Seminar zur Motiven SS 2026

Motives Seminar-Invited Lectures

This semester the motives seminar will be a series of invited talks. We meet in WSC-S-U-3.01, 16-18 Uhr (c.t.). Talks will also be available via Zoom; the Zoom link will be in the lecture announcement. If you are not on our mailing list and would like the link, please send me a request at marc.levine@uni-due.de, similarly if you are interested in giving a talk.

Schedule

April 14, Xiaowen Dong (Essen). Unstable relations for motivic Hopf maps and motivic Toda brackets

Abstract In classical topology we have the famous Hopf maps $\eta_{top}: S^3\rightarrow S^2$ and $\nu_{top}:S^7\rightarrow S^4$. The first Hopf map $\eta_{top}$ is the canonical map from the unit sphere in $\mathbb{C}^2$ to the complex projective space $\mathbb{C}\mathbb{P}^1$. Instead of complex numbers we can also use quaternions and get the second Hopf map $\nu_{top}$ in the same way. Furthermore if we consider the stabilizations of both Hopf maps, we have the relations $2\eta_{top}=0$ and $\eta_{top}\nu_{top}=0$ in classical stable homotopy theory.

In motivic homotopy theory we have analogues of the topological Hopf maps. For this talk we work over the base scheme $\mathrm{Spec} \ \mathbb{Z}$. The first motivic Hopf map $\eta$ is defined to be the canonical projection from $\mathbb{A}_{\mathbb{Z}}^2-\{0\}$ to $\mathbb{P}_{\mathbb{Z}}^1$. Moreover, this map can also be constructed via the Hopf construction on the scheme $\mathbb{G}_{m}$. The Hopf construction on $\mathrm{SL}_{2}$ gives us the second motivic Hopf map $\nu$. The complex realizations of $\eta$ and $\nu$ are the corresponding topological Hopf maps.

One can consider the $\mathbb{P}^1$-stabilizations of $\eta$ and $\nu$ in the motivic stable homotopy category SH$(\mathbb{Z})$. One also has the endomorphism $h:=\langle 1\rangle+\langle -1\rangle$ of the unit $\mathbb{S}_{\mathbb{Z}}$ in SH$(\mathbb{Z})$, with complex realisation $2\in\pi_{0}(\mathbb{S})=\mathbb{Z}$; here the endomorphisms $\langle \pm 1\rangle$ of $\mathbb{S}_{\mathbb{Z}}$ are the $\mathbb{P}^1$-stabilizations of the automorphisms $[x_0:x_1]\mapsto [x_0:\pm x_1]$ of $\mathbb{P}_{\mathbb{Z}}^1$. Dugger, Isaksen and Morel prove the relations $h\eta=\eta h=0$ and $\eta\nu=\nu\eta=0$ in SH$(\mathbb{Z})$.

In this talk, we will discuss versions of these relations in the motivic unstable homotopy category $\mathcal{H}(\mathbb{Z})$, and show how one obtains non-trivial examples of Toda brackets from these nullhomotopic relations.

April 21, Sil Linskens (Universität Regensburg). A 2-categorical approach to building six functor formalisms

Abstract The idea of six functors formalisms originates in Grothendieck's work on duality for étale cohomology of schemes. Much more recently, a simple and powerful definition of this structure was given using the theory of higher categories. This has greatly improved our ability to work with such structures. However it does not simplify our task of constructing six functor formalisms, and in fact a priori it makes it much harder. Nevertheless, work of Liu-Zheng formalized the most important construction principle, which goes back to the original work of Artin, Grothendieck and Verdier on the six functor formalism on étale cohomology. I will explain a new approach to this construction principle which is joint work with Bastiaan Cnossen and Tobias Lenz. To do this we recast the problem as that of computing a certain universal ($\infty$,2)-category, which we then do by combining methods from parametrized and ($\infty$,2)-category theory.

April 28, Svetlana Makarova (University of Melbourne).Contractions and flops via moduli of non-pure sheaves.

Abstract: When introducing the notion of moduli problems, perhaps the most trivial example one may consider is the "functor of points" of a variety $X$, whose "classifying space" (moduli space) is $X$ itself. Its very close relative is the stack of ideal sheaves of length 1 subschemes: when $X$ is normal and of dimension at least 2, this stack is the product of $X$ with $B\mathbb{G}_m$. Despite the apparent triviality, modifying this stack gives rise to interesting birational phenomena, about which I will speak. I will describe a certain enlargement $U$ of the stack of ideal sheaves that yields, via taking the good moduli space, a birational contraction of $X$, and an open substack of $U$ that yields a surgery diagram via non-GIT wall-crossing. If time permits, I will explain how the interpretation of the surgery as a fine moduli of sheaves helps prove instances Kawamata’s DK-hypothesis in this setting. This is joint work with Andres Fernandez Herrero.

May 5, Andrei Konovalov, t.b.a.

May 12, N.N.

May 26, N.N.

June 23, Louisa Bröring (Essen) t.b.a.

July 7, N.N.

July 14, N.N.

July 21, N.N.