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Doosung Park

Publications and source records attributed to Doosung Park.

32 records · Page 2Linked to original sources
arXiv2024

Log motivic nearby cycles

We define the log motivic nearby cycles functor. We show that this sends the motive of a proper smooth scheme over the fraction field of a DVR to the motive of the boundary of a log smooth model assuming absolute purity, which is unconditional in the equal characteristic case. In characteristic $0$, we show that the $\infty$-categories of motives over the standard log point and rigid analytic motives are equivalent, and we relate log motivic nearby cycles functor with Ayoub's motivic nearby cycles functor.

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arXiv2024

Inverting log blow-ups in log geometry

In the category of log schemes, it is unclear how to define the blow-ups for non-strict closed immersions. In this article, we introduce the notion of divided log spaces. We obtain the category of divided log spaces by locally inverting log blow-ups in the category of log schemes. We show that blow-ups exist for closed immersions of log smooth divided log spaces. This is an ingredient of the motivic six-functor formalism for log schemes.

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arXiv2024

Logarithmic Gysin sequences for regular immersions

For a regular immersion of schemes $Z\to X$ and a cohomology theory of fs log schemes, we formulate the logarithmic Gysin sequence using the "logarithmic compactification" $(\mathrm{Bl}_Z X,E)$ instead of the open complement $X-Z$, where $E$ is the exceptional divisor. We show that all $\mathbb{A}^1$-invariant cohomology theories produced from motivic spectra and various non $\mathbb{A}^1$-invariant cohomology theories like Nygaard completed prismatic cohomology admit logarithmic Gysin sequences.

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arXiv2023

On the log motivic stable homotopy groups

We compare the log motivic stable homotopy category and the usual motivic stable homotopy category over a perfect field admitting resolution of singularities. As a consequence, we show that the log motivic stable homotopy groups are isomorphic to the usual motivic stable homotopy groups.

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arXiv2023

Real topological Hochschild homology of schemes

We prove that real topological Hochschild homology THR for schemes with involution satisfies base change and descent for the Z/2-isovariant étale topology. As an application, we provide computations for the projective line (with and without involution) and the higher dimensional projective spaces.

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arXiv2023

A Hochschild-Kostant-Rosenberg theorem and residue sequences for logarithmic Hochschild homology

This paper incorporates the theory of Hochschild homology into our program on log motives. We discuss a geometric definition of logarithmic Hochschild homology of derived pre-log rings and construct an André-Quillen type spectral sequence. The latter degenerates for derived log smooth maps between discrete pre-log rings. We employ this to show a logarithmic version of the Hochschild-Kostant-Rosenberg theorem and that logarithmic Hochschild homology is representable in the category of log motives. Among the applications, we deduce a generalized residue sequence involving blow-ups of log schemes.

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arXiv2023

Logarithmic Prismatic Cohomology via Logarithmic THH

Inspired by Bhatt-Morrow-Scholze's work on ${\rm THH}$, we define Nygaard-completed log prismatic cohomology based on log topological Hochschild homology via filtrations on log ${\rm THH}$ and its variants. Moreover, of independent interest, we describe log ${\rm THH}$ for quasiregular semiperfectoids as a $1$-parameter deformation of ordinary, non-logarithmic Hochschild homology.

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arXiv2021

Triangulated categories of logarithmic motives over a field

In this work we develop a theory of motives for logarithmic schemes over fields in the sense of Fontaine, Illusie, and Kato. Our construction is based on the notion of finite log correspondences, the dividing Nisnevich topology on log schemes, and the basic idea of parameterizing homotopies by $\overline{\square}$, i.e. the projective line with respect to its compactifying logarithmic structure at infinity. We show that Hodge cohomology of log schemes is a $\overline{\square}$-invariant theory that is representable in the category of logarithmic motives. Our category is closely related to Voevodsky's category of motives and $\mathbb{A}^{1}$-invariant theories: assuming resolution of singularities, we identify the latter with the full subcategory comprised of $\mathbb{A}^{1}$-local objects in the category of logarithmic motives. Fundamental properties such as $\overline{\square}$-homotopy invariance, Mayer-Vietoris for coverings, the analogs of the Gysin sequence and the Thom space isomorphism as well as a blow-up formula and a projective bundle formula witness the robustness of the setup.

math.AG↗
arXiv2021

Motivic interpretation of Albanese varieties of smooth varieties

For every noetherian smooth and separated scheme over an algebraically closed field, we define its derived Albanese in Voevodsky's triangulated category of effective Nisnevich motives. To justify our definition, we relate the derived Albanese with the Albanese scheme. We also prove that the derived Albanese satisfies the Nisnevich descent property.

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arXiv2021

Intermediate Jacobians and the slice filtration

For every $n$-dimensional smooth projective variety $X$ over ${\bf C}$, the motive $M(X)$ is expected to admit a Chow-Künneth decomposition $M_0(X)\oplus\cdots \oplus M_{2n}(X)$. Inspired by the slice filtration of $M(X)$ we propose the definitions of $M_2(X)$ and $M_{2n-2}(X)$.

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arXiv2020

Description of mixed motives

Assuming the Künneth type standard conjecture, we propose a way to describe objects of mixed motives explicitly. We study their formal properties, and we associate mixed motives to schemes smooth and separated over a field. This serves as a universal cohomology theory. We also discuss $\ell$-adic realizations, and we discuss an unconditional construction of 2-motives and their properties.

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arXiv2017

Triangulated categories of motives over fs log schemes

We introduce the notion of log motivic triangulated categories, which is the theoretical framework for understanding the motivic aspect of cohomology theories for fs log schemes. Then we study the Grothendieck six operations formalism for log motivic triangulated categories.

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arXiv2017

Equivariant cd-structures and descent theory

We construct the equivariant version of cd-structures, and we develop descent theory for topologies comes from equivariant cd-structures. In particular, we reprove several results of Cisinski-Déglies on the étale descent, qfh-descent, and h-descent. Since the étale topos, qfh-topos, and h-topos do not come from usual cd-structures, such results cannot be produced by usual cd-structures. We also apply equivariant cd-structures to study several topologies on the category of noetherian fs log schemes.

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