higher Chern forms and classes?
This is a big topic, which should be covered in the union of many standard texts (Chern, Griffiths-Harris, Milnor-Stasheff...). I'll list a few answers off the top of my head.
Suppose that $L$ is ...
- 35.2k
7
votes
Accepted
How do you get the spectral curve from a Calabi-Yau?
In general there is no way to extract a spectral curve from a Calabi-Yau threefold.
In the study of strings on Calabi-Yaus, one object of interest is the periods, i.e. integrals of the holomorphic ...
- 2,087
7
votes
Accepted
Multiple mirrors phenomenon from SYZ and HMS perspective
Some relation between these two ambiguities is indeed part of the expected but still largely conjectural big picture.
Let $X$ be a compact Calabi-Yau manifold, viewed in a symplectic way. Let $M_X$ ...
- 6,920
6
votes
(1/2) K3 surface or half-K3 surface: Ways to think about it?
Of course what you've written is too vague to be a definition, but I can guess what they're talking about. In low-dimensional topology there's a 4-manifold called $E(1)$; this is a rational complex ...
- 7,005
5
votes
Mirror symmetry for K3 fibered Calabi-Yau threefolds
Morally speaking, a K3 fibration on one side of the mirror correspondence should correspond to a K3 degeneration on the other side. In the physics literature, I think this observation goes back to ...
- 426
5
votes
Accepted
Is there any relativistic interpretation on considering Kaehler-Einstein metrics and Calabi-Yau manifolds?
First, a purely mathematical remark: it is not so easy to construct Riemannian Ricci-flat metrics on compact manifolds. Ricci flat Kähler (= Calabi-Yau) metrics give a large class of examples and are "...
- 6,920
4
votes
Accepted
Mirror partners of some Calabi-Yau threefolds
Your two examples are actually of very different characters. The first has Hodge numbers $h^{2,1} = 3$ and $h^{1,1} = 51$; the second is rigid. This means that in the first case you're looking for a ...
- 426
3
votes
Multiple mirrors phenomenon from SYZ and HMS perspective
Two algebraic varieties are called Fourier-Mukai partners if their bounded derived categories of
coherent sheaves are equivalent.
A "trivial" example of Fourier-Mukai partners is given by ...
- 314
3
votes
Explicitly computing Donaldson-Thomas invariants (of CY 3-folds)
This answer might just be a list of references, but I hope it helps. The most explicit computations of which I am aware exploit a torus action on the Calabi-Yau 3-fold in question, where the calculus ...
- 271
3
votes
Examples of Calabi-Yau manifolds with $\mathbb{T}^2$ symmetry
As far as I understand, such metrics can be obtained by the so-called "Calabi-anszats":
http://www.numdam.org/article/ASENS_1979_4_12_2_269_0.pdf
Namely, one can construct a Calabi-Yau metric on the ...
- 14.3k
2
votes
How many Coulomb branches do we (conjecturally) know?
To motivate other people to give other answers, I will sketch the example coming from class S 4d theories $\mathcal{T}(\Sigma,G)$ attached to a marked compact Riemann surface $\Sigma$ and a ...
- 5,701
2
votes
When is the birational Torelli problem for CY threefolds true?
I think the main idea in both papers you mention is that two birational Calabi-Yau threefolds with Picard number 1 are necessarily isomorphic. The use of this argument in such a situation is quite old ...
- 7,300
2
votes
Constraints on the base of an elliptically fibered Calabi-Yau threefold
I found some very old notes in which I worked out some restrictions on the base space for elliptically fibered Calabi-Yau 3-folds, part of it will apply in any dimension. There are some references to ...
- 515
1
vote
How to find a rational $\mathbb{F}_{\!q}$-curve on a quite classical Calabi–Yau threefold?
Further intersecting $T$, which is $3$-dimensional in its affine model inside $\Bbb A^5$, with a generic variety of codimension two should lead to a curve. (It may be that i do not catch the point of ...
- 1,976
1
vote
Large Complex Structure Limit of Calabi-Yau family and uniqueness of limit
In Physics, the large complex structure limit of $X$ is dual to a large volume limit of the mirror Calabi-Yau $X^v$. More precisely, different phases of $X^v$ connected by flops correspond to ...
- 73
1
vote
Explicit metrics on non-compact Calabi-Yau threefolds
The simplest example would be on the, non-small resolution, $\mathbf{K}_{\mathbb{CP}^1 \times\mathbb{CP}^1}$ of $\mathbb{CP}^1 \times\mathbb{CP}^1$. This the C-Y metric is the Calabi Ansazt using the ...
- 496
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