By Christina Birkenhake
This paintings is on the crossroads of a couple of mathematical parts, together with algebraic geometry, numerous complicated variables, differential geometry, and illustration idea. the point of interest of the ebook is on complicated tori, one of the easiest of advanced manifolds, that are vital within the above components.
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Additional resources for Complex tori
From now on we will neglect torsion in H even (X; Z), so by an abuse of notation, when we say that an element λi of H 2i (X; Q) lies in H 2i (X; Z), we really mean that λi lies in the image of H 2i (X; Z) in H 2i (X; Q). By the Hirzebruch–Riemann–Roch Theorem [40, Th. 19) 1 where td(T X) is the Todd class of T X, which is 1+ 12 c2 (T X) as X is a Calabi–Yau ∨ 3-fold, and (λ0 , λ1 , λ2 , λ3 ) = (λ0 , −λ1 , λ2 , −λ3 ), writing elements of H even (X; Q) as (λ0 , . . , λ3 ) with λi ∈ H 2i (X; Q). The Chern character is additive over short exact sequences.
Thus there exists y ∈ Y (K) with (π1 )∗ (y) = w and (π2 )∗ (y) = w . 1) thus gives (−1)n νW (w) = νY (y) = (−1)n νW (w ), so that (−1)n νW (w) = (−1)n νW (w ). Hence νX (x) is well-deﬁned. Therefore there exists a unique function νX : X(K) → Z with the property in the proposition. It remains only to show that νX is locally constructible. For ϕ, W, n as above, ϕ∗ (νX ) = (−1)n νW and νW constructible imply that νX is constructible on the constructible set ϕ∗ (W (K)) ⊆ X(K). But any constructible subset S of X(K) can be covered by ﬁnitely many such subsets ϕ∗ (W (K)), so νX |S is constructible, and thus νX is locally constructible.
A sheaf C is called constructible if there is a locally ﬁnite stratiﬁcation X = j∈J Xj of X in the complex analytic topology, such that C|Xj is a Q-local system for all j ∈ J, and all the stalks Cx for x ∈ X are ﬁnite-dimensional Q-vector spaces. A complex C • of sheaves of Q-modules on X is called constructible if all its cohomology sheaves H i (C • ) for i ∈ Z are constructible. b (X) for the bounded derived category of constructible complexes Write DCon b on X. It is a triangulated category.
Complex tori by Christina Birkenhake