derived categories of twisted sheaves on calabi-yau manifolds

Let τ : H ∗ (X, C) → H ∗ (X, C) be the map given by
τ(v0, v1, v2, . . . , v2n) = (v0, iv1, −v2, . . . , i 2n v2n),
where i = √ −1 ∈ C and n = dimC X.
For any v ∈ H ∗ (X, C) define v ∨ by the formula
v ∨ = τ(v). td(X).τ( td(X)) −1 .
On H∗ (X, C) consider the bilinear pairing given by
(v, w) = v ∨ .w;
this pairing will be called the generalized Mukai product on H ∗ (X, C).
Remark 3.1.3. Just for a reminder, we include here the first four terms in the usual
expansion <strong>of</strong> the Todd class <strong>of</strong> a vector bundle E on a complex manifold X:
td(E ) = 1 + 1
2 c1 + 1
12 (c2 1 + c2) + 1
24 c1c2
− 1
720 (c4 1 − 4c 2 1c2 − 3c 2 2 − c1c3 + c4) + . . . ,
where we set ci = ci(E ). Whenever we refer to td(X), we actually refer to td(TX),
the Todd class <strong>of</strong> the tangent bundle <strong>of</strong> X.
Remark 3.1.4. Note that if X satisfies the following condition:
(TD) td(X) consists only <strong>of</strong> classes <strong>of</strong> dimension divisible by 4
(which happens, for example, if c1(X) = 0 and dim X ≤ 4, which are all the cases
we’re interested in), the definition <strong>of</strong> v ∨ simplifies to the easier form v ∨ = τ(v),
which agrees with that <strong>of</strong> [25, Chapter 6] along the even dimensional part <strong>of</strong>
the cohomology. Also, note that although the classes involved can have complex
coefficients, if v, w are rational or real, the product (v, w) is again rational or real,
because X has even (real) dimension.
Remark 3.1.5. If u ∈ H k (Y, C), v ∈ H n−k (Y, C), the product (u, v) only differs
from the usual one (given by cupping and integrating along Y ) only by a factor <strong>of</strong>
i k , which depends only on k. We’ll use this fact later, when studying Calabi-Yau
threefolds.
Lemma 3.1.6. Let X, Y be complex manifolds, both satisfying (TD), let πX :
X × Y → X be the projection onto the first factor, and let u ∈ H ∗ (X × Y, C),
v, v ′ ∈ H ∗ (X, C). The following formulas hold:
X
(v.v ′ ) ∨ = v ∨ .v ′∨
,
π ∗ X(v ∨ ) = (π ∗ Xv) ∨ ,
πX,∗(u ∨ ) = (−1) dimC Y πX,∗(u) ∨ ;
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