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NOTES ON VANISHING CYCLES AND APPLICATIONS

Part of: Theory of singularities and catastrophe theory Singularities Differential equations in the complex domain Local theory

Published online by Cambridge University Press:  29 October 2020

LAURENŢIU G. MAXIM Open the ORCID record for LAURENŢIU G. MAXIM [Opens in a new window]
LAURENŢIU G. MAXIM*
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, MadisonWI53706, USA
*
e-mail: maxim@math.wisc.edu
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Abstract

Vanishing cycles, introduced over half a century ago, are a fundamental tool for studying the topology of complex hypersurface singularity germs, as well as the change in topology of a degenerating family of projective manifolds. More recently, vanishing cycles have found deep applications in enumerative geometry, representation theory, applied algebraic geometry, birational geometry, etc. In this survey, we introduce vanishing cycles from a topological perspective and discuss some of their applications.

Keywords

hypersurface singularitiesMilnor fibrationvanishing cyclesnearby cyclesperverse sheavescharacteristic classes

MSC classification

Primary: 14B05: Singularities
Secondary: 14B07: Deformations of singularities 32S05: Local singularities 32S20: Global theory of singularities; cohomological properties 32S25: Surface and hypersurface singularities 32S30: Deformations of singularities; vanishing cycles 32S50: Topological aspects 34M35: Singularities, monodromy, local behavior of solutions, normal forms 58K30: Global theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 109 , Issue 3 , December 2020 , pp. 371 - 415
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Finnur Larusson

The author was supported by the Simons Foundation Collaboration Grant #567077 and by the Sydney Mathematical Research Institute.

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