I read the note on the proof of the theorem of Borel Weil Bott by Jacob Lurie (see http://www.math.harvard.edu/~lurie/papers/bwb.pdf), and I stuck to the spot he stuck said that for one $ mathbb {P} ^ 1 $-bundle up $ pi: E longrightarrow S $ and any trunk group $ { cal L} $ on $ E $ with fiber grade $ n $According to the theory of descent, we have the following isomorphism:

$$ pi _ * { cal L} cong R ^ 1 pi _ * ({ cal L} times K ^ { times n + 1}) ~~ (*) $$

from where $ K $ after him is the relative canonical bundle.

I realize that the slices on both sides are vector bundles $ S $ with fibers $ H ^ 0 ( mathbb {P} ^ 1, { cal L} | _ { mathbb {P} ^ 1}) $ and $ H ^ 1 ( mathbb {P} ^ 1, { cal L} | _ { mathbb {P} ^ 1} otimes K _ { mathbb {P} ^ 1} ^ { otimes n + 1} ) $ They are (not canonical) isomorphic to Serre Duality. Note that this applies if we have two line bundles whose fiber degrees add up $ -2 $ However, I do not believe that the isomorphism (*) holds for such a pair of wires.

My question is, what is the assumption to be able to hold (*)? What content of descent did he use? I am also confused as to why Serre Duality takes this form, rather than the usual one $ { cal L} ^ { vee} times K $, Can we get the similar conclusion on the basis of this usual form, d. H.

$$ pi _ * { cal L} cong R ^ 1 pi _ * ({ cal L} ^ { vee} otimes K)? $$