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Let X be a Calabi-Yau 3-fold, and $D\subset X$, smooth,Fano divisor. Since $K_X=0$ we have $N_D^X=K_D$. I have seen the following fact in many papers:

By deforming X within Kahler moduli, we can shrink D to a point to get a singular 3-fold Y.

Here are my questions?

1-Why is that possible?(Why it is not possible for non-Fano surfaces)

2-How can we find the type of singularity?(Explicitly for a given surface D)

I have to mention that problem is local, you can consider the total space of $K_D$ as local model for neighborhood of D, instead of whole X.

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2 Answers 2

up vote 5 down vote accepted

Corrected. In the first version I stated that CY 3-fold containing Fano surfaces are unknown, this is not at all true as Mohammad pointed out. For example, one can take a product $E\times E\times E=X$ with $E$ an elliptic curve admitting a $\mathbb Z_3$ action, and then take crepant resolution of the quotient of diagonal action $X/\mathbb Z_3$. On the resolved manifold there will be $27=3^3$ copies of $\mathbb CP^2$. On the contrary, as Zhiyu pointed out, on smooth quintic in $\mathbb CP^4$ one can not find a plane $\mathbb CP^2$, since the second integral co-homology of the quintic is generated by a hyperplane section (in the previous "correction" of this answer I was claiming the opposite).

The rest of the answer was correct, and here it is.

It is also important that $D$ is Fano. In this case the normal budnle to it is negative (i.e. its inverse is ample), and so we can contract holomorphically the divisor by Grauert theorem. Notice that you can find a Kahler form that shrinks $D$ to zero only if it can be contracted holomorphically.

In order to see what singularity you get you just need to study case by case Fano surfaces. For example in the case of $\mathbb CP^2$ you get obrifold singularity $\mathbb C^3/\mathbb Z_3$ (with diagonal action of $\mathbb{Z}_3$ on $\mathbb C^3$). In the case of $\mathbb CP^1\times \mathbb CP^1$ you get the sinuglariy $x^2+y^2+z^2+t^2=0 / \pm 1$. In the case when $D$ is a cubic surface you get just $P_3(x,y,z,t)=0$, where $P_3$ is a homogenious polynomial of degree $3$. All other cases of cours were studied and some descripiton of singularities exists. In praticular you can find the model of the singularity if you construct an anti-canonical embedding of your Fano surface in some $\mathbb CP^n$.

No, why can you deform the Kahler form? This is because on the total space of the canonical bundle of a Fano surface the set of Kahler forms is connected, and in particular some of these forms are in the cohomology class $-D+\pi^*\omega$, where D is the zero section, and $w$ is the pullback of a Kahler form from the total space of the $K$ with zero section contracted. Replacing $-D$ with $-tD$ $(0< t\le 1)$ you get a one parameter family cohomology classes that can be represented by Kahler forms. Finally, for $t=0$ you get a cohomology class vanishing on $D$ (by definition).

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The examples of such CY's are known, for example see a paper of Yau and others on Local Gromov witten invariants. –  Mohammad F. Tehrani May 30 '10 at 5:30
    
Thank Mohammad, you are comletely right, I did not thought enough –  Dmitri May 30 '10 at 9:33
    
Where can I find the statement and the proof of Grauert theorem? –  Mohammad F. Tehrani Jun 2 '10 at 2:32

You cannot find a smooth quintic containing $P^2$ since by Lefschetz hyperplane theorem every divisor in the quintic is a complete intersection of the quintic with a hyperplane.

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Zhiyu, thanks for this remark! –  Dmitri May 30 '10 at 17:02

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