Let $X$ be a compact Kähler manifold, then there exists a canonical Spin$^\mathbb{C}$ bundle on $X$ for any given complex line bundle $L$: \begin{align*} S^+=\bigoplus_i\Lambda^{2i}(X;L)\text{ and } S^-=\bigoplus_i\Lambda^{2i+1}(X;L) \end{align*} And once we put a metric $A$ on $L,$ we get a dirac operator \begin{align*} D:=\sqrt{2}(\bar\partial_A+\bar\partial^*_A):\Gamma(S^+)\rightarrow \Gamma(S^-) \end{align*}
All the languages available mention that the index of this operator is \begin{align*} \int_X \text{ch}(L)Td(X) \end{align*} and this is equal to \begin{align*} \sum\limits_i (-1)^i\text{ dim}_{\mathbb{C}}H^i(X,L) \end{align*} by Hirzebruch–Riemann–Roch theorem. Now I have some doubts here, any clarification is most welcome.
So, what is Td$(X)$ here is it Td$(T^{1,0}(X))?,$ the Todd class of the holomorphic tangent bundle of $X$ or Todd class of the complexified tangent bundle of $X$? Secondly, this index is supposed to agree with the usual index I know about i.e., \begin{align*} \text{dim$_\mathbb{R}$ Ker}(D)-\text{dim$_\mathbb{R}$ Coker}(D) \end{align*} Is that correct or I am supposed to count the dimension w.r.t. $\mathbb{C}$ everywhere?