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Smoothing of real algebraic hypersurfaces by rigid isotopies

Alexander Nabutovsky (1991)

Annales de l'institut Fourier

Define for a smooth compact hypersurface M n of R n + 1 its crumpleness κ ( M n ) as the ratio diam R n + 1 ( M n ) / r ( M n ) , where r ( M n ) is the distance from M n to its central set. (In other words, r ( M n ) is the maximal radius of an open non-selfintersecting tube around M n in R n + 1 . ) We prove that any n -dimensional non-singular compact algebraic hypersurface of degree d is rigidly isotopic to an algebraic hypersurface of degree d and of crumpleness exp ( c ( n ) d α ( n ) d n + 1 ) . Here c ( n ) , α ( n ) depend only on n , and rigid isotopy means an isotopy passing only through hypersurfaces of degree...

Some applications of the theory of harmonic integrals

Shin-ichi Matsumura (2015)

Complex Manifolds

In this survey, we present recent techniques on the theory of harmonic integrals to study the cohomology groups of the adjoint bundle with the multiplier ideal sheaf of singular metrics. As an application, we give an analytic version of the injectivity theorem.

Some consequences of perversity of vanishing cycles

Alexandru Dimca, Morihiko Saito (2004)

Annales de l’institut Fourier

For a holomorphic function on a complex manifold, we show that the vanishing cohomology of lower degree at a point is determined by that for the points near it, using the perversity of the vanishing cycle complex. We calculate this order of vanishing explicitly in the case the hypersurface has simple normal crossings outside the point. We also give some applications to the size of Jordan blocks for monodromy.

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