The logarithmic $h$- and $v$-topologies
We introduce the $h$- and $v$-topologies in the context of logarithmic geometry and discuss their applications to log \'etale cohomology, log differential forms, and log motives.
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arXiv subjects
Nikolai Opdan
Publications and source records attributed to Nikolai Opdan.
Logarithmic motives with compact support
We develop a theory of motives with compact support for logarithmic schemes over a field. Starting from the notion of finite logarithmic correspondences with compact support, we define the logarithmic motive with compact support analogous to the classical case for schemes. We then establish an analog of a Gysin sequence and, assuming resolution of singularities, a K{\"u}nneth formula. This implies that our theory is $\overline{\square}$-invariant, which presents a critical feature that is absent in the classical case. Further assuming resolution of singularities, we prove a duality theorem for log schemes which we use to establish a cancellation theorem for log schemes whose underlying scheme is proper. Moreover, we discuss new homology and cohomology theories for log smooth fs logarithmic schemes based on our results.
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Lectures on the cohomology of reciprocity sheaves
These are the notes accompanying three lectures given by the second author at the Motivic Geometry program at CAS, which aim to give an introduction and an overview of some recent developments in the field of reciprocity sheaves.
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Introduction to Framed Correspondences
We give an overview of the theory of framed correspondences in motivic homotopy theory. Motivic spaces with framed transfers are the analogue in motivic homotopy theory of $E_{\infty}$-spaces in classical homotopy theory, and in particular they provide an algebraic description of infinite $\mathbb{P}^1$-loop spaces. We will discuss the foundations of the theory (following Voevodsky, Garkusha, Panin, Ananyevskiy, and Neshitov), some applications such as the computations of the infinite loop spaces of the motivic sphere and of algebraic cobordism (following Elmanto, Hoyois, Khan, Sosnilo, and Yakerson), and some open problems.
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