6.3 Some conjectures about singular sets [03UV]

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6.3 Some conjectures about singular sets

Our conjectures are in fact rather “wishes”, i.e. they are desired properties of $B^{sing}=B\setminus B^{sm}$ . For simplicity we assume that $B^{sing}$ is a stratified set (say, CW complex) of dimension less or equal than $n-1$ .

Conjecture 5

We have a decomposition $B^{sing}=B^{sing}_{n-1}\cup B^{sing}_{\leq n-2}$ , where $B^{sing}_{\leq n-2}$ consists of strata of dimension less or equal than $n-2$ , $B^{sing}_{n-1}$ is the union of strata of dimension $n-1$ , and locally near every point $x\in B^{sing}_{n-1}$ the ${\bf Z}$ -affine structure is modeled by the “book” $\cup_{i\in I}{{\bf R}}^{n-1}\times{{\bf R}}_{\geq 0}$ . Here $I$ is a finite set, all half-spaces have a common plane ${{\bf R}}^{n-1}\times\{0\}$ and $x$ belongs to this plane. ${\bf Z}$ -affine structure on $B^{sm}=\sqcup_{i\in I}{{\bf R}}^{n-1}\times{{\bf R}}_{>0}$ is the natural one.

This conjecture gives a local model for a singular ${\bf Z}$ -affine structure at a singular component of codimension one. Let us discuss the case of higher codimension. We start with the following definition.

Definition 7

A ${\bf Z}$ -affine structure with singularities on $B$ is given by:

1.

a closed subset $B^{pre-sing}\subset B$ of a compact space $B$ ;
2.

a ${\bf Z}$ -affine structure on the open set $B\setminus B^{pre-sing}$ .

One can think about closed set of “potential singularities” $B^{pre-sing}$ as containing the actual set of singularities $B^{sing}$ ).

Definition 8

A continuous path $\gamma(t),t\in[0,1]$ in the space of ${\bf Z}$ -affine structures with singularities on a given compact space $B$ is given by:

1.

a continuous path $B_{t}^{pre-sing}$ in the space of all compact subsets of $B$ ,
2.

a ${\bf Z}$ -affine structure on $B\setminus B_{t}^{pre-sing}$ for all $t\in[0,1]$

Notice that for each $t_{0}\in(0,1)$ and $x_{0}\in B\setminus B_{t_{0}}^{pre-sing}$ we can choose neighborhoods $U_{t_{0}}$ of $t_{0}$ and $U_{x_{0}}$ of $x_{0}$ such that $U_{x_{0}}\subset B\setminus B_{t}^{pre-sing}$ for all $t\in U_{t_{0}}$ . Then we require that:

3.

if $U_{t_{0}}$ and $U_{x_{0}}$ are sufficiently small then the induced ${\bf Z}$ -affine structure on $U_{x_{0}}$ does not depend on $t\in U_{t_{0}}$ .

Notice that in the case when the homotopy type of $B\setminus B_{t}^{pre-sing}$ remains unchanged the representation $\rho_{t}:\pi_{1}\left(B\setminus B_{t}^{pre-sing}\right)\to GL(n,{\bf Z})\ltimes{\bf R}^{n}$ stays the same.

We are going to give an example of a non-trivial path in the next subsection. We expect that singularities which appear in the collapse of Calabi-Yau manifolds satisfy the following

Conjecture 6

If $B^{sing}=B^{pre-sing}$ is of codimension at least two in $B$ , then there is a continuous path $\gamma(t)$ in the space of ${\bf Z}$ -affine structures with singularities which connects a given structure with the one coming from a PL compactification, and such that for all $t\in[0,1]$ we have $codim(B_{t}^{pre-sing})\geq 2$ and $\gamma(t)$ has Finiteness and Independence properties.