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The more modern approach to the question adressed by Friedman is via logarithmic geometry. Most relevant for your question is the paper of Kawamata and Nammikawa, "Logarithmic deformations of normal c …

Provided that the facets of $P$ have integral normal vectors, this is always the case. To construct such a tropical variety, first let $\Sigma$ be the normal fan to $P$. This comes along with a piecew …

Yes, this is possible. For an example of a Calabi-Yau threefold with such differences of curves, see my paper with Pavanelli http://arxiv.org/pdf/math/0512182.pdf. I am sure there are much simpler exa …

For the first question, given a sharp monoid $P$ (i.e., the group of units of $P$ is the zero group), there is a log structure on any scheme $X$ given by $M_X={\mathcal O}_X^*\oplus P$ with the struct …

Let $X$ be a Calabi-Yau three-fold with only ordinary double points. One can always find a small resolution $Y\rightarrow X$ where $Y$ is a (not necessarily Kaehler) complex manifold. Let $C_1,\ldots, …

Let me go from the weakest to strongest sense in which the conjecture should be true. First, at the purely topological level, it is true for any Calabi-Yau variety with a toric degeneration whose dua …

I don't think I defined the Hodge numbers in this way. Rather, the argument in Section 1 shows that the Hodge numbers agree with the dimensions of the terms in the $E_2$ page of the Leray spectral seq …

I'll fill in a few details here; more can be found in the references that Daniel gave. Suppose first that $f:X\rightarrow B$ is a special Lagrangian $T^n$ fibration with only smooth fibres. If we jus …

Both the elliptic curve and K3 case can be calculated explicitly, the first more than the second. First let's recall how the structures arise. We have the 2-form $\omega$ and the $n$-form $\Im \Omeg …

It is true that if $X\subseteq Y$ is a normal crossings divisor, then $Y$ has a log structure whose sheaf of monoids is the sheaf of regular functions invertible outside of $X$. It is also true that t …

The crucial point for the second question is the following. In arbitrary dimension, it is not true that $\pi^{-1}(B_0)$ and $\pi^{-1}(B_0)^{\vee}$ are homeomorphic as fibre bundles. This is essentiall …