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Related to the previous item, another equivalent and more topological definition in complex dimension 2 is the following: a Stein surface is a complex surface X with a real-valued Morse function f on X such that, away from the critical points of f, the field of complex tangencies to the preimage = is a contact structure that induces an ...
Let X be a topological space and A, B be two subspaces whose interiors cover X. (The interiors of A and B need not be disjoint.) The Mayer–Vietoris sequence in singular homology for the triad (X, A, B) is a long exact sequence relating the singular homology groups (with coefficient group the integers Z) of the spaces X, A, B, and the intersection A∩B. [8]
Here the natural (flat) connection on S ⊗ O X induced by the flat connection on S and the flat connection d on O X, and O X is the sheaf of holomorphic functions on X, and is the sheaf of 1-forms on X. This natural flat connection is a Gauss–Manin connection ∇ and can be described by the Picard–Fuchs equation.
The Naval Vessel Register (NVR) is the official inventory of ships and service craft in custody of or titled by the United States Navy. It contains information on ships and service craft that make up the official inventory of the Navy from the time a vessel is authorized through its life cycle and disposal.
For example, the two embedded circles in a figure-eight shape provide examples of one-dimensional cycles, or 1-cycles, and the 2-torus and 2-sphere represent 2-cycles. Cycles form a group under the operation of formal addition, which refers to adding cycles symbolically rather than combining them geometrically.
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where H 0 (X,A) is the group A(X) of global sections of A on X. For example, if the group H 1 (X,A) is zero, then this exact sequence implies that every global section of C lifts to a global section of B. More broadly, the exact sequence makes knowledge of higher cohomology groups a fundamental tool in aiming to understand sections of sheaves.
H 0 (V,Ω n) then carries over to any base field , when Ω is taken to be the sheaf of Kähler differentials and the power is the (top) exterior power , the canonical line bundle . The geometric genus is the first invariant p g = P 1 of a sequence of invariants P n called the plurigenera .
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