A Koszul complex in quaternionic analysis and its applications

Abstract

Let $n⩾1,\hat{I}\text{c◯}\subset {\mathbb{H}}^n$ be a domain. We construct a Koszul-type complex for the ideal sheaf ${\mathcal{I}}_X^{(k)}$ of $k$-regular functions vanishing on $X=\{(q_0,q_1,\cdots ,q_{n-1})\in \hat{I}\text{c◯}:q_0=0\}$ in several quaternionic variables: $$0\to {\mathcal{R}}^{(k+2)}\stackrel{{\tilde{\mathcal{L}}}^{(k)}}{\to }{\mathcal{R}}^{(k+1)}\oplus {\mathcal{R}}^{(k+1)}\stackrel{{\mathcal{L}}^{(k)}}{\to }{\mathcal{I}}_X^{(k)}\to 0,$$ where $k⩾0$, ${\mathcal{R}}^{(k)}$ is the sheaf of $k$-regular functions on $\hat{I}\text{c◯}$, ${\tilde{\mathcal{L}}}^{(k)}=(-L_1^{(k+2)},L_0^{(k+2)}{)}^T$, ${\mathcal{L}}^{(k)}=(L_0^{(k+1)},L_1^{(k+1)})$, 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 $\hat{I}\text{c◯}\cap \{q_0=0\}=\varnothing$ 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 $H^3(\hat{I}\text{c◯},\mathbb{R})=0$. Furthermore, a cohomological vanishing criterion is given for $H^1(\hat{I}\text{c◯},{\mathcal{I}}_X^{(k)})$; under this criterion, every $k$-regular function on $\{q_0=0\}\cap \hat{I}\text{c◯}$ extends to a $k$-regular function on $\hat{I}\text{c◯}$.