Filtrations in $\mathbb{C}$-motivic stable homotopy theory

Problem Statement
This paper addresses the computation of effective, connective, and very effective filtrations within the $\mathbb{C}$-motivic, 2-complete, cellular stable homotopy category ($SH_{\mathbb{C}}$). The primary motivation is to verify and generalize Voevodsky's slice conjectures, which posit that the slices of specific motivic spectra (such as the sphere spectrum, algebraic cobordism $MGL$, and motivic modular forms $mmf$) are determined by purely topological data derived from the classical Adams-Novikov spectral sequence (ANSS) of their Betti realizations. While these conjectures have been proven for specific cases (e.g., $MGL$, the sphere over algebraically closed fields, and Landweber exact spectra), a unified computational framework for a broader class of motivic analogues, particularly the motivic modular forms spectrum $mmf$, was sought.

Methodology
The author employs the filtered spectrum model for the $\mathbb{C}$-motivic stable homotopy category, established by Gheorghe-Isaksen-Krause-Ricka (GIKR). This model relies on an equivalence of categories between $SH_{\mathbb{C}}$ and the category of modules over the filtered spectrum $\Gamma_{\star}(S)$, where $\Gamma_{\star}: SH_{cl} \to SH_{fil}$ is the $\mathbb{C}$-motivic analogue functor.

The methodology proceeds through the following steps:

1. Modeling Covers via Filtered Spectra: The author constructs explicit filtered spectrum models for the effective, connective, and very effective covers of a motivic analogue $\Gamma_{\star}(X)$, where $X$ is a classical spectrum. These models are defined by modifying the totalization construction of $\Gamma_{\star}(X)$ (specifically, truncating the Bousfield-Kan spectral sequences at specific antidiagonals or applying connective covers in specific weights).
2. Homotopy Characterizations: Before applying the models, the paper establishes homotopy-theoretic characterizations of the respective covers. It proves that for spectra $Y$ that are "approximable" (a condition satisfied by $\Gamma_{\star}(X)$ when $X$ is bounded below), the covers are uniquely determined by specific properties of their homotopy groups (e.g., vanishing conditions in certain coweights or isomorphisms under the action of $\tau$).
3. Slice Computation: Using the explicit filtered models, the author computes the slices (the cofibers of the cover maps) by analyzing the associated Bousfield-Kan spectral sequences (BKSS).
4. Spectral Sequence Comparison: The paper analyzes the effective slice spectral sequence for $\Gamma_{\star}(X)$ and compares it to the classical ANSS of $X$. This involves a "half-speed" re-indexing and the use of the décalage functor to relate the slice spectral sequence differentials to those of the ANSS.

Key Contributions and Results

• Generalized Slice Theorem (Theorem 1.2 / Corollary 7.11): For any classical spectrum $X$ that is bounded below and has even $MU$-homology, the $q$-th effective slice of its motivic analogue $\Gamma_{\star}(X)$ is given by:
  $$s_q(\Gamma_{\star}(X)) \simeq \bigvee_{s+f=2q} \Sigma^{s,q} M E^{s,f}_2$$
  where $E^{s,f}_2$ is the $E_2$-page of the classical ANSS for $X$, and $M$ is the functor taking a group to the $\mathbb{C}$-motivic Eilenberg-MacLane spectrum. This result recovers Voevodsky's conjectures for $S$, $MGL$, $MU$, $ku$, and $ko$, and provides a new computation for $mmf$ (the motivic modular forms spectrum).
• Effective Slice Spectral Sequence (Theorem 1.3 / Theorem 8.1): The paper establishes a precise relationship between the effective slice spectral sequence of $\Gamma_{\star}(X)$ and the classical ANSS of $X$. The $E_1$-page of the slice spectral sequence is obtained by freely adjoining $\tau$ to the $E_2$-page of the ANSS. Furthermore, the differentials are related by the formula: $d_r(x) = \tau^r y$ in the slice spectral sequence if and only if $d_{2r+1}(x) = y$ in the ANSS.
• Models for Connective and Very Effective Covers: The author provides explicit filtered spectrum models for the connective and very effective covers of $\Gamma_{\star}(X)$, allowing for the computation of their respective slices.
• Landweber Exactness: It is shown that for Landweber exact spectra, the effective and very effective covers coincide, recovering known results for motivic Landweber exact spectra.
• Homotopy Characterizations: The paper provides necessary and sufficient conditions on homotopy groups (involving the $\tau$-action and vanishing in specific coweights) that characterize effective, connective, and very effective covers within the cellular subcategory.

Significance
The paper claims significance primarily in its ability to unify and extend previous computations of motivic slices. By leveraging the filtered spectrum model, the author demonstrates that the complex structure of motivic slices for a wide class of spectra (including the previously open case of $mmf$) is entirely governed by the classical Adams-Novikov spectral sequence. The work confirms Voevodsky's conjectures in the 2-complete $\mathbb{C}$-motivic setting for all spectra of the form $\Gamma_{\star}(X)$ where $X$ is bounded below with even $MU$-homology. Additionally, the explicit relationship between the effective slice spectral sequence and the classical ANSS provides a computational tool that translates motivic problems into classical topological ones, recovering results by Levine and others while offering new insights into the structure of motivic modular forms. The paper remains modest regarding its scope, noting that it is restricted to the cellular subcategory and the 2-complete setting, and that the "even $MU$-homology" assumption is required for the closed-form description of the slices, though homotopy groups can still be computed without it.