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Stable motivic homotopy theory at infinity

2021

In this paper, we initiate a study of motivic homotopy theory at infinity. We use the six functor formalism to give an intrinsic definition of the stable motivic homotopy type at infinity of an algebraic variety. Our main computational tools include cdh-descent for normal crossing divisors, Euler classes, Gysin maps, and homotopy purity. Under $\ell$-adic realization, the motive at infinity recovers a formula for vanishing cycles due to Rapoport-Zink; similar results hold for Steenbrink's limiting Hodge structures and Wildeshaus' boundary motives. Under the topological Betti realization, the stable motivic homotopy type at infinity of an algebraic variety recovers the singular complex at in…

[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG][MATH.MATH-AT] Mathematics [math]/Algebraic Topology [math.AT]Mathematics::Algebraic TopologyMathematics - Algebraic GeometryMathematics::Algebraic GeometryMathematics::K-Theory and Homology[MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT]Mathematics::Category TheoryFOS: MathematicsAlgebraic Topology (math.AT)[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]Mathematics - Algebraic TopologyPrimary: 14F42 19E15 55P42 Secondary: 14F45 55P57Algebraic Geometry (math.AG)
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