Its known ( see " The birational geometry of degenerations") that there exist a smooth one parameter family (i.e. total space is smooth) of two dimensional complex toris over unit disk whose central fiber is a normal crossing union of (say four for example) some copies of $\mathbb{P}^2$ blown up at 3 points of a triangle; i.e.

$$\pi:X \to \Delta \subset \mathbb{C},$$ such that $ X_t=\pi^{-1}(t), t\neq 0,$ is smooth and and is a complex tori; $X_0= \cup V_i$ and the singular locus of $X_0$ restricted to each $V_i$ is a cycle of 6 minus 1 curves. It is also known that in this case, the dual graph of $X_0$ is $S^1\times S^1$.

Authors mention that this degeneration can be realized via toroidal embeddings (I assume it means using toric varieties) but there is no explicit example.

Does any body know any explicit example, presenting such toroidal embedding?

Just to know: The total space $X$ mentioned above is a Kulikov model whose smooth fibers are complex tori

  • The paper of Mumford "An analytic construction of degenerating abelian varieties over complete rings" might be useful. – ulrich Mar 13 '13 at 5:04

Watch out: toroidal $\neq$ toric !

It is not possible to realize this situation in a toric variety, at least not so that $\pi$ is a toric morphism, because toric varieties are rational by definition and complex tori are not. (I am assuming that by complex torus you mean a compact quotient of $\mathbb C^g$).

A toroidal embedding is an open subset $U\subseteq X$ in a normal variety $X$, such that for every closed point $x\in X$ there exist a toric variety $\overline T$, a point $t\in\overline T$, and an isomorphism of complete local $k$-algebras $\widehat {\mathscr O}_{X,x}\simeq \widehat{\mathscr O}_{\overline T,t}$ such that the ideal of $X\setminus U$ maps isomorphically to the ideal of $\overline{T}\setminus T$.

In other words, a toroidal embedding is something that locally analytically looks like the embedding of the open dense alberaic torus of a toric variety. I suppose the reference you are citing meant that it is possible to make the $\pi^{-1}(t\neq 0)\hookrightarrow X$ embedding to be toroidal.

I think that Theorem 2.1 of Weak semistable reduction in characteristic 0 by Abramovich-Karu produces a toroidal embedding for you. If not, then it should at least give you an idea of how to do it. In fact, section 1 of that paper collects the basics about toroidal embeddings, so you should check it out anyway.

  • OK, I thought X can be embedded in some higher dimensional toric variety (not that I meant X-->\Delta is toric itself) Now can you say how an example like that can be built, that might be enlighting toward the definition of toroidal embeddings. – Mohammad F. Tehrani Mar 13 '13 at 4:12
  • @Mohammad: take a look at the reference I gave you. – Sándor Kovács Mar 13 '13 at 4:27
  • This and similar papers, even the "Toroidal embeddings I" book, talk about properties of a toroidal embeddings; non of them shows starting from a given polyhedra, how to build an embedding. I know that this is an open problem, but I am just looking for couple of examples for which the procedure is known, e.g. for the one mentioned in my question. – Mohammad F. Tehrani Mar 19 '13 at 19:29
  • Mohammad: this paper, in particular Thm 2.1 proves in general what you want to do. Take your example and run their proof through on your example. That should give you what you want. – Sándor Kovács Mar 19 '13 at 21:15

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