Theorem 12.32.277C35E15A510

Proof. Step 1: weights. Give $c_-$ weight $-1$, $c_+$ weight $1$, and $h,w,u$ weight zero. These weights come from the ruling action and are compatible with the quadratic relation and the lattice action. Using the relation to eliminate products $c_-c_+$, a polynomial of weight $-k$, $k\geq 1$, can be written as $c_-^ kF(z,h,w,u)$, and a polynomial of weight $k$ as $c_+^ kF(z,h,w,u)$, where $F$ has degree $r-k$ in tensor degree $r$; weight zero consists of the polynomials $F(z,h,w,u)$. These expressions are unique. Indeed, the substitution $c_+\mapsto h(h+\rho w)/c_-$ embeds the algebra of lemma 12.32.1 into

the reduced expressions involve distinct powers of $c_-$, and their coefficients vanish only if the original coefficients vanish, because $h(h+\rho w)$ is a nonzero polynomial. This proves that the weight decomposition is a direct sum.

Step 2: coefficient bundles. Let $U_3$ be the rank-three bundle on $E$ of linear polynomials in $h,w,u$ with the transformation rule

By the transformation of $c_-$ and $c_+$ in lemma 12.32.1, the block of weight $-k$ is $H^0(E,A^ k\otimes \operatorname {Sym}^{r-k}U_3)$, and the block of weight $k$ is $H^0(E,A^{-k}\otimes \operatorname {Sym}^{r-k}U_3)$; the inverse translation of $u$ appears when invariance is written out. The signs of the twists agree with those in lemma 12.31.1.

Step 3: reduction to the tangent bundle. Since $q_0\ne 0$, put $H=h+(K/q_0)w$. Then the translation of $u$ is $u\mapsto u-mq_0H$, and consequently

where the variable $h$ of the bundle $U$ defined above is replaced by $H$. The symmetric powers of $U_3$ are the direct sums of the bundles $w^ j\operatorname {Sym}^{s-j}U$, $0\leq j\leq s$. The vanishing for negative twists in lemma 12.31.2 eliminates every positive weight, and the same lemma identifies the algebra of sections of weight zero with $\mathbb C[H,w]$.

In each negative weight, expanding in powers of $w$ identifies the remaining spaces with the corresponding blocks for the tangent bundle in lemma 12.31.1, with a polynomial variable $w$ adjoined. Sending the Euler variable $h$ of those blocks to $H$ and their leading factor $y^ k$ to $c_-^ k$ gives bijections in every tensor degree and weight. Products of surviving blocks multiply their coefficients and add their powers of $c_-$, and the eliminated positive weights contribute no products. The bijections therefore preserve multiplication and give the asserted isomorphism of graded algebras. $\square$