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Donaldson-Thomas invariant is expressed as the weighted Euler characteristicof the so-called Behrend (constructible) function. In \cite{Behrend} Behrendintroduced a DT-type invariant for a morphism. Motivated by this invariant, weextend the motivic Hirzebruch class to naive Donaldson-Thomas type analogues.We also discuss a categorification of the DT-type invariant for a morphism froma bivariant-theoretic viewpoint, and we finally pose some related questions forfurther investigations.

Naive motivic Donaldson-Thomas type Hirzebruch classes and some problems