derived categories of twisted sheaves on calabi-yau manifolds

The reason for the notation is that we have ΦE· ·
Y →X ◦ ΦF Z→Y = ΦE· ◦F ·
Z→X , (see,
for example, [4, 1.4]) and similarly for the transforms on the level <strong>of</strong> rational
cohomology.
Pro<strong>of</strong>. We have
v(E · ◦ F · ) = ch(E · ◦ F · ). td(X × Z)
= ch(RpXZ,∗(Lp ∗ L
·
XY E ⊗ Lp ∗ Y ZF · )). td(X × Z)
= pXZ,∗(ch(Lp ∗ L
·
XY E
⊗ Lp ∗ Y ZF · ).p ∗ Y td(Y )). td(X × Z)
∗
td(X).pZ td(Z))
= pXZ,∗(p ∗ XY ch(E · ).p ∗ Y Z ch(F · ).p ∗ Y td(Y ).p ∗ X
= pXZ,∗(p ∗ XY v(E · ).p ∗ Y Zv(F · ))
= v(E · ) ◦ v(F · )
Proposition 3.1.11. Let X and Y be complex projective manifolds, and assume
that they satisfy condition (TD). For U · ∈ Db coh (X × Y ), let
U ·∨ = RHom(U · , OX×Y ).
Then, for every c ∈ H ∗ (X, C), c ′ ∈ H ∗ (Y, C), we have
U ·
(c, ϕY →X(c ′ )) = (−1) dimC X U
(ϕ ·∨
X→Y (c), c ′ ),
where the pairings are the Mukai products on X and on Y , respectively.
Pro<strong>of</strong>. Let u = v(U · ), and note that u∨ = v(U ·∨ ). We have
U ·
(c, ϕY →X(c ′
)) = c
X
∨ U ·
.ϕY →X(c ′ )
= c
X
∨ .πX,∗(u.π ∗ Y c ′ )
= πX,∗(π
X
∗ Xc ∨ .u.π ∗ Y c ′ )
= π
X×Y
∗ Xc ∨ .u.π ∗ Y c ′
= ((π
X×Y
∗ Xc).u ∨ ) ∨ .π ∗ Y c ′
= πY,∗(((π
Y
∗ Xc).u ∨ ) ∨ ).c ′
= (−1) dimC X
= (−1) dimC X
πY,∗((π
Y
∗ Xc).u ∨ ) ∨ .c ′
Y
U ·∨
ϕX→Y (c) ∨ .c ′
= (−1) dimC X (ϕ U ·∨
X→Y (c), c ′ ).
(We have used Lemma 3.1.6 at various stages <strong>of</strong> the computation.)
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