Relative coinvariant generators and actual Virasoro matrices

Choose the finite homogeneous positive-weight representative set $A$ and finite incoming generating set from F2, and let $E$ bound the weights in $A$. The finite current bundle $F_E=V_{\le E,C}\otimes K_{C\mathrm{/}B}$ has a filtration with graded quotients finite sums of $K_{C\mathrm{/}B}^{1-r}$, $0\le r\le E$. Write $D$ for the sum of marked sections. After restricting to a fixed neighborhood, the genera and marked-point distributions are constant. F1 therefore gives one $k_0$ for which every fiber $H^1(F_E(kD))$ vanishes for $k\ge k_0$. Put $K={\pi }_{\ast }{F}_E(k_{0}D)$, a fixed finite holomorphic bundle. Choose the connected coordinate polydisc $U$ so that F3 makes $H^1(T,K{\mathrm{\mid }}_T)=0$ for every coordinate subpolydisc $T\subset U$.

For every $\mathrm{\ell }>k_0$, F1 gives the exact sequence $0\to K\to {\pi }_{\ast }{F}_E(\mathrm{\ell }D)\to P_{k_0,\mathrm{\ell }}\to 0$. Its kernel is the same K for every arbitrarily large $\mathrm{\ell }$. Thus F3 lifts every global principal-part section on T to a global current section on $C_T$, simultaneously in the sense that the same base T works for all pole orders. In particular the principal part $a{t}_i^{-\mathrm{\ell }}d{t}_i$ at one incoming point, zero at the others and with all intermediate orders set to zero, has such a lift; its error poles have order at most $k_0$. No simultaneous frame or acyclicity assertion for infinitely many different direct-image bundles is needed.

We construct the needed meromorphic vector fields. On a base coordinate polydisc the exact tangent sequence, twisted by $kD$, is $0\to T_{C\mathrm{/}B}(kD)\to T_C(kD)\to {\pi }^{\ast }{T}_B(kD)\to 0.$ On each fiber the first term is $K^{-1}(kD)$, the last is a finite sum of $\mathcal{O}(kD)$, and the middle is their extension. For sufficiently large k all three have zero first cohomology by line duality and F1. Apply F1 to them. Their direct-image bundle map is fiberwise surjective because the kernel has zero fiber H1. A finite maximal minor supplies a local holomorphic right inverse. Apply it to the section ${\pi }^{\ast }{\mathrm{\partial }}_a$, whose base component has no pole. This yields a global section $\widetilde{{\mathrm{\partial }}_a}$ on $C_U$ with only finite marked poles and with differential exactly ${\pi }^{\ast }{\mathrm{\partial }}_a$. Shrink U for the finitely many coordinate fields; the acyclicity condition in step 1.1 persists on its coordinate subpolydiscs.

Fix $L\ge E+k_0$. By F2's strict-index spanning, there is a uniform auxiliary degree bound $\nu (L)$ such that every homogeneous vector above that bound is a sum of $a_{(-\mathrm{\ell })}^{(i)}{w}^{\circ }$, $a\in A$, $\mathrm{\ell }>L$. The global lift in step 2.1 acts on $w^{\circ }$ as that term plus a finite remainder. Its remainder modes have degree at most $E+k_0-1$, so its total degree is strictly smaller than the original vector, exactly as in F2's fixed-curve argument. All coefficients and reductions are holomorphic on T because the lift is a section on the whole $C_T$. Induction on the nonnegative auxiliary degree expresses every constant vector of W, modulo $J_{\mathrm{p}\mathrm{r}\mathrm{e}}(T)$, as an $\mathcal{O}(T)$-linear combination of the fixed finite space $G=W_{\le \nu (L)}$. Every section of $W\otimes \mathcal{O}(T)$ is a finite sum of such vectors with holomorphic coefficients. This proves the asserted global generation on T. The auxiliary grading controls this finite reduction only; it is not used as an infinitesimal conformal dilation.

Its local vertical coefficient $h_{a,i}$ is meromorphic in $t_i$ with a finite negative Laurent cutoff and holomorphic in b. On a fixed input vector w the proposed Virasoro residue $A_{a}w$ is a finite sum: finitely many negative modes come from that pole bound, and sufficiently large positive conformal modes kill w by module truncation. Thus it lies in a fixed finite-dimensional span of conformal-mode images of w, with holomorphic coefficients on U. The operators ${\mathrm{\nabla }}_a={\mathrm{\partial }}_a-A_a$ are consequently actual sheaf differential operators on the algebraic tensor sheaf, obeying the Leibniz rule.

To prove ideal preservation, let $\sigma =u(t,b)dt$ be a finite-state global meromorphic current. Differentiating the finite-coordinate action along the flow of $\widetilde{{\mathrm{\partial }}_a}$ gives its globally defined Lie derivative, locally ${\mathcal{L}}_{\widetilde{{\mathrm{\partial }}_a}}\sigma =[({\mathrm{\partial }}_a+h{\mathrm{\partial }}_t+h^{\mathrm{\prime }})u-{\sum }_{r\ge 0}\frac{h^{(r+1)}}{(r+1)!}L(r)u]dt\mathrm{.}$ For justification, a centered infinitesimal coordinate change is $s\mapsto s+ϵ(h(t+s)-h(t))$; F4 differentiates its state operator to ${\sum }_{r\ge 0}{h}^{(r+1)}L(r)\mathrm{/}(r+1)!$. Pullback differentiates the coefficient by ${\mathrm{\partial }}_a+h{\mathrm{\partial }}_t$, and the relative one-form contributes $h^{\mathrm{\prime }}$. The state operator enters with the displayed negative sign in the pullback Lie derivative. Coordinate functoriality in F4 makes these local formulas agree on overlaps away from D, and the finite state sum gives only finite poles along D. This constructs the global current Lie derivative, rather than assuming it exists.

For each state cutoff n, increase the allowed marked pole order k until all fibers $H^1(F_n(kD))$ vanish. F1 gives the canonical holomorphic bundle $H_{n,k}={\pi }_{\ast }{F}_n(kD)$ and its base change on the fixed neighborhood. Every finite-state finite-pole current is included in one of these sufficiently positive twists, so Q is the cokernel of ${⨁}_{n,k}(H_{n,k}\otimes W\otimes {\mathcal{O}}_B)⟶W\otimes {\mathcal{O}}_B,$ where the first map is the residue action and k ranges over such positive twists. Pullback commutes with these direct sums, tensors and cokernels. F1 identifies each pulled-back $H_{n,k}$ with its corresponding direct image; the residue formula in F4 identifies the action. Hence the resulting cokernel is exactly the coinvariant sheaf of the pulled-back family. In particular base change to a point gives the fixed-curve quotient, not merely an abstract vector space of the same dimension. This uses no unproved coherence of Q.

Module residue Jacobi gives $[A_a,J(\sigma )]=J\left([{\sum }_{r\ge 0}\frac{h^{(r+1)}}{(r+1)!}L(r)u-{\mathrm{\partial }}_t(hu)]dt\right)\mathrm{.}$ Indeed the $r=-1$ Virasoro term is $hY(Tu,t)=h{\mathrm{\partial }}_{t}Y(u,t)$ for a constant state; integration by parts supplies $-{\mathrm{\partial }}_t(hu)$ including the derivative of every coefficient of u. All sums are finite on each input as in step 3.2. Therefore ${\mathrm{\nabla }}_a(J(\sigma )w)=J(\sigma ){\mathrm{\nabla }}_{a}w+J({\mathcal{L}}_{\widetilde{{\mathrm{\partial }}_a}}\sigma )w\mathrm{.}$ The current in the last term is a permitted global meromorphic current by step 3.3, so ${\mathrm{\nabla }}_a$ preserves $J_{\mathrm{p}\mathrm{r}\mathrm{e}}(T)$ for every open T. In particular it descends to Q. The central contribution is retained: for $u=\omega$, the $L(2)\omega =(c\mathrm{/}2)\mathbf{1}$ term contributes the actual third-derivative term in this same formula. No projective-connection or Hodge-line normalization has been inferred.

Choose a basis $e_1,\dots ,e_d$ of G. By step 3.2 the finitely many ${\mathrm{\nabla }}_a{e}_i$ have a fixed finite-dimensional W-span with holomorphic coefficients on U. Apply the global reduction of step 3.1 to each of those finitely many constant vectors, obtaining actual holomorphic coefficient functions $({\mathrm{\Omega }}_a{)}_{ij}$ on U and relations ${\mathrm{\nabla }}_a{e}_i-{\sum }_j({\mathrm{\Omega }}_a{)}_{ij}{e}_j\in J_{\mathrm{p}\mathrm{r}\mathrm{e}}(U)\mathrm{.}$ These finite holomorphic matrices are the required output of the geometric construction. They may depend on the chosen reductions and lifts. No integrability, scalar curvature or nodal regular-singularity property is asserted. Together with steps 3.1–4.2 this proves every assertion of the Statement.