# Examples of Maximal degeneration of Deligne on Calabi-Yau degeneration

Pierre Deligne in his celebrated paper entitling "Local behavior of Hodge structures at infinity" introduced Maximal degenerations of Calabi-Yau manifolds.

Let $\pi:X\to \mathbb C^*$ be a family of degeneration of of Calabi-Yau fibers. We have from D. Barlet asymptotic formula

$$\int_{X_s}\Omega_s\wedge\bar\Omega_s=C(\log |s|)^m|s|^{2k}(1+O(1))$$

for some $C \in \mathbb C^∗$, $k ∈ \mathbb Z, 0 \leq m \leq n = dim(X)$ and we can formulate $m$ via log-canonical threshold, see Proposition 2.1 of this paper of Mourougane

In another view we want the degeneracy index $m=δ(X,X_0)$ defined by Halle–Nicaise be exactly $n=dim X$

We say that $X$ has maximal degeneration at $s = 0$, if in the formula above we have $m = n$.

Later this definition become very important in Miror symmetry for example in Siebert-Gross program or Kontsevich-Soibelman program.

I am woundring if there is any concrete example for maximal degeneration on moduli space of Calabi-Yau fibers. Moreover, which properties are known that central fiber $X_0$ must have , to get such maximal degeneration ?

• OK, the definition of Deligne is slightly weaker than of this definition I mentioned. Moreover, I assumed Calabi-Yau manifold , Vadim Paper is on Calabi-Yau Scheme, which if we reduce his result over CY manifold we get nothing – user21574 Feb 3 '17 at 18:26
• If central fiber be snc then the monodromy representation is unipotent, but inverse is not correct in general. and hence Kontsevitch-Soiblman definition of maximal degeneracy is different from Deligne. Your examples does not work for Kontsvitch-Soibleman maximal degeneracy. – user21574 Feb 12 '17 at 1:17
• If $X$ is smooth and of dimension $n + 1$ and if you integrate a $\mathscr {C}^\infty$ form along the $\pi$ fibers the power of the logarithm can not exceed $n$ and it is unlikely maximal degeneracy led to GH limit and in fact GH limit is not depend to maximal degeneracy of asymptotic formula of integral of volumes on fibers. – user21574 Feb 12 '17 at 1:18
• Moreover, to have a non-zero term in $\vert s \vert^{2k.} (Log | s |)^n$ in the asymptotic formula of fiberwise integral of volume form the monodromy must have a block of Jordan of size $n-1$ for the eigenvalue 1 (on the cohomology of degree n of the Milnor fiber). This corresponds to a block of Jordan unipotent of maximum size. – user21574 Feb 12 '17 at 1:20
• Kontsevich-Soibelman program is unlikely to be correct for Gromov-Hausdorff limit of Calabi-Yau varieties – user21574 Feb 12 '17 at 1:25