My article on the subject:
To summarize:
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If we have “var” fields then the type of the field cannot be determined until the expression is analyzed, and that happens after we already need to know the type of the field.
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What if there are long chains, or even cycles in those references? All of those algorithms would have to be rewritten and tested in a world where top-level type information is being determined from them rather than being consumed by them.
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If you have “var” fields then the initializer could be of anonymous type. Suppose the field is public. There is not yet any standard in the CLR or the CLS about what the right way to expose a field of anonymous type is.