derived categories of twisted sheaves on calabi-yau manifolds

Definition 3.1.8. Let X and Y be complex projective manifolds, and let U · ∈
Db coh (X × Y ). Define the cohomological integral transform associated to U · U ·
, ϕY →X ,
by
U ·
ϕ
Proposition 3.1.9. The diagram
commutes.
Pro<strong>of</strong>. We have
ϕ
Y →X : H ∗ (Y, C) → H ∗ (X, C),
U ·
Y →X(v) = πX,∗(v(U · ).π ∗ Y (v)).
D b coh(Y )
ΦU ·
Y →X ✲ D b coh(X)
H ∗ v
❄ U ϕ
(Y, C)
·
v
❄
Y →X ✲ ∗
H (X, C)
U ·
ϕY →X(v(E · )) = πX,∗(π ∗ Y (v(E · )).v(U · ))
= πX,∗(π ∗ Y (ch(E · ). td(Y )). ch(U · ). td(X × Y ))
td(X))
= πX,∗(ch(Lπ ∗ Y E · ). ch(U · ).π ∗ Y td(Y ).π ∗ X
= πX,∗(ch(Lπ ∗ L
·
Y E ⊗ U · ). td(TπX )).td(X) = ch(RπX,∗(Lπ ∗ L
·
Y E ⊗ U · )). td(X)
U ·
= ch(ΦY →X(E · )). td(X)
U ·
= v(ΦY →X(E · )),
which is what we wanted. Here, TπX is the relative tangent bundle <strong>of</strong> πX : X×Y →
X, which is obviously isomorphic to π ∗ Y TY .
Proposition 3.1.10. Let X, Y and Z be complex projective manifolds, and let
E · ∈ Db coh (X ×Y ), F · ∈ Db coh (Y ×Z). Denote by pXY , pY Z and pXZ the projections
from X × Y × Z to X × Y , Y × Z and X × Z, respectively, and define E · ◦ F · ∈
D b coh
(X × Z) by
E · ◦ F · = RpXZ,∗(Lp ∗ L
·
XY E ⊗ Lp ∗ Y ZF · ).
Similarly, for e ∈ H ∗ (X × Y, C) and f ∈ H ∗ (Y × Z, C) consider
Then we have
e ◦ f = pXZ,∗((p ∗ XY e).(p ∗ Y Z f)).
v(E · ◦ F · ) = v(E · ) ◦ v(F · ).
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