Timeline for What is a moduli space of Calabi-Yau threefolds?
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12 events
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| Nov 30, 2021 at 21:00 | comment | added | Jason Starr | Yes, that is correct. None of these arguments is my proof -- I have seen this argument repeated in lectures many times (unfortunately I cannot immediately think of a written reference, but I am sure that there is a reference). | |
| Nov 30, 2021 at 20:55 | comment | added | Cranium Clamp | Wonderful. So any compact Kahler manifold with $ h^{2,0} = 0 $ is projective. Your proof shows that it has nothing to do with being a threefold or $ h^{1,0} = 0 $. | |
| Nov 30, 2021 at 20:46 | comment | added | Jason Starr | These are projective because the second cohomology group $H^2(X,\mathbb{C})$ is purely $(1,1)$. Since $H^2(X,\mathbb{Q})$ is dense in $H^2(X,\mathbb{C})$, and since the Kaehler cone is open in $H^2(X,\mathbb{C})$, it follows that there are elements of $H^2(X,\mathbb{Q})$ that are in the Kaehler cone. Then, by the Kodaira embedding theorem, a suitably positive and divisible integer multiple of this class equals the first Chern class of a very ample invertible sheaf. This is how you can embed the Calabi-Yau threefold in $\mathbb{P}^n$, and thus get a point of $H_{n,d}$. | |
| Nov 30, 2021 at 20:44 | comment | added | Cranium Clamp | I understand now. Thank you so much for your patience. I guess my exercise for now will be to prove that any CY3 with $ h^{1,0} = 0 = h^{2,0} $ is projective and then proceed. | |
| Nov 30, 2021 at 20:27 | comment | added | Jason Starr | Since the Hodge numbers $h^{1,0}$ and $h^{2,0}$ vanish, the stacks of Calabi-Yau threefolds can be glued together from "charts" that are of the form $H_{n,d}/\text{Aut}(\mathbb{P}^n)$, where $H_[n,d}$ is the open subscheme of the Hilbert scheme parameterizing closed subschemes of $\mathbb{P}^n$ that are smooth Calabi-Yau threefolds of degree $d$. So the stacks are locally quite close to the scheme $H_{n,d}$. By the Tian-Todorov Theorem, this scheme is smooth and quasi-projective. | |
| Nov 30, 2021 at 20:12 | comment | added | Cranium Clamp | I'm sorry, I should mention that I'm just a PhD student who is more or less ignorant about stacks. The only stack I've dealt with is the moduli stack of elliptic curves. | |
| Nov 30, 2021 at 20:02 | comment | added | Jason Starr | I do not understand your comment. Certainly simply connected Calabi-Yau manifolds of dimension 3 have vanishing Hodge numbers $h^{p,0}$ for $p$ equal to 1 or 2. This makes their moduli stacks easier to understand, not harder to understand. | |
| Nov 30, 2021 at 19:02 | comment | added | Cranium Clamp | So my 'simply connected' assumption rules out your first two classes of examples, right? And yes, I'm mainly looking for the irreducible ones. Looks like we don't have much understanding of those yet? | |
| Nov 30, 2021 at 0:43 | comment | added | Jason Starr | There are better MO users than me to discuss this, but according to the Beauville-Bogomolov decomposition (closely related to the Berger-de Rham decomposition), such manifolds are finite group quotients of a complex torus, a product of a K3 surface and an elliptic curve, or an "irreducible Calabi-Yau manifold", i.e., one for which the Hodge numbers $h^{p,0}$ equal $0$ for $p=1,2$. For the last class, all of these are projective. So you can construct moduli stacks of these manifolds using Artin's algebraization theorems, cf. "Versal deformations and algebraic stacks". | |
| Nov 29, 2021 at 23:52 | comment | added | Cranium Clamp | @Jason Starr thank you for this information. I would like to know how this space is constructed though. For example, I know about the moduli space of curves of genus g, or the moduli of stable sheaves on a surface. There are techniques to construct such spaces, via GIT. Is there something like this for CY3s? | |
| Nov 29, 2021 at 23:46 | comment | added | Jason Starr | By the Tian-Todorov Theorem, the local Kuranishi spaces are all smooth. So connected components are irreducible. The main open conjecture about the components is the Kawamata-Morrison Conjecture. | |
| Nov 29, 2021 at 23:32 | history | asked | Cranium Clamp | CC BY-SA 4.0 |