Air Force Bases [U.S.] [V. 2 - Outside U.S.] by R. Mueller

By R. Mueller

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Let F be a vector bundle over U , and j∗ F its push-forward. Then the restriction of j∗ F to the fibers of π is reflexive. Proof: Let Z = π −1 ({y}) be a fiber of π. Since S is flat over Y and of codimension at least 2, we have j∗ (OZ∩U ) = OZ . Clearly, for an open embedding γ : T1 −→ T2 and coherent sheaves A, B on T1 , we have γ∗ (A ⊗ B) = γ∗ A ⊗ γ∗ B. Thus, for all coherent sheaves A on U , we have j∗ A ⊗ OZ = j∗ (A ⊗ OZ∩U ). 2) This implies that j∗ (F Z ) = j∗ F Z . It is well known ([OSS], Ch.

Proof: [V3] (see also [V3-bis]). ¨ 4. 7 is the following algebraic computation. 8: Let M be a simple hyperk¨ahler manifold, and H a hyperk¨ahler structure on M . 2). Let g ⊂ End(H ∗ (M )) be the Lie algebra generated by subalgebras aH ⊂ End(H ∗ (M )), for all hyperk¨aher structures H on M . Then (i) The algebra g is naturally isomorphic to the Lie algebra so(V ⊕ H), where V is the linear space H 2 (M, R) equipped with the Bogomolov– Beauville pairing, and H is a 2-dimensional vector space with a quadratic form of signature (1, −1).

In other words, M is Mumford–Tate generic if for all n ∈ Z>0 , the n-th power M n is generic. Clearly, Mumford–Tate generic implies generic. 14: Let M be a compact manifold, H a hyperk¨ahler structure on M and S be the set of induced complex structures over M . Denote by S0 ⊂ S the set of L ∈ S such that (M, L) is Mumford-Tate generic with respect to H. Then S0 is dense in S. Moreover, the complement S\S0 is countable. 15: Let M be a compact holomorphically symplectic manifold. Assume that M is of general type with respect to a hyperk¨ahler structure H.

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