Let $E\to X$ be a holomorphic vector bundle. I wonder whether there exists a way to compute a representative of the Todd class $Td(E)\in H^*(X)$ of $E$ (i.e. a proper differential form in $\Omega^*(X)$).
For example, we can compute a representative of the Chern character $ch(E)$ by choosing a metric and computing $\exp(\frac{i}{2\pi}F)$ for the associated Chern connection. Now the Todd class is constructed using the splitting principle and so somme injective map between cohomology rings. Therefore I do not see how to get a concrete representative. Is there indeed a way?
Consider the function on $M_n(\mathbb{C})$ defined for $t\in \mathbb{R}-\{0\}$ by $$B\mapsto \frac{\det(tB)}{\det(I-e^{-tB})}.$$ that we write it as a power series in $t$ $$\frac{\det(tB)}{\det(I-e^{-tB})}=\sum T_k(B)t^k.$$ Let $(E,\nabla)\to X$ be a vector bundle endowed with a connection, we put $$td_k(E,\nabla):=T_k\left(\frac{i}{2\pi}F_{\nabla}\right)$$ and $$td(E,\nabla)=\sum td_k(E,\nabla)=\frac{\det\left(\frac{i}{2\pi}F_{\nabla}\right)}{\det\left(I-\exp(-\frac{i}{2\pi}F_{\nabla})\right)}.$$