First-order families of positive real linear monads

Use the dual-number scheme, relative projective-space charts, finite locally free sheaves and twisting sheaves as defined, respectively, in the four suppliers listed under Definition references, with the inherited AC convention for the twisting-sheaf construction. Let $A=\mathbb{C}[ϵ]\mathrm{/}({ϵ}^2)$ and equip it with the conjugate-linear involution $\overline{a+bϵ}=\bar{a}+\bar{b}ϵ$. Its reduction is the quotient map $A\to \mathbb{C}$ sending $ϵ$ to zero. For every $r\in \mathbb{R}$, the map ${\rho }_r:A\to A$ fixing $\mathbb{C}$ and sending $ϵ$ to $rϵ$ preserves this involution.

A first-order family of positive real linear monads of dimensions $(k,2k+N,k)$ is a three-term complex of locally free sheaves on ${\mathbb{P}}_A^3=\mathrm{Proj}A[{\lambda }_1,{\lambda }_2,{\mu }_1,{\mu }_2]$, $\mathcal{O}(-1){\otimes }_A{U}_A\stackrel{{\alpha }_A}{\to }\mathcal{O}{\otimes }_A{S}_A\stackrel{{\beta }_A}{\to }\mathcal{O}(1){\otimes }_A{U}_A^{\mathrm{\prime }},$ where $U_A,S_A,U_A^{\mathrm{\prime }}$ are free $A$-modules of ranks $k,2k+N,k$, each map is homogeneous linear in $({\lambda }_1,{\lambda }_2,{\mu }_1,{\mu }_2)$, ${\beta }_A{\alpha }_A=0$, ${\alpha }_A$ is a locally split injection, ${\beta }_A$ is a locally split surjection, and $F_A=\ker {\beta }_A\mathrm{/}\mathrm{im}{\alpha }_A$ is locally free of rank $N$. The two short exact sequences defined by the kernel and image split locally; hence their formation and the middle cohomology commute with reduction $A\to \mathbb{C}$. The reduced complex is therefore an ordinary linear monad on ${\mathbb{P}}_{\mathbb{C}}^3$.

The first-order locally-free/base-change range used here further requires that $H^0({\mathbb{P}}_A^3,F_A)$ is a free $A$-module and reduction identifies it with $H^0({\mathbb{P}}_{\mathbb{C}}^3,F_0)$, where $F_0=F_A{\otimes }_A\mathbb{C}$. The image in $S_A$ of the map from the four coefficient copies of $U_A$ under ${\alpha }_A$ is a direct summand and reduces to the corresponding coefficient-image subspace of $S_0$. For each real twistor line $\mathrm{\ell }$ in the reduced projective space, the restriction of $F_A$ to ${\mathrm{\ell }}_A=\mathrm{\ell }{\times }_{\mathbb{C}}\mathrm{Spec}A$ is trivial of rank $N$, and reduction identifies its global sections with $H^0(\mathrm{\ell },F_0{\mathrm{\mid }}_{\mathrm{\ell }})$. These conditions state the precise global, coefficient-image, and line-section base-change range; no arbitrary analytic family or algebraization assertion is included.

Extend the coordinate map $j({\lambda }_1,{\lambda }_2,{\mu }_1,{\mu }_2)=(-{\bar{\lambda }}_2,{\bar{\lambda }}_1,-{\bar{\mu }}_2,{\bar{\mu }}_1)$ semilinearly using the involution of $A$; write $\tilde{\sigma }$ for its projective action over $A$ and $\sigma$ for its reduction. Since $j^2=-1$, both projective actions square to the identity and preserve each real twistor line. A real twistor line means the projectivization of a $j$-invariant complex two-plane in ${\mathbb{C}}^4$. Equivalently these are the finite lines ${\mu }_1=z_1{\lambda }_1-{\bar{z}}_2{\lambda }_2,{\mu }_2=z_2{\lambda }_1+{\bar{z}}_1{\lambda }_2,(z_1,z_2)\in {\mathbb{C}}^2,$ together with ${\mathrm{\ell }}_{\mathrm{\infty }}=\{{\lambda }_1={\lambda }_2=0\}$. Indeed a nonzero vector $v$ and $jv$ are complex-linearly independent because $j^2=-1$; they span the invariant plane. If its $\lambda$ projection is nonzero, that projection and its quaternionic conjugate give an invertible two-by-two matrix with determinant $\mathrm{\mid }{\lambda }_1{\mathrm{\mid }}^2+\mathrm{\mid }{\lambda }_2{\mathrm{\mid }}^2$, so it has exactly the displayed graph form; if the projection is zero the plane is the $\mu$ plane. Each fixed-line restriction above means the corresponding constant thickening ${\mathrm{\ell }}_A$.

For an $A$-locally free sheaf $G$, write $D_A(G):=({\tilde{\sigma }}^{\ast }\bar{G}{)}^{\vee }$, where $\bar{G}$ has its scalar action conjugated by $a\mapsto \bar{a}$ and $(-{)}^{\vee }=\mathcal{H}o{m}_{\mathcal{O}}(-,\mathcal{O})$. On the reduction, write $D(F_0):=({\sigma }^{\ast }\overline{F_0}{)}^{\vee }$. A first-order real structure is an ${\mathcal{O}}_{{\mathbb{P}}_A^3}$-linear isomorphism $h_A:F_A\to D_A(F_A)$ fixed by the canonical double-dual identification $D_A^2(F_A)\cong F_A$. Its reduction $h_0:F_0\to D(F_0)$ gives fiber pairings $b_x(v,w):=h_{0,x}(v)(\bar{w}),v\in (F_0{)}_x,w\in (F_0{)}_{\sigma x},$ which are linear in $v$, conjugate-linear in $w$, and satisfy $b_{\sigma x}(w,v)=\overline{b_x(v,w)}$. For a $\sigma$-invariant real line $\mathrm{\ell }$ and $s,t\in H^0(\mathrm{\ell },F_0{\mathrm{\mid }}_{\mathrm{\ell }})$, the induced line form is the constant value of the function $x\mapsto b_x(s(x),t(\sigma x))$; constancy is part of the stated real-structure condition. It is first-slot-linear and Hermitian by the displayed symmetry. Positivity means $b_{\mathrm{\ell }}(s,s)>0$ for every nonzero $s$. A unitary framing at infinity is an isometric isomorphism $F_A{\mathrm{\mid }}_{{\mathrm{\ell }}_{\mathrm{\infty }}}\cong (A{\otimes }_{\mathbb{C}}W){\otimes }_A{\mathcal{O}}_{{\mathrm{\ell }}_{\mathrm{\infty }}}$, where ${\mathrm{\ell }}_{\mathrm{\infty }}=\{{\lambda }_1={\lambda }_2=0\}$ and $W={\mathbb{C}}^N$ has its standard first-slot-linear Hermitian form.

An isomorphism of such families is an $A$-linear chain isomorphism that intertwines the $\sigma$-Hermitian pairings and the framing. No order or positivity is assigned to elements of $A$ with nonzero nilpotent part: positivity is required only after reduction. The base-change scope in this definition consists exactly of reduction $ϵ\mapsto 0$ and the involution-preserving maps ${\rho }_r$ for real $r$.

Brantner, Lecture 1, “First order deformations”

The Stacks Project, Definition 33.16.1

The affine scheme of dual numbers

Relative projective space from standard charts

Locally free sheaves of finite rank

Twisting sheaf on Proj