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A Voevodsky motive associated to a log scheme

Georgii Shuklin

Abstract

For each fs log scheme $(X,\mathcal M_X)$ over a field $k$ we construct a geometrical Voevodsky motive $[X]^{log}\in DM_{gm}(k,\mathbb Q)$. We prove that, for $k=\mathbb C$, the Betti realization of $[X]^{log}$ is the log Betti cohomology of $(X, \mathcal M_X)$. We give applications to motivic tubular neighborhoods, limit motives and the monodromy filtrations.

A Voevodsky motive associated to a log scheme

Abstract

For each fs log scheme over a field we construct a geometrical Voevodsky motive . We prove that, for , the Betti realization of is the log Betti cohomology of . We give applications to motivic tubular neighborhoods, limit motives and the monodromy filtrations.
Paper Structure (18 sections, 61 theorems, 369 equations)

This paper contains 18 sections, 61 theorems, 369 equations.

Table of Contents

  1. Introduction
  2. Presheaf of virtual log structures
  3. Category of virtual log schemes
  4. Free $\mathbb E_\infty$-algebras
  5. Rational cohomology of a Kato-Nakayama space
  6. From log structures to motivic sheaves
  7. The homological motive of a log scheme
  8. Betti realization of $[X]^{log}$
  9. Logarithmic $h$-descent
  10. Logarithmic motivic cohomology
  11. Motivic cohomology of a log projective line
  12. Log motivic cohomology as a motivic sheaf
  13. Log smooth schemes and motivic tubular neighborhoods
  14. Limit motives via log geometry
  15. Motivic monodromy filtration
  16. ...and 3 more sections

Key Result

Proposition 1.2

There is a canonical quasi-isomorphism

Theorems & Definitions (105)

  • Example 1.1
  • Proposition 1.2
  • Conjecture 1.3
  • Conjecture 1.4
  • Proposition 1.5
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • Definition 2.4
  • Remark 2.5
  • ...and 95 more