derived categories of twisted sheaves on calabi-yau manifolds

D
elliptic fibration
H
contraction to
flop
Q;
4H-D
contraction to
complete intersection
<strong>of</strong> type (2,4); flop
11H-3D
elliptic fibration
Figure 6.1: The Kähler cones in Example 6.2.2
multi-section). One can take H for a multisection. There is one more interesting
Calabi-Yau threefold that can be constructed from X (see Sections 4.5 and 6.6),
namely X 2 . We’d like to be able to claim that X 2 is not birational to X, and for
this we analyze the Kähler cones <strong>of</strong> X.
One can draw the Kähler cones for X (Figure 6.2.2). The walls are given by
1. D – map to P 2 ; elliptic fibration;
2. H – map to P 4 , image is a quintic with 53 ordinary double points; passing
through this wall is a flop;
3. 4H − D – map to P 5 , image is a complete intersection <strong>of</strong> type (2, 4) with 41
ordinary double points; passing through this wall is a flop;
4. 11H − 3D – map to P 2 ; elliptic fibration.
Note that the first elliptic fibration is embedded in P 2 × P 4 , while the second one
in P 2 × P 5 (although the fibers are degree 5 each time). If we could prove that
X 2 can also be embedded in P 2 × P 4 , this would prove that X 2 is not birational
to X. We are unable to do this in the course <strong>of</strong> this work due to lack <strong>of</strong> time.
Example 6.2.3. The construction is almost identical to that in the previous example,
except that we take the base <strong>of</strong> the fibration to be S = P 1 × P 1 . (We want
to do this in order to have KS divisible by 2; see Section 6.7.) The ambient space
is now P 1 × P 1 × P 4 , and M is a 5 × 5 skew-symmetric matrix <strong>of</strong> polynomials <strong>of</strong>
tri-degree (aij, bij, 1), where aij = 0 except when i = 4 or j = 4, in which case
aij = 1, and bij = 0 except when i = 5 or j = 5, in which case bij = 1.
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