Unique chain lifting and sigma-Hermitian monad duality

Put $K=\ker \beta$ and $K^{\mathrm{\prime }}=\ker {\beta }^{\mathrm{\prime }}$. Local split exactness gives short exact sequences of holomorphic vector bundles $0\to K\to S\otimes \mathcal{O}\to U^{\mathrm{\prime }}\otimes \mathcal{O}(1)\to 0,0\to U\otimes \mathcal{O}(-1)\to K\to F\to 0$ and the corresponding primed sequences.

For a degree-one matrix $P(z)$ set $P^{\mathrm{\#}}(z)=P(jz{)}^{†}$. Since $j$ is antilinear and $j^2=-1$, $P^{\mathrm{\#}}$ is holomorphic linear and $(P^{\mathrm{\#}}{)}^{\mathrm{\#}}=-P$. With $D(\alpha ,\beta )=(-{\beta }^{\mathrm{\#}},{\alpha }^{\mathrm{\#}})$, direct substitution gives $D^2(\alpha ,\beta )=(-({\alpha }^{\mathrm{\#}}{)}^{\mathrm{\#}},(-{\beta }^{\mathrm{\#}}{)}^{\mathrm{\#}})=(\alpha ,\beta )$. The signed dual monad is

Here $(\bar{E}{)}^{\vee }$ is the complex-linear dual of the conjugate coefficient space; $†$ is the matrix of this conjugate-dual operation in the chosen bases. Ordinary duals in the Hom calculations below are denoted $E^{\ast }$. A $\sigma$-Hermitian self-pairing is a bundle isomorphism $h_F:F\to D(F)$ fixed by this bidual operation.

For holomorphic vector bundles $E,T$, define $\mathcal{H}om(E,T)$ on an open set to consist of holomorphic fiberwise complex-linear bundle maps. In the frames of [F1] these are holomorphic matrices, transforming by the target transition matrix on the left and the inverse source transition matrix on the right; this identifies $\mathcal{H}om(E,T)=E^{\ast }\otimes T$. Split local frames show that $\mathcal{H}om$ preserves each locally split short exact bundle sequence, covariantly in its second variable and contravariantly in its first. Degree-zero sheaf cohomology is global sections by the kernel at the start of an injective resolution, so $H^0(\mathcal{H}om(E,T))={\mathrm{Hom}}_{\mathrm{h}\mathrm{o}\mathrm{l}}(E,T)$. Finite constant coefficient factors give finite direct sums, whose cohomology is the corresponding finite sum: finite sums of injective sheaves remain injective since $\mathrm{Hom}(-,{⨁}_j{I}_j)={⨁}_j\mathrm{Hom}(-,I_j)$ for finitely many $j$; taking a finite direct sum of injective resolutions and applying the additive global-sections functor proves the assertion, with independence from the supplied replacement by [F3]. Put $T=U_1\otimes \mathcal{O}(-1)$ and $\mathcal{A}=\mathcal{H}om(K,T)=K^{\ast }\otimes U_1\otimes \mathcal{O}(-1)$. Dualizing the unprimed first sequence in step 1.1 and tensoring by $T$ gives $0\to (U^{\mathrm{\prime }}{)}^{\ast }\otimes U_1\otimes \mathcal{O}(-2)\to S^{\ast }\otimes U_1\otimes \mathcal{O}(-1)\to K^{\ast }\otimes U_1\otimes \mathcal{O}(-1)\to 0.$ By [F2] and the finite-sum observation, the middle sheaf has zero $H^0,H^1$ and the left sheaf has zero $H^1,H^2$. The long exact sequence [F3] therefore gives $H^0(\mathcal{A})=H^1(\mathcal{A})=0$. Apply $\mathcal{H}om(K,-)$ to the locally split primed second sequence of step 1.1 to obtain $0\to \mathcal{A}\to \mathcal{H}om(K,K^{\mathrm{\prime }})\to \mathcal{H}om(K,F^{\mathrm{\prime }})\to 0$. Its long exact sequence [F3] has segment $0\to H^0(\mathcal{A})\to {\mathrm{Hom}}_{\mathrm{h}\mathrm{o}\mathrm{l}}(K,K^{\mathrm{\prime }})\to {\mathrm{Hom}}_{\mathrm{h}\mathrm{o}\mathrm{l}}(K,F^{\mathrm{\prime }})\to H^1(\mathcal{A})$. Both outer groups vanish, so the indicated map is bijective. Consequently the specified composite $K\to F\stackrel{f}{\to }{F}^{\mathrm{\prime }}$ has a unique holomorphic lift $\varphi :K\to K^{\mathrm{\prime }}$.

Set $T=S_1\otimes \mathcal{O}$ and compose $\varphi :K\to K^{\mathrm{\prime }}$ with the primed inclusion $K^{\mathrm{\prime }}\hookrightarrow T$. To extend this map across the unprimed inclusion $K\hookrightarrow S\otimes \mathcal{O}$, apply $\mathcal{H}om(-,T)$ to the unprimed first sequence of step 1.1. Its local splitting and the frame calculation in step 2.1 give $0\to \mathcal{B}\to \mathcal{H}om(S\otimes \mathcal{O},T)\to \mathcal{H}om(K,T)\to 0$, where $\mathcal{B}=\mathcal{H}om(U^{\mathrm{\prime }}\otimes \mathcal{O}(1),T)=(U^{\mathrm{\prime }}{)}^{\ast }\otimes S_1\otimes \mathcal{O}(-1)$. The finite-coefficient vanishings [F2] give $H^0(\mathcal{B})=H^1(\mathcal{B})=0$. The long exact sequence [F3] therefore makes the restriction map ${\mathrm{Hom}}_{\mathrm{h}\mathrm{o}\mathrm{l}}(S\otimes \mathcal{O},T)\to {\mathrm{Hom}}_{\mathrm{h}\mathrm{o}\mathrm{l}}(K,T)$ bijective. It follows that there is a unique holomorphic middle map $g:S\otimes \mathcal{O}\to S_1\otimes \mathcal{O}$ whose restriction to $K$ is the inclusion composed with $\varphi$.

The restriction $g\alpha$ lands in $\mathrm{im}{\alpha }^{\mathrm{\prime }}$ because its class in $F^{\mathrm{\prime }}$ is $f$ applied to the zero class of $\alpha$. Since ${\alpha }^{\mathrm{\prime }}$ is injective, this defines a unique map $p:U\otimes \mathcal{O}(-1)\to U_1\otimes \mathcal{O}(-1)$. Since ${\beta }^{\mathrm{\prime }}g$ vanishes on $K$, it factors uniquely through $(S\otimes \mathcal{O})\mathrm{/}K\cong U^{\mathrm{\prime }}\otimes \mathcal{O}(1)$, defining the third chain map. The chain equations hold by construction. Every matrix entry of these maps is a global holomorphic section of $\mathcal{O}$, hence is constant by the stated hypothesis $H^0({\mathbb{P}}^3,\mathcal{O})=\mathbb{C}$. Any other lift agrees on $K$ by step 2.1 and on $S$ by step 3.1; injectivity of ${\alpha }^{\mathrm{\prime }}$ and surjectivity of $\beta$ then force equality of both end maps.

By step 4.1, $h_F$ has a unique chain lift $h=(p,H,r):M\to D(M)$. The displayed diagram records its two chain equations; it is notation only, and the equations and uniqueness are established in the text. Thus the squares assert $H\alpha =-{\beta }^{\mathrm{\#}}p$ and $r\beta ={\alpha }^{\mathrm{\#}}H$. Its dual $D(h)$ lifts the same self-pairing, so uniqueness gives $h=D(h)$; in particular $H=H^{†}$ and the end maps are adjoints under the pairing's identifications. The lift is an isomorphism: lift $h_F^{-1}$ by step 4.1 and use uniqueness to identify both composites with the identity lift. The middle map $H$ is therefore nondegenerate. In bases identifying the end spaces by this lift, the chain equation is $H\alpha (z)=-\beta (jz{)}^{†}$. Replacing $z$ by $jz$ and using linearity in homogeneous coordinates yields both displayed identities in the Statement.