23
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 ...

- 6,498

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 ...

- 2,947

12
votes

Accepted

### Singularities of the moduli stack of Calabi-Yau threefolds

Yes, Calabi-Yau manifolds have unobstructed deformations. This is due to Tian and Todorov; there is a nice algebraic proof in a paper by Kawamata, J. Algebraic Geom. 1 (1992), no. 2, 183–190.

- 35.4k

12
votes

Accepted

### central charge and Calabi-Yau dimension

Given a $N=(2,2)$ two dimensional superconformal field theory (SCFT), one can construct two topological field theories called the $A$ and $B$ models. To each of these topological field theories, one ...

- 6,720

12
votes

### Can Calabi-Yau manifolds have nonabelian discrete symmetry groups?

Assuming you actually meant "Kähler" and not "Calabi-Yau":
In the book Fundamental Groups of Compact Kähler Manifolds by Amorós et al., on page 6 (example 1.11) it is asserted that every finite ...

- 22.1k

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}...

- 14.6k

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 ...

- 6,720

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 ...

- 1,054

8
votes

### Can Calabi-Yau manifolds have nonabelian discrete symmetry groups?

If you work one dimension down, at the level of K3 surfaces, there's a very pretty classification of finite group actions preserving the holomorphic form due to Mukai. In that classification, the ...

- 426

7
votes

### Can Calabi-Yau manifolds have nonabelian discrete symmetry groups?

The main theorem of Fine and Panov's 'The diversity of symplectic Calabi-Yau six-manifolds' (http://arxiv.org/abs/1108.5944) implies that, in particular, every finitely presented group arises as the ...

- 23.2k

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 $M_X$ ...

- 6,720

7
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 ...

- 32.5k

6
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 ...

- 1,987

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 ...

- 6,660

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 "...

- 6,720

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

4
votes

### How to construct (another) Landau-Ginzburg model for a compete intersection Calabi-Yau?

Yes. One way outlined in the work of Abouzaid-Auroux-Katzarkov's work (http://arxiv.org/pdf/1205.0053.pdf) is to look at the space $\mathbb{P}^n\times\mathbb{C}^r$ blown up along the codimension 2 ...

- 3,025

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 ...

- 307

3
votes

### A technical question in Feix's construction of hyperkahler metric on cotangent bundles

I just read your question. Funny, I was also trying to read this paper carefully not long ago, and I was annoyed by exactly that kind of problem.
I think the problem is to say that "the space of ...

- 1,327

3
votes

### Unique Kahler-Einstein metric $g$ with $\mathrm{Ricc}(g)=-g$ when first Chern class $C_1(M)<0$: $\mathrm{Ricc}(h)=-g\,\Rightarrow\,h=cg$ for $c>0$?

so here I guess $M$ is a compact Kähler manifold.
Thanks to Yau theorem, we know that there exists a unique Kähler metric $h$ in each Kähler cohomology class such that $\mathrm{Ric}(h)=-g$ (more ...

- 2,587

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 ...

- 13.8k

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

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 ...

- 21

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 ...

- 426

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