What is the first work that studies, refers to, or mentions the Hodge characteristic?
The Hodge polynomial is the unique ring homomorphism $$ P_{hdg}:K_0(\mathbf{Var}/\mathbb{C)}\to \mathbb{Z}[u,v,u^{-1},v^{-1}], $$ from the Grothendieck's ring of varieties to the ring of Laurent polynomials with two variables, that sends a smooth proper complex variety $X$ to the polynomial $$ P_{hdg}(X)=\sum_{p,q\geq 0} h^{p,q}(X) u^p v^q, $$ where $h^{p,q}(X)=\dim_\mathbb{C}\mathrm{H}^q(X,\Omega_X^p)$.
There is a more refined version of the above ring homomorphism, called the Hodge characteristic, whose target is the Grothendieck ring of Hodge structures $K_0(HS)$ instead of $\mathbb{Z}[u,v,u^{-1},v^{-1}]$.
I am mainly interested in references for the first occurrence of the Hodge characteristic. Still, references about the first occurrence of the Hodge polynomial would be appreciated.
The proof of the existence of these homorphisms is given in (Srinivas, 2002)'s The Hodge characteristic. From the introduction of Srinivas's work, I got the impression that the existence of these homomorphisms was only used by Kontsevich, but not proven before. Is that correct? If so, which one of Kontsevich's works does Srinivas refer to? I did not find it in the list of references, and Google was not helpful on this occasion.
On the other hand, Gillet and Soulé's 'Descent, motives and K-theory' indicates, on page 27 of the arXiv preprint and on page 28 of the published version, that the invariants $h^{p,q}(X)$ (for $X$ any complex variety) where already mentioned on page 191 in (Grothendieck,1989)'s 'Récoltes et semailles'. Which makes me wonder if the Hodge polynomial was implicitly/explicitly introduced/used before Kontsevich's proof.
Thank you in advance!
