E \to X

is a smoothly varying family of positive-definite Hermitian forms

h_{x}

\nabla

that is compatible with

h

and whose

(0, 1)

\nabla^{0, 1} = \overset{ˉ}{\partial}_{E}

Choose a local holomorphic frame

e

and write

h = h (e, e) > 0

θ = h^{- 1} \partial h = \partial lo g h, \nabla e = θ \otimes e .

e^{'} = g e

g

, then

h^{'} = ∣ g ∣^{2} h

and

θ^{'} = \partial lo g h^{'} = θ + g^{- 1} \partial g,

which is exactly the connection-form transformation law. The local formulas therefore define a global connection.

Identify

O (- 1)

C^{n + 1}

restricts to each tautological line. On the affine chart

U_{i} = {Z_{i} \neq = 0}

, put

w_{j} = Z_{j} / Z_{i}

for

j \neq = i

and use the holomorphic frame

s_{i} (w) = (w_{0}, \dots, w_{i - 1}, 1, w_{i + 1}, \dots, w_{n}) .

Then

h_{i} (s_{i}, s_{i}) = 1 + \sum_{j \neq = i} ∣ w_{j} ∣^{2}, θ_{i} = \frac{\sum _{j \neq = i} w _{j} d w _{j}}{1 + \sum _{j \neq = i} ∣ w _{j} ∣ ^{2}} .

F_{\nabla} = \overset{ˉ}{\partial} θ_{i} = - \partial \overset{ˉ}{\partial} lo g (1 + ∣ w ∣^{2}) .

Consequently Chern-Weil theory gives the closed representative

c_{1} (O (- 1)) = [\frac{i F _{\nabla}}{2 π}] = [- \frac{i}{2 π} \partial \overset{ˉ}{\partial} lo g (1 + ∣ w ∣^{2})] = - [\frac{ω _{FS}}{2 π}],

where

ω_{FS}

Solved by gpt-5.6-sol high.

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