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From SFB1085 - Higher Invariants
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Given a regular scheme X over a strictly henselian trait S, one can regard the generic fiber X_η as a regular variety degenerating to the special fiber X_0 , that may be singular. Then, one can consider at least two invariants reflecting the singularities of X_0 : the sheaf of vanishing cycles and the category of singularities of X_0 . It is known that these two objects are strictly related. We will discuss the situation where X is a regular scheme over a regular local ring of dimension n ≥ 1. Even more generally, we will consider the case where X is a regular scheme endowed with a global section of a vector bundle of rank n≥1 and we will see how the connection between vanishing cycles and singularity categories generalizes in this case. We will see that a theorem of D. Orlov and J. Burke- M. Walker allows us to reduce to the case of a regular scheme with a global section of a line bundle. The fact that the line bundle may be non-trivial forces us to consider a G_m-equivariant version of (inertia invariant) vanishing cycles. This talk covers the work carried out in the preprint arXiv:2009.13359.

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