Doosung Park

Doosung Park
University of Zurich | UZH · Institut für Mathematik

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10
Publications
715
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4
Citations
Citations since 2016
10 Research Items
4 Citations
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2016201720182019202020212022012345
2016201720182019202020212022012345

Publications

Publications (10)
Preprint
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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 Andr\'e-Quillen type spectral sequence. The latter degenerates for derived log smooth maps between discrete pre-log rings. We employ this to show a...
Article
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This document is a short user’s guide to the theory of motives and homotopy theory in the setting of logarithmic geometry. We review some of the basic ideas and results in relation to other works on motives with modulus, motivic homotopy theory, and reciprocit
Preprint
Full-text available
This document is a short user's guide to the theory of motives and homotopy theory in the setting of logarithmic geometry. We review some of the basic ideas and results in relation to other works on motives with modulus, motivic homotopy theory, and reciprocity sheaves.
Preprint
Full-text available
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 idea of parameterizing homotopies by $\overline{\square}$, i.e. the projective line with respect t...
Preprint
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For any noetherian scheme $X$ smooth and separated over an algebraically closed field $k$, we describe the Albanese variety ${\rm Alb}\,X$ of $X$ in the world of ${{\rm DM}^{e\! f\! f}(k)}$. As an application, we explain the structure of the Picard functor ${\rm Pic}_{X/k}^0$.
Article
Let $X$ be an $n$-dimensional connected scheme smooth and projective over ${\bf C}$. We decompose the motive $\underline{\rm Hom}({\bf L}^{n-2},M(X))$ using intermediate Jacobians. We also construct a morphism $M_{2n-2}(X)\rightarrow M(X)$ induced by a conjectural Chow-K\"unneth decomposition of $M(X)$.
Article
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Assuming the K\"unneth type standard conjecture, we propose the definition of mixed motives. 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...
Article
Using the localization property, we construct a triangulated category of motives over quasi-projective T-schemes for any coefficient where T is a noetherian separated scheme, and we prove the Grothendieck six operations formalism. We also construct integral \'etale realization of motives.
Article
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.
Article
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-D\'eglies on the \'etale descent, qfh-descent, and h-descent. Since the \'etale topos, qfh-topos, and h-topos do not come from usual cd-structures, such resul...

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Project (1)
Project
In this project we set forth a theory of log motives over fields based on the notion of a finite correspondence between log schemes. Palpable properties such as box homotopy invariance, Mayer-Vietoris for coverings, and a symmetric monoidal structure witness the robustness of the setup. Some of the finer properties are concerned with a projective bundle theorem, a blow-up distinguished triangle, and a Gysin distinguished triangle. Beyond this we show that Voevodsky’s category of derived motives is the A1-localization of the category of log motives, and that Hodge cohomology of log schemes is representable in our framework.