Pome2604E6F7AC76Lemma 12.31.1.2604E6F7AC76 For every integer $r\geq 0$, the space $H^0(M,\operatorname {Sym}^ rT_ M)$ is the direct sum of finitely many weight spaces, with weights $-r\leq \ell \leq r$. The weight-zero space is $H^0(E,\operatorname {Sym}^ rU)$. For $1\leq k\leq r$, the space of weight $-k$ is
and the space of weight $k$ is $H^0(E,A^{-k}\otimes \operatorname {Sym}^{r-k}U)$. Multiplication adds tensor degrees and weights.
Proof. On the cover $\mathbb C\times \mathbb P^1$, the cotangent coordinates of the two ruling charts are related by
and the lattice acts by
both formulas follow by preserving the one-form $u\, \mathrm{d}z+y\, \mathrm{d}t$. In the other ruling chart the action is
which is holomorphic at $v=0$ and agrees with the formula in the finite chart. Both formulas preserve $h=ty=-v y_\infty $.
A function that is homogeneous of degree $r$ on the fibers is a sum of terms $g_ b(z,t)y^ bu^{r-b}$, $0\leq b\leq r$, with $g_ b$ holomorphic on $\mathbb C^2$. Since $y=-v^2y_\infty $, holomorphy at $v=0$ says that $v^{2b}g_ b(z,1/v)$ is holomorphic there; there is no cancellation between different $b$, because the monomials in $u_\infty ,y_\infty $ are linearly independent. Expanding in $t$ therefore gives
with entire coefficients.
Give $t$ weight $1$, $y$ weight $-1$ and $u$ weight $0$, so that $t^ ay^ bu^{r-b}$ has weight $\ell =a-b$. The chart changes and the lattice action preserve this grading; equivalently, it is the grading by the cotangent lift of the global action $t\mapsto ct$, $c\in \mathbb C^*$. The bounds $0\leq a\leq 2b$ and $b\leq r$ give $-r\leq \ell \leq r$.
For $\ell =-k<0$, a homogeneous function of weight $\ell $ can be written uniquely as $y^ kF(z,h,u)$ with $F$ homogeneous of degree $r-k$. Its invariance is equivalent to
which is the transformation rule for sections of $A^ k\otimes \operatorname {Sym}^{r-k}U$. Conversely, each such polynomial defines a function that is holomorphic in both ruling charts, because its exponents satisfy $0\leq a\leq 2b$. For $\ell =k>0$, the condition $a\leq 2b$ shows that a function of weight $\ell $ can be written uniquely as $t^ kh^ kF(z,h,u)$, with $F$ of the same remaining degree $r-k$. Its invariance involves the factor $b_\lambda ^{-k}$, which gives $A^{-k}\otimes \operatorname {Sym}^{r-k}U$. Weight zero consists exactly of the homogeneous polynomials $F(z,h,u)$ with the transformation rule of $\operatorname {Sym}^ rU$. Products of polynomial functions add degrees and weights, which proves the last assertion. $\square$
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