A Koszul complex in quaternionic analysis and its applications

Abstract

Let $n\geqslant 1, Ω\subset\mathbb{H}^n $ be a domain. We construct a Koszul-type complex for the ideal sheaf $\mathcal{I}_X^{(k)}$ of $k$-regular functions vanishing on $X=\{(q_0, q_1, \cdots, q_{n-1})\in Ω: q_0=0\}$ in several quaternionic variables: $$0\to \mathcal{R}^{(k+2)}\xrightarrow{\widetilde{\mathscr{L}}^{(k)}} \mathcal{R}^{(k+1)}\oplus\mathcal{R}^{(k+1)}\xrightarrow{\mathscr{L}^{(k)}} \mathcal{I}_X^{(k)}\to 0,$$ where $k\geqslant 0$, $\mathcal{R}^{(k)}$ is the sheaf of $k$-regular functions on $Ω$, $\widetilde{\mathscr{L}}^{(k)}=(-L_1^{(k+2)},L_0^{(k+2)})^{T}$, $\mathscr{L}^{(k)}=(L_0^{(k+1)},L_1^{(k+1)})$, and $L_0^{(k)},L_1^{(k)}$ are multiplication-like operators on $k$-regular functions. This gives the quaternionic analogue of the classical Koszul complex. And we present the long exact sequence in cohomology for the case $Ω\cap\{q_0=0\}=\emptyset$ with explicit differential connecting maps, by applying the Cauchy-Fueter complex and cohomological methods. As an application, in the special case $n=1, k=1$, the operator pair $(L_0^{(1)}, L_1^{(1)})$ is shown to be surjective if and only if $H^3(Ω, \mathbb{R})=0$. Furthermore, a cohomological vanishing criterion is given for $H^1(Ω,\mathcal{I}_X^{(k)})$; under this criterion, every $k$-regular function on $\{q_0=0\}\capΩ$ extends to a $k$-regular function on $Ω$.