Can I define an abstract class for all derived Singletons in this way?

Self implemented singletons are an anti-pattern. The need to inherit, and lock your classes into a specific form goes away if you just implement a factory.

No you can't because when you want to use new T() you should have a public constructor for it, and it's different from singleton definition. Because in singleton you should have a private constructor, and in this case (public one) everyone can create new instance of your object and it's not a singleton.

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