Allied Journal of Medical Research

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Allied Journal of Medical Research 44 7897 074717

Vortex Vanishing Theorm

 

There are some vanishing theorems for simple normal crossing pairs, which will play important roles in the study of higher dimensional algebraic varieties. We note that the notion of simple normal crossing pairs includes here the case when the ambient variety itself has several irreducible components with simple normal crossings.

In algebraic geometry, a vanishing theorem gives conditions for coherent cohomology groups to fade.

Andreotti–Grauert vanishing theorem Grauert–Riemenschneider vanishing theorem Kawamata–Viehweg vanishing theorem Kodaira vanishing theorem Mumford vanishing theorem Ramanujam vanishing theorem Nadel vanishing theorem Fujita vanishing theorem

In the late 1980s, Green and Lazarsfeld studied the cohomology of topologically trivial line bundles on smooth complex projective varieties (and compact K¨ahler manifolds). The generic vanishing theorem is one among their main results: it says that, under certain conditions on the variability , the cohomology of a generically chosen line bundle is trivial in all degrees less than the dimension of the variety. This theorem, and related results by Green and Lazarsfeld, are a really useful gizmo within the study of irregular varieties and abelian varieties.

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