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I was told that Calabi-Yau's can be birational to each other but not isomorphic (biholomorphic).

But I've never seen explicit examples. Can anybody here show me one?

(E.g. maybe an explicit example of a flop between Calabi-Yaus?)

edited Sep 9 at 6:42

asked Sep 9 at 2:07

temp
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What does "isomorphic" mean in this context? – Igor Rivin Sep 9 at 4:22

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Presumably it means "biholomorphic". – YangMills Sep 9 at 4:46

Try looking at the paper "The movable fan of the Horrocks--Mumford quintic" by Michael Fryers. That gives an explicit example of a CY 3-fold with (IIRC) precisely 8 birational models. – Artie Prendergast-Smith Sep 9 at 18:40

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One other small comment is that your "e.g." is really more than an "e.g.": any two birational Calabi--Yaus (as long as they're smooth, or a bit more generally, have only terminal singularieties) are related by a sequence of flops. (Maybe you already knew that.) – Artie Prendergast-Smith Sep 9 at 18:41

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Ah, a quick search shows it is proved for any dimension by Kawamata in 2007. – temp Sep 9 at 23:08

show 3 more comments

Atsushi Kanazawa · Answer 1 · 2012-09-09 05:53:35Z UTC

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You may be interested in Lee and Oguiso's paper Connecting certain rigid birational non-homeomorphic Calabi--Yau threefolds via Hilbert scheme. This gives a pair of CY3s you want (with additional interesting properties).

answered Sep 9 at 5:53

Atsushi Kanazawa
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Sasha · Answer 2 · 2012-09-09 07:29:04Z UTC

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Take a quintic hypersurface in $P^4$ with several (say $n$) ordinary double points. Each of them locally analytically has 2 small resolution. Combining those you can construct $2^n$ global small resolutions. All of them are birational Calabi-Yau threefolds.

answered Sep 9 at 7:29

Sasha
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Why are they not isomorphic to each other? – temp Sep 9 at 23:02

Usually some of them are projective and the other are not. It depends on existence of Weil divisors passing through all singular points. – Sasha Sep 10 at 2:52

@temp: Suppose $X$ and $X'$ are connected by flopping a single rational curve $C$ to $C'$. Since they are birational, the divisor class groups of $X$ and $X'$ are naturally isomorphic, and you can use this to show that they're not biholomorphic. For example, if $D$ is an effective divisor on $X$, satisfying $D\cdot C = 1$, then its proper transform on $X'$ (which I'll also call $D$), satisfies $D\cdot C' = -1$. The second Chern class also changes: $$ c_2(X')\cdot D = c_2(X)\cdot D + 2 $$ – Rhys Davies Sep 10 at 7:26

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@Rhys Davies: this argument only shows that the birational isomorphism is not biregular, or that $X$ and $X'$ are not isomorphic OVER their common contraction. Sometimes, manifolds $X$ and $X'$ related by a flop are ABSTRACTLY isomorphic, for example two small resolutions of a quadratic cone are. – Sasha Sep 10 at 10:21

What if I want to find projective examples that are birational but not isomorphic? – temp Sep 20 at 7:49

Examples of Calabi-Yau that are birational to each other?

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