Toroidal embedding

Question

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

Answer 1

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.

Answer 2

naf's comment is correct. This example can be seen as an instance of Mumford's construction of degenerations of abelian varieties into unions of toric varieties. Here is a "purely toric" construction of such a degeneration, where the analytic toroidal structure is explicit (you may already know this construction, but I thought it would be useful to post for completeness):

Take the $\mathbb{Z}^2$-periodic tiling of $\mathbb{R}^2$ by the translates of the triangulation of the unit square $[0,1]\times[0,1]$ into two triangles, by decomposing the square along its diagonal. Let's call this tiling $\mathcal{T}$. Now take $\mathcal{F}:={\rm Cone}(\mathcal{T})$ which results from placing this tiling of $\mathbb{R}^2=\{(x,y,1)\in \mathbb{R}^3\}$ at height $1$ in $\mathbb{R}^3$, then taking the positive span of each triangle in $\mathbb{R}^3$. The result is a decomposition of the positive half-space $\{(x,y,z)\,:\,z>0\}$ into standard affine cones, i.e. cones which are $SL_3(\mathbb{Z})$-equivalent to the positive octant.

Then $\mathcal{F}$ is a normal fan and so the resulting toric variety $\widetilde{X}:=X(\mathcal{F})$ is smooth. Note that it is an complex-analytic toric manifold, but not an algebraic variety, because it has infinitely many toric affine charts. The map $\mathbb{R}^3\to \mathbb{R}$ which projects to the third coordinate induces a map of fans $\mathcal{F}\to \mathcal{G}$ where $\mathcal{G}$ is the fan in $\mathbb{R}$ consisting of the positive axis. Hence, there is a torus-equivariant holomorphic map $\widetilde{X}\to \mathbb{C}$. The fiber $\widetilde{X}_0$ over $0\in \mathbb{C}$ is the toric boundary of $\widetilde{X}$. It consists of an infinite $\mathbb{Z}^2$-periodic quilt of Cremona surfaces as in your question. This is easily seen by observing that the quotient fan at any ray of $\mathcal{F}$ is the fan of the Cremona surface.

Now observe that $\mathbb{Z}^2$ acts by translation on $\mathbb{R}^2$ and this action extends to an action $\mathbb{Z}^2\to SL_3(\mathbb{Z})$ by integral-affine shearing maps on $\mathbb{R}^3$. Furthermore, this action preserves both the fan $\mathcal{F}$ and the projection to $\mathcal{G}$. So we get an action of $\mathbb{Z}^2$ on $\widetilde{X}$ by biholomorphisms, which preserves the map to $\mathbb{C}$. Furthermore, by the above description, the action is easily seen to be free on $\widetilde{X}_0$. It is easily checked that there is an analytic disc $\Delta\subset \mathbb{C}$ around the origin over which the action of $\mathbb{Z}^2$ is free and properly discontinuous. Let's call the inverse image of this disc in $\widetilde{X}$ by $\widetilde{X}^{<1}$.

Now take the quotient $X:=\widetilde{X}^{<1}/(2\mathbb{Z})^2$. It has a holomorphic map $X\to \Delta$, and the central fiber $X_0$ is the quotient of the $\mathbb{Z}^2$-periodic quilt of Cremona surfaces by the action of $(2\mathbb{Z})^2$. So its central fiber has dual complex $S^1\times S^1$. The space $X$ is a smooth complex threefold, because it was the free quotient of a group of biholomorphisms. To understand the general fiber $X_t$ for $0\neq t\in \Delta$, we know that the fiber $\widetilde{X}_t$ was $(\mathbb{C}^*)^2$ and that $\mathbb{Z}^2$ acted on this fiber freely by translations. The quotient is therefore a two-dimensional complex torus (in fact, an abelian variety).

The toric charts on $X$ are clear from this picture, as $X$ was constructed as the free quotient of an open subset of a toric variety.