Csdrhrt

In differential geometry/topology one can construct, given a function space (or fibre bundle), the jet space (or bundle) by considering the k-th order Taylor expansions.

Now I was wondering if something similar exists with respect to Laurent series for complex functions/sections?

If $M^{2m}$ and $N^{2n}$ are smooth manifolds, then the $k$-jet space $J^k(M, N)$ is an affine bundle over $M times N$ with fiber over $(x, y)$ being the space of $k$-order Taylor expansion of smooth functions (or rather, smooth germs) $f : U_x to V_y$ where $U_x$ and $V_y$ are local charts around $x$ and $y$ respectively. This is noncanonically isomorphic to the vector space $J^k(Bbb R^{2m}, Bbb R^{2n})$, or $k$-order Taylor polynomials of maps $Bbb R^{2m} to Bbb R^{2n}$ at the origin.

Suppose $M$ and $N$ admit and has been specified individual complex structures. Then there is a subspace $mathscr{R}^k subset J^k(M, N)$ such that $mathscr{R}^k$ similarly fibers over $M times N$ with fibers being the subspace $J^k_c(Bbb C^m, Bbb C^n)$ of $J^k(Bbb R^{2m}, Bbb R^{2n})$ consisting of $k$-order Taylor polynomials of holomorphic functions $Bbb C^m to Bbb C^n$.

Funnily enough, this space is not as huge as you think: the whole Taylor series of a holomorphic function is determined by it's $1$-jet. That is, if $f$ is germ of a holomorphic function $U_x subset M to V_y subset M$, which is equivalent to the "Cauchy-Riemann differential relation" $j^1 f in mathscr{R}^1$ (note here that $mathscr{R}^1$ is cut out by polynomial equations of the form $partial u_i/partial x_lambda = partial v_i/partial y_mu$ and $partial u_i/partial y_lambda = -partial v_i/partial x_mu$, i.e., the Cauchy-Riemann equations), then the $k$-jet prolongation $mathscr{R}^1 to mathscr{R}^k$ sending $j^1 f$ to $j^k f$ is a local parametrization for any $k$.

On a different note, it's interesting to ask when this particular differential relation (or the "complex jet space" you wanted) admits an $h$-principle. Namely, the space of holomorphic maps $C^omega(M, N)$ are precisely base of holonomic sections $s : M to J^1(M, N)$ such that $s(M) subset mathscr{R}^1$. When is the inclusion $text{Hol}(mathscr{R}^1) to text{Sec}(mathscr{R}^1)$ of the space of holonomic (or integrable) sections into the full space of sections a homotopy equivalence? I believe this is true if $M$ is a Stein manifold which in $dim M = 2$ is equivalent to $M$ being a noncompact Riemann surface, and that seems to be the appropriate context for $h$-principles to hold (in general noncompactness in the domain is very necessary for any sort of $h$-principle to hold). Maybe experts can comment more on this.