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Calabi-Yau-logy for physicists: Obtaining an example of a Calabi-Yau six-fold

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Can you give an example of a Calabi-Yau six-fold?

asked Nov 6, 2016 in Mathematics by juancho (860 points) [ no revision ]

1 Answer

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I think that the most simple example is generated by the octic in \(CP^7\) with an Euler number 720608    .  Do you agree?
 

answered Nov 6, 2016 by juancho (860 points) [ revision history ]
edited Nov 6, 2016 by juancho

If by 6-fold you mean of complex dimension 6, then it's correct. More generally, an hypersurface of degree $n+1$ in $CP^n$  is a Calabi-Yau of complex dimension $n-1$. Still more generally, complete intersections of degrees $(d_1,..., d_k)$ in $CP^n$ with $d_1+...+d_k=n+1$ are Calabi-Yau of complex dimensions $n-k$. For example, complete intersections of degrees $(2,7)$, $(3,6)$ or $(4,5)$  in $CP^8$ are Calabi-Yau of complex dimension $6$, with Euler numbers respectively $575876$, $325188$ and $185120$ (if I am not mistaken).

@40227, many thanks for your answer.  You are very right, the Euler numbers that you are providing are correct.  Now I will find examples with triple intersections.   All the best.

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