Nearby cycles, monodromy and specialization

Six lessons construct nearby and vanishing cycles, fix monodromy and variation signs, prove proper-on-support pushforward compatibility, compare normal and conormal sections, compute quadratic tests, and identify vanishing cycles with positive real support. Forty-one exercises have complete solutions.

• Nearby cycles and the two monodromy triangles · editable source
• Proper pushforwards of nearby and vanishing cycles · editable source
• Nearby cycles through the normal deformation · editable source
• Complex nearby cycles as normal and conormal sections · editable source
• Quadratic cycles and the holomorphic microsupport test · editable source
• Vanishing cycles as positive real support · editable source

A route through the calculations

Use the finite-support sequence and ramification examples to fix signs first. Follow the support carrier through pushforward, then compare the cover with normal deformation. The slit and polar computations lead to conormal sections; the covered quadratic ball supplies the degree needed for holomorphic detection. Positive real support gives a second geometric description and its own counterexample outside the complex setting.

The source and proof guide identifies the exact human passages checked and the programme prerequisites still required. In particular, a comparison merely stated by reference in a free paper is not counted as a supplied proof.

The proofs retain their stated finite-dimensional topology, sheaf-operation, weak constructibility, small-ball stabilization, analytic normal-cone, Fourier and microlocal coefficient-model prerequisites. Weak-coefficient arguments permit arbitrary bounded modules; the perfect subcategory is specified separately.

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