Positive real linear monad normal forms

First the holomorphic monad maps have homogeneous linear entries. Indeed, in the $m=1$ sequence of [F10], $H^1({\mathbb{P}}^r,\mathcal{O})=0$ by [A1], so restriction $H^0({\mathbb{P}}^r,\mathcal{O}(1))\to H^0({\mathbb{P}}^{r-1},\mathcal{O}(1))$ is surjective and its kernel is the defining hyperplane coordinate times $H^0(\mathcal{O})=\mathbb{C}$. Starting with $r=0$, where $\mathcal{O}(1)$ on a point has one-dimensional section space, induction shows that every section is a linear combination of the homogeneous coordinates: subtract the global coordinate lifts of its restriction, then the remaining section is a constant multiple of the defining coordinate. This applies to entries of both $\mathrm{Hom}(\mathcal{O}(-1),\mathcal{O})$ and $\mathrm{Hom}(\mathcal{O},\mathcal{O}(1))$. Thus no algebraic/analytic comparison is needed to introduce their four coefficient maps. By [A1] and [F9], the hyperplane sequences and analytic scalar theorem give exactly the negative-twist vanishings required by [F1]. Apply [F1] to the pairing isomorphism; it gives the unique constant Hermitian lift $H$ with the signed identities there. Thus no algebraic cohomology group is being substituted for the analytic lifting groups.

The first-order families here are the algebraic locally free sheaves on ${\mathbb{P}}_A^3$ specified in [F6]. The algebraic cohomology calculation [F11] gives $H^0({\mathbb{P}}_A^3,\mathcal{O})=A$ and the required vanishing of $\mathcal{O}(-1)$ and $\mathcal{O}(-2)$ over $A$, with the monomial modules and maps commuting with coefficient homomorphisms. The lifting calculation in [F1] uses only locally split kernel/image sequences, the long exact sequence, these cohomology groups, and the fact that global maps between equal twists are their constant coefficient matrices. The same calculation over $A$ therefore lifts the family pairing uniquely to an $A$-linear constant chain isomorphism with Hermitian middle matrix $H_A$; its inverse exists because its reduction is invertible. In both kernel/image sequences the vanishings identify $H^0(F_A)$ with the constant coefficient kernel $L_A$ of ${\beta }_A$. The specified global-section base-change condition makes the reduction of $L_A\hookrightarrow S_A$ injective; a full-rank minor is then a unit, so $L_A$ is a direct summand.

By [F2], the global constant subspace $L$ is positive and $S=L\oplus M$ orthogonally, and the reduced monad on $M$ preserves the full framing. Apply [F3] to that reduced monad; it proves $H{\mathrm{\mid }}_M$ positive. Since $H{\mathrm{\mid }}_L$ is positive and the sum is orthogonal, $H$ is positive definite on all of $S$. In particular the argument also covers $k=0$, where $M=0$.

By hypothesis the coefficient-image module $M_A$ is a direct summand. The coefficient identities give $M_A^{\perp }=L_A$; for a direct summand in a finite free module with a perfect pairing, the restriction map to its dual is split-surjective, so taking orthogonal complements twice returns that summand. Hence $M_A=L_A^{\perp }$ and $S_A=L_A\oplus M_A$. For each reduced real line $\mathrm{\ell }$, its $\mathrm{ȷ}$-invariant two-plane gives $\mathrm{\ell }\cong {\mathbb{P}}_{\mathbb{C}}^1$ and ${\mathrm{\ell }}_A\cong {\mathbb{P}}_A^1$. The restricted kernel/image sequence has kernel term $U_A{\otimes }_A{\mathcal{O}}_{{\mathrm{\ell }}_A}(-1)$; by [F11], $H^0({\mathrm{\ell }}_A,{\mathcal{O}}_{{\mathrm{\ell }}_A}(-1))=0$. Thus global sections of $K_A{\mathrm{\mid }}_{{\mathrm{\ell }}_A}$ inject into those of $F_A{\mathrm{\mid }}_{{\mathrm{\ell }}_A}$. A constant vector $l\in L_A\subseteq S_A$ gives a section of $K_A{\mathrm{\mid }}_{{\mathrm{\ell }}_A}$, and if its image in $H^0({\mathrm{\ell }}_A,F_A{\mathrm{\mid }}_{{\mathrm{\ell }}_A})$ vanishes then that section, and hence $l$, vanishes. The line-section base-change hypothesis identifies the reduced section space, so the reduced line positivity gives positivity on $L_0$. The internal-positivity proof applies to the reduction on $M_0$, so its reduced form is positive too. On infinity the family framing identifies the remaining orthogonal complement isometrically with $W_A$.

Write the four coefficient maps of $\alpha$ as $P={\alpha }_{{\lambda }_1}$, $Q={\alpha }_{{\lambda }_2}$, $R={\alpha }_{{\mu }_1}$, and $T={\alpha }_{{\mu }_2}$. From the first-slot convention and $j^2=-1$, the corresponding coefficients of $\beta$ are $Q^{†}H,-P^{†}H,T^{†}H,-R^{†}H$ in that order. The ${\mu }_1^2,{\mu }_2^2,{\mu }_1{\mu }_2$ coefficients of $\beta \alpha =0$ give $T^{†}HR=R^{†}HT=0$ and $R^{†}HR=T^{†}HT$. The maps $R,T$ are injective by exactness at the two coordinate points of ${\mathrm{\ell }}_{\mathrm{\infty }}$. Positivity of $H$ therefore makes their common Gram matrix $G$ positive definite. Replacing the paired end bases by $G^{-1\mathrm{/}2}$ normalizes both maps to isometries with orthogonal images. Choosing an orthonormal basis of $S$ then identifies $R=-{\iota }_1$, $T=-{\iota }_2$ and $S=(V\oplus V)\oplus W_0$, with $W_0$ their orthogonal complement.

At infinity the maps are ${\alpha }_{\mathrm{\infty }}=-{\mu }_1{\iota }_1-{\mu }_2{\iota }_2$ and ${\beta }_{\mathrm{\infty }}={\mu }_2{\iota }_1^{†}-{\mu }_1{\iota }_2^{†}$. For every $[{\mu }_1:{\mu }_2]\in {\mathbb{P}}^1$, the kernel of ${\beta }_{\mathrm{\infty }}$ in $V\oplus V$ is the line-valued copy $\{({\mu }_{1}v,{\mu }_{2}v):v\in V\}$, which is exactly the image of ${\alpha }_{\mathrm{\infty }}$; $W_0$ is killed by ${\beta }_{\mathrm{\infty }}$ and meets that image trivially. Therefore the restricted cohomology and its section pairing are canonically $W_0\otimes {\mathcal{O}}_{{\mathrm{\ell }}_{\mathrm{\infty }}}$ and its standard positive form. The specified unitary framing identifies $W_0$ with $W$ without discarding any flat summand.

Relative to $S=V\oplus V\oplus W$, write $P=(B_1,B_2,J{)}^T$ and $Q=(Q_1,Q_2,Q_W{)}^T$. The coefficient of ${\lambda }_1{\mu }_1$ in $\beta \alpha =0$ is $Q^{†}HR+T^{†}HP=0$, so $Q_1=-B_2^{†}$; the coefficient of ${\lambda }_1{\mu }_2$ is $Q^{†}HT-R^{†}HP=0$, so $Q_2=B_1^{†}$. Define $I^{†}=Q_W$. These definitions give exactly the displayed $\alpha$ and, by the lifted reality identity, the displayed $\beta$.

The ${\lambda }_1^2$ coefficient of $\beta \alpha =0$ is $Q^{†}HP=0$. Substitution of the blocks gives $-[B_2,B_1]+IJ=[B_1,B_2]+IJ=0$. The ${\lambda }_1{\lambda }_2$ coefficient is $Q^{†}HQ-P^{†}HP=0$. Expanding it gives $[B_1,B_1^{†}]+[B_2,B_2^{†}]+I{I}^{†}-J^{†}J=0$, which is ${\mu }_{\mathbb{R}}=0$. Conversely, these two moment equations make all coefficients of $\beta \alpha$ vanish, and direct substitution yields $\beta \alpha ={\lambda }_1^2{\mu }_{\mathbb{C}}+{\lambda }_1{\lambda }_2{\mu }_{\mathbb{R}}-{\lambda }_2^2{\mu }_{\mathbb{C}}^{†}=0.$

Let a normalized framed chain isomorphism have domain, middle and target maps $g_{-},g_0,g_{+}$. It preserves $H$, so $g_0$ is unitary. Comparing the two $\mu$ coefficients forces $g_0{\mathrm{\mid }}_{V\oplus 0}=g_{-}$ and $g_0{\mathrm{\mid }}_{0\oplus V}=g_{-}$; unitarity then preserves $W_0$. The infinity framing is fixed, so $g_0{\mathrm{\mid }}_{W_0}=1$. Thus $g_0=\mathrm{diag}(g,g,1)$ for one $g\in U(V)$, and the $\lambda$ coefficients force $B_i^{\mathrm{\prime }}=g{B}_i{g}^{-1}$, $I^{\mathrm{\prime }}=gI$, and $J^{\mathrm{\prime }}=J{g}^{-1}$. Surjectivity of $\beta$ makes $g_{+}$ uniquely determined and equal to $g$. Conversely, each such $g$ defines the displayed pairing-preserving framed chain isomorphism. Hence there are no residual transformations beyond simultaneous $U(V)$.

On the affine chart $[{\lambda }_1:{\lambda }_2:{\mu }_1:{\mu }_2]=[1:0:z_1:z_2]$, the raw maps are $a_z$ and $b_z^{†}$ from [F5]. Since the original $\alpha ,\beta$ have their required ranks at every point, [F5] gives costability and stability. Thus the extracted zero-moment datum is regular. Conversely, for stable and costable data, [F5] proves full homogeneous rank, composition zero, and local splitting, so the raw formula does define the required monad.

Put $B_1^{\mathrm{\prime }}=t_1{B}_1$, $B_2^{\mathrm{\prime }}=t_2{B}_2$, $I^{\mathrm{\prime }}=I{h}^{-1}$, and $J^{\mathrm{\prime }}=t_1{t}_{2}hJ$. Substituting $({\lambda }_1,t_1{t}_2{\lambda }_2,t_1{\mu }_1,t_2{\mu }_2)$ into the primed $A,C$ gives $A^{\mathrm{\prime }}=t_{1}A$ and $C^{\mathrm{\prime }}=t_{2}C$. Substitution into ${\alpha }^{\mathrm{\prime }}$ then gives ${\alpha }^{\mathrm{\prime }}(g_{t}z)(t_1{t}_2{)}^{-1}=\mathrm{diag}(t_2^{-1},t_1^{-1},h)\alpha (z)\mathrm{.}$ The same substitution into ${\beta }^{\mathrm{\prime }}$ gives ${\beta }^{\mathrm{\prime }}(g_{t}z)\mathrm{diag}(t_2^{-1},t_1^{-1},h)=\beta (z)$. The factors therefore are the stated chain maps; on the framed flavor coordinates $h=\mathrm{diag}(e_a^{-1})$ supplies the stated weights.

After lifting reduced orthonormal bases, the common Gram matrix of the two $\mu$ coefficient maps has the form $I+ϵK$ with $K=K^{†}$. The change $T=I-\frac{1}{2}ϵK$ satisfies $T^{†}(I+ϵK)T=I$ using ${ϵ}^2=0$ and $\overline{ϵ}=ϵ$. Replacing the source basis by this change and the target by its paired change normalizes both $\mu$ maps over $A$ without ordering nilpotent elements or asserting nilpotent positivity. For a normalized framed family chain isomorphism with maps $g_{-},g_0,g_{+}$, the ${\mu }_1$ and ${\mu }_2$ coefficients force $g_0{\iota }_1={\iota }_1{g}_{-}$ and $g_0{\iota }_2={\iota }_2{g}_{-}$. The fixed framing gives $g_0{\mathrm{\mid }}_{W_0}=1$, so $g_0=\mathrm{diag}(g_{-},g_{-},1)$. Since ${\iota }_1$ is isometric and $g_0$ preserves the middle Hermitian form, $g_{-}^{†}{g}_{-}=1$. The $\lambda$ coefficients then force exactly the transformations of $B_i,I,J$, and conversely each $g_{-}\in U(k,A)$ defines a normalized framed chain isomorphism. Thus the residual group is $U(k,A)$ and coefficient extraction is independent of the normalized basis on isomorphism classes.

For any such raw datum and finite $z$, write $a_z=(B_1-z_1,B_2-z_2,J{)}^T$ and $b_z^{†}=(-B_2+z_2,B_1-z_1,I)$. The two moment equations give $b_z^{†}{a}_z=0$ and $a_z^{†}{a}_z=b_z^{†}{b}_z=:f_z$. Regularity makes both maps full rank, so $f_z$ is positive definite. On ${\mathrm{\ell }}_z$ the monad is $\alpha =a_z{\lambda }_1+b_z{\lambda }_2$ and $\beta =b_z^{†}{\lambda }_1-a_z^{†}{\lambda }_2$. The cross Gram terms vanish, and $a_z,b_z$ identify two orthogonal copies of $V$ after applying $f_z^{-1\mathrm{/}2}$. In these orthonormal coordinates the end maps on the first two copies are $({\lambda }_1,{\lambda }_2{)}^T$ and $(-{\lambda }_2,{\lambda }_1)$, the Euler complex $V\mathcal{O}(-1)\to (V\oplus V)\mathcal{O}\to V\mathcal{O}(1)$; the complementary term is $E_z\mathcal{O}$. Hence $H^0(F{\mathrm{\mid }}_{{\mathrm{\ell }}_z})=E_z$ with its positive induced form, and $F{\mathrm{\mid }}_{{\mathrm{\ell }}_z}$ is trivial. At infinity the same calculation in step 4.1 gives $W\otimes \mathcal{O}$ and its framing. The maps in step 6.1 are exactly the required locally split end maps.

The zero-level quotient result [F8] applies because step 6.1 established stability and both zero moments; it identifies each regular complex orbit with its unique unitary orbit. The explicit argument in step 5.2 is stronger for the framed real monads: it identifies their actual normalized isomorphisms with $U(V)$ and fixes the inverse extraction orbit. If a constant flat summand is present, [F2] places it in the framed $W_0$ complement and the same fixed-frame argument leaves it unchanged. When $k=0$, every map into or out of $V=0$ is zero and $U(0)$ has one element. Because all coefficient images lie in $M=L^{\perp }$, $J$ has no component in the framed flat subspace $L\subset W$ and $I$ vanishes on that subspace. Thus flat directions remain as unused directions of $W$.

The coefficient equations in steps 4.2 and 5.1 are polynomial identities, so over $A$ they give the same block form and both moment equations. Conversely, take an $A$-valued zero-moment datum whose reduction is regular. Every full-rank minor of its reduced raw maps lifts to a unit in the local rings of the first-order thickening; thus the end maps are locally split and the cohomology is locally free. On a fixed real line the matrices $a_z,b_z$ have full-rank reductions, so their common Gram $f_z$ is invertible over $A$. Their orthogonal images split off two free copies of $V_A$, leaving a free rank-$N$ complement; the line complex is the Euler complex plus that constant complement, so its sections commute with reduction. The global section and coefficient-image conditions are exactly the additional range conditions imposed in the Statement. The standard middle form induces a nondegenerate real pairing, positive after reduction. Thus such data give families in the stated range. The residual group is $U(k,A)$ by step 6.3. Reduction and each map $ϵ\mapsto rϵ$ commute with coefficient extraction and with the correction $T$. This proves first-order naturality for precisely the stated base changes.

The finite monad resolution and its locally free cohomology give the algebraic Chern character and Chern classes in [F7]. Its same finite line resolution yields the topological classes after the hyperplane cycle class is identified with the Euler class of the complex normal line. The positive incidence-fiber Thom calculation in [F7] gives ${\pi }^{\ast }u=H_{\mathrm{t}\mathrm{o}\mathrm{p}}^2$. If the topological bundle is ${\pi }^{\ast }E$, naturality gives ${\pi }^{\ast }{c}_2(E)=k{H}_{\mathrm{t}\mathrm{o}\mathrm{p}}^2=k{\pi }^{\ast }u$; since ${\pi }^{\ast }u$ is a generator, $c_2(E)=ku$. This is conditional on the pullback hypothesis and asserts no Ward correspondence.