Toroidal compactifications

Does anyone know if there is an intrinsic definition of a toroidal compactification (over $\mathbb{C}$)?

Something like: Let $X$ be an algebraic variety over $\mathbb{C}$. Then $X \subset \bar{X}$ is a toroidal compactification if for every $x \in \bar{X}\setminus X$ there is an analytic neighborhood $N(x)$ of $x$ such that $N(x)$ is isomorphic to a toric variety. If there is, could someone give me any reference for it?

• What is intrinsic? Isn't your definition intrinsic enough? – Alex Degtyarev Dec 6 '14 at 19:18
• I don't know if they one I wrote down is correct. This is the definition of a toroidal embedding. Is every toroidal compactification a toroidal embedding? – cata Dec 6 '14 at 20:17
• This is a great question, I don't know if I've ever seen a definition written down. Certainly there should be a toroidal embedding $X \hookrightarrow \bar{X}$, which is the definition you wrote down. Of course one should also have $\bar{X}$ be compact (proper, if not over $\mathbb{C}$). I don't believe there are any other requirements, although here I am not too certain. – rghthndsd Dec 7 '14 at 2:24