Nonlinear Equations - UFRJ

[SEC. 5.5: COMPACTIFICATIONS 67
4. the kernel of P V (x) DV (x) is tangent to the group action,
where P W denotes the orthogonal projection onto W ⊥ . (The derivative
is with respect to x).
Example 5.16. H d is a non-degenerate fewspace of equations for
P n = C n+1 /C × , with χ(h) = h d .
Example 5.17. Let n = n−1+· · ·+n s −s and Ω = {x ∈ C n+s : x i =
0 for some i}. In the multi-homogeneous setting, the homogenization
group (C × ) s acts on M = C n+s \ Ω by
(x 1 , . . . , x s ) h (h 1 x 1 , . . . , h s x s )
and the multiplicative character for F i is
χ i (h) = h di1
1 hdi2 2 · · · h dis
s
By tracing through the definitions, we obtain:
Lemma 5.18. Let F be a fewspace of equations on M/H with character
χ. Then,
V (hx) = χ(h)V (x)
K(hx, hy) = |χ(h)| 2 K(x, y)
h ∗ ω = ω.
In particular, ω induces a form on M/H.
All this may be summarized as a principal bundle morphism:
H
⊂ >
χ
−−−−→ C ×
⊂ >
M
⏐
↓
M/H
V
−−−−→ F ∗ \ {0}
⏐
↓
v
−−−−→ P(F ∗ )
This diagram should be understood as a commutative diagram.
The down-arrows are just the canonical projections.
The quotient M/H is endowed with the possibly degenerate Hermitian
metric given by ω F .