Parshin's conjecture revisited

Parshin's conjecture and motivic cohomology with compact support

We discuss Parshin's conjecture on rational K-theory over finite fields and its implications for motivic cohomology with compact support.

HPD-invariance of the Tate, Beilinson and Parshin conjectures

We prove that the Tate, Beilinson and Parshin conjectures are invariant under Homological Projective Duality (=HPD). As an application, we obtain a proof of these celebrated conjectures (as well as of the strong form of the Tate conjecture) in the new cases of linear sections of determinantal varieties and complete intersections of quadrics. Furthermore, we extend the original conjectures of Tate, Beilinson and Parshin from schemes to stacks and prove these extended conjectures for certain low-dimensional global orbifolds.

Proof of the Parshin's Conjecture

We prove the Parshin's conjecture on the rational triviality of the higher algebraic $K$-theory of smooth projective varieties over finite fields. This is known to imply the Beilinson-Soulé conjecture for the fields of positive characteristic. Especially it implies that for a field $F$ of char $p$ and $n>0$, the only rationally non-trivial weight appearing in $K_n(F)\otimes \mathbb{Q}$ can be $n$, thus $K_n(F)\otimes \mathbb{Q}=K_n^M(F)\otimes \mathbb{Q}$ where $K_n^M(F)$ is the Milnor $K$-theory.

Algebraic cobordism revisited

We define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities. The main result is the equivalence of double point cobordism to the theory of algebraic cobordism previously defined by Levine and Morel. Double point cobordism provides a simple, geometric presentation of algebraic cobordism theory. As a corollary, the Lazard ring given by products of projective spaces rationally generates all nonsingular projective varieties modulo double point degenerations. Double point degenerations arise naturally in relative Donaldson-Thomas theory.…

K-theory of toric varieties revisited

After surveying higher K-theory of toric varieties, we present Totaro's old (c. 1997) unpublished result on expressing the corresponding homotopy theory via singular cohomology. It is a higher analog of the rational Chern character isomorphism for general toric schemes. In the special case of a projective simplicial toric scheme over a regular ring one obtains a rational isomorphism between the homotopy K-theory and the direct sum of m copies of the K-theory of the ground ring, m being the number of maximal cones in the underlying fan. Apart from its independent interest, in retrospect, Totaro…

重新审视PreLie operad内部的精细结构

Fine structures inside the PreLie operad revisited

本文证明了Chapoton 2010年的猜想：在线性物种的对称幺半范畴中，pre-Lie operad作为李代数由循环李元素物种上的自由operad自由生成。