24 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 ...
YangMills's user avatar
  • 6,656
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 ...
jdc's user avatar
  • 2,984
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}...
Piotr Achinger's user avatar
12 votes
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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 ...
Mark Gross's user avatar
11 votes
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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 ...
user25309's user avatar
  • 6,820
10 votes
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$K_X+B \equiv 0$ 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 ...
Chen Jiang's user avatar
  • 1,084
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 ...
Donu Arapura's user avatar
  • 34.2k
7 votes
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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 ...
Clay Cordova's user avatar
  • 1,987
7 votes
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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 $M_X$ ...
user25309's user avatar
  • 6,820
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 ...
Jonny Evans's user avatar
  • 6,935
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 ...
Ursula's user avatar
  • 426
5 votes
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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 "...
user25309's user avatar
  • 6,820
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 ...
Ursula's user avatar
  • 426
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 ...
Aurelio's user avatar
  • 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 ...
aglearner's user avatar
  • 14k
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 ...
Sergey's user avatar
  • 314
2 votes

Find the Picard Fuchs operator of a four parameter fundamental period

The standard strategy would be to apply the GKZ method or the Griffiths-Dwork technique, as outlined in Cox and Katz's Mirror Symmetry and Algebraic Geometry. Is there a reason neither of these is ...
Ursula's user avatar
  • 426
2 votes

BCOV's holomorphic anomaly equation at genus one

There is an introduction to the BCOV theory from math perspective in this artcle by Kanazawa and Zhou. In the case $h^{1,1}=1$, it is checked in page 16 that the $\mathcal{F}_1$ (non-holomorphic ...
Dimitris's user avatar
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 ...
doetoe's user avatar
  • 515
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 ...
Pulcinella's user avatar
  • 5,506
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 ...
Libli's user avatar
  • 7,200
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 ...
dan_fulea's user avatar
  • 1,821
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 ...
Salix's user avatar
  • 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 ...
Craig's user avatar
  • 486

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