# Schedule of the Workshop "K-theory in algebraic geometry and number theory"

## Monday, May 15

10:15 - 10:50 |
Registration & Welcome coffee |

10:50 - 11:00 |
Opening remarks |

11:00 - 12:00 |
Ben Antieau: Negative and homotopy K-theoretic extensions of the theorem of the heart |

12:00 - 13:50 |
Lunch break |

13:50 - 14:50 |
Joseph Ayoub: Hodge theory for the conservativity conjecture |

15:00 - 16:00 |
Tom Bachmann: The generalized slices of hermitian K-theory and applications |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:30 |
Gonçalo Tabuada: Additive invariants of orbifolds |

afterwards |
Reception |

## Tuesday, May 16

09:30 - 10:30 |
Peter Scholze: On topological cyclic homology |

10:30 - 11:00 |
Group photo and coffee break |

11:00 - 12:00 |
Matthew Morrow: p-adic vanishing cycles and the relative de Rham-Witt complex |

12:00 - 15:00 |
Lunch break and discussions |

15:00 - 16:00 |
Ryomei Iwasa: Chern classes with modulus: toward the comparison of relative K-theory and relative motivic cohomology |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:30 |
Lie Fu: K-theoretical and motivic hyperKähler resolution conjecture |

19:00 - |
Conference dinner at the Restaurant Meyer's (Clemens-August-Str. 51a) |

## Wednesday, May 17

09:30 - 10:30 |
Paul Arne Østvær: A^{1} contractible varieties |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Jens Hornbostel: Chow-Witt groups of classifying spaces |

12:00 - 15:00 |
Lunch break and discussions |

15:00 - 16:00 |
Thomas Nikolaus: Polynomial functors and algebraic K-theory |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:30 |
Oliver Röndigs: The first and second stable homotopy groups of motivic spheres over a field |

## Thursday, May 18

09:30 - 10:30 |
Amalendu Krishna: Torsion in the 0-cycle groups |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Marco Schlichting: Homology stability and the K-theory of quadratic forms |

12:00 - 15:00 |
Lunch break and discussions |

15:00 - 16:00 |
Christian Dahlhausen: Improved Milnor K-theory of valuation rings of local fields |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:30 |
Satoshi Kondo: Norm compatible systems in motivic cohomology of Drinfeld modular schemes |

## Friday, May 19

09:30 - 10:30 |
Annette Huber-Klawitter: Beyond homotopy invariance: the case of differential forms |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Shuji Saito: Rigid analytic K-theory |

12:00 - |
Lunch break, farewell |

# Abstracts

## Ben Antieau: Negative and homotopy K-theoretic extensions of the theorem of the heart

Inspired by work of Neeman, Barwick proved that the K-theory of a stable infinity-category with a bounded t-structure agrees with the K-theory of its heart in non-negative degrees. Joint work with David Gepner and Jeremiah Heller extends this to an equivalence of nonconnective K-theory spectra when the heart satisfies certain finiteness conditions such as noetherianity. Applications to negative K-theory and homotopy K-theory of ring spectra are provided, which were the original motivation for our work.

## Joseph Ayoub: Hodge theory for the conservativity conjecture

The conservativity conjecture predicts that a correspondance between Chow motives is invertible if and only if its class in cohomology is invertible. I will discuss some aspects of a program aiming at proving the conservativity conjecture. More specifically, I'll try to explain how Hodge theory is used.

## Tom Bachmann: The generalized slices of hermitian K-theory and applications

The generalized slice filtration is a systematic way of filtering motivic spectra (and, consequently, constructing spectral sequences). Its definition is a minor modification of Voevodsky’s slice filtration, first proposed by Spitzweck-Oestvaer. We show that the generalized slices of the Hermitian K-theory motivic spectrum can be computed completely, in terms of motivic cohomology and (a version of) generalized motivic cohomology. This answers a question of Schlichting. As a consequence, we obtain a (strongly convergent) spectral sequence computing Hermitian K-theory from generalized motivic cohomology, similar to the motivic spectral sequence for algebraic K-theory. As further applications we compute the coefficients of generalized motivic cohomology (in terms of the coefficients of ordinary motivic cohomology) and show that the motivic cohomology mod 2 of the generalized slice-connective cover of Hermitian K-theory is given by A//A(1), again paralleling the topological picture. This answers a question of Isaksen.

The first half of this talk will focus on definitions and the interplay of the results. In the second half we will sketch the proof of the main theorem, i.e. the computation of the slices of KO.

## Christian Dahlhausen: Improved Milnor K-theory of valuation rings of local fields

Milnor K-theory is an interesting cohomology theory for fields and also for local rings with infinite residue fields. Dealing with arbitrary local rings, it is better to work with improved Minor K-theory which was introduced by Gabber and developed by Kerz. We study this improved Milnor K-theory of the valuation ring of a local field and prove a Gersten type conjecture for it.

## Lie Fu: K-theoretical and motivic hyperKähler resolution conjecture

Inspired by string theory, Yongbin Ruan proposed the so-called Cohomological HyperKähler Resolution Conjecture (CHRC) which says that the orbifold/Chen-Ruan cohomology ring of a Gorenstein orbifold is isomorphic to the cohomology ring of a hyperKähler resolution. CHRC, as well as its non-hyperKähler version on crepant resolutions, attracted a lot of research work ever since. I would like to report a series of joint work with Zhiyu Tian and Charles Vial on the motivic version of the HyperKähler Resolution Conjecture, as an approach of study the Chow rings, or more generally the motives, of holomorphic symplectic varieties. Our Motivic HyperKähler Resolution Conjecture (MHRC) predicts that the orbifold Chow ring/motive of a singular holomorphic symplectic variety (in the sense of Beauville-Namikawa) is isomorphic to the Chow ring/motive of a symplectic resolution. We confirm the MHRC in the cases of surfaces, Hilbert schemes of K3 surfaces, Hilbert schemes of abelian surfaces and generalized Kummer varieties. As applications, in each of the cases, (1) we deduce Jarvis-Kaufmann-Kimura's K-theoretic HyperKähler Resolution Conjecture; (2) We provide a multiplicative Chow-Künneth decomposition as the candidate to Beauville's splitting of Bloch-Beilinson filtrations on Chow rings; (3) We improve the known results on Beauville-Voisin conjecture.

## Jens Hornbostel: Chow-Witt groups of classifying spaces

We compute the Chow-Witt rings of the classifying spaces for the symplectic and special linear groups. The computations for the symplectic groups show that Chow-Witt groups are a symplectically oriented ring cohomology theory. As a consequence of the computations for the special linear groups, we can show that an oriented vector bundle of odd rank splits off a trivial summand if its top Chern class and its top integral Stiefel-Whitney class are trivial.

This is joint work with Matthias Wendt.

## Annette Huber-Klawitter: Beyond homotopy invariance: the case of differential forms

Joint work with S. Kelly.

The aim of the project is the introduction of an improved version of differential forms for singular varieties with applications in birational geometry in mind. The case of positive characteristic is of particular importance. The methods are inspired by the theory of motives, but without homotopy invariance.

## Ryomei Iwasa: Chern classes with modulus: toward the comparison of relative K-theory and relative motivic cohomology

Binda-Saito have constructed the relative motivic cohomology of a modulus pair in terms of algebraic cycles. It is expected that there is an isomorphism (or isogeny) from the relative K-group to the relative motivic cohomology after tensing Q, which extends Bloch’s Riemann-Roch theorem. In a joint work with Wataru Kai, we have constructed Chern class maps from the relative K-group to the relative motivic cohomology, which would be an ingredient of the comparison. In this talk, I explain the construction of these Chern classes and show some of their properties.

If time permits, I will talk about my work in progress on the cycle class map with modulus.

## Satoshi Kondo: Norm compatible systems in motivic cohomology of Drinfeld modular schemes

Joint work with Seidai Yasuda.

1. We construct elements in the d-th K-group with rational coefficient of the integral model of the moduli of rank d Drinfeld modules that are related L-function of automorphic forms. This is a follow-up result of our 2012 result where we constructed elements for Drinfeld modular varieties (not integral model).

2. We show that elements constructed in a similar manner in the motivic cohomology with integral coefficients are norm compatible (form an Euler system). That is, via the norm map, we obtain expression in terms of Hecke operators that gives the L-factor. (We do not have a similar result for K-theory.)

## Amalendu Krishna: Torsion in the 0-cycle groups

The knowledge of torsion in the Chow group of 0-cycles has been an important part of study of algebraic cycles since the time Roitman proved his famous theorem. In this talk, we shall revisit this problem and, in particular, prove an extension of Roitman-Milne theorems to singular affine varieties. We shall then show how this extension allows one to get information about the torsion in the Chow group of 0-cycles with modulus on smooth affine varieties. We shall also derive some applications to the study of top rank vector bundles on affine varieties.

## Matthew Morrow: p-adic vanishing cycles and the relative de Rham-Witt complex

A description of p-adic vanishing cycles, on a smooth p-adic scheme, in terms of a logarithmic de Rham-Witt complex was given in 2005 by Geisser and Hesselholt; the proof was based on Bloch-Kato's famous work on p-adic vanishing cycles and Milnor K-theory. I will explain some related results which follow from the joint work "Integral p-adic Hodge theory" with Bhargav Bhatt and Peter Scholze.

## Thomas Nikolaus: Polynomial functors and algebraic K-theory

Besides being abelian groups, the K-theory groups of a commutative ring admit more structure: they form a graded commutative ring and admit Lambda-operations. We present an approach to understand these structures on the K-theory groups as coming form homotopy coherent structures on the K-theory spaces. This refines older work by Grayson, Hiller, Kretzer and Quilllen. As a result we find a notion of spectral lambda ring and discuss the consequences. This work ist joint with Clark Bawick, Saul Glasman and Akhil Mathew.

## Paul Arne Østvær: A^{1} contractible varieties

We'll discuss A^{1} homotopy theory and examples of smooth affine varieties which are contractible in this setting.

## Oliver Röndigs: The first and second stable homotopy groups of motivic spheres over a field

The talk will report on joint work (partly in progress) with Markus Spitzweck and Paul Arne Østvær. This work describes the 1-line and the 2-line of stable homotopy groups of the motivic sphere spectrum over a field of characteristic not two. More precisely, the kernel of the unit map to an appropriate connective cover of hermitian K-theory can be described in these lines via Milnor K-theory and motivic cohomology. The main computational tool is Voevodsky's slice spectral sequence.

## Shuji Saito: Rigid analytic K-theory

Let K be a field with a complete non-archimedean absolute value |⋅| and R = {x ∈ K | |x| ≤ 1} and fix π ∈ R with |π| < 1. Let X be a (formal) scheme over R and write X_{n} = X ⊗_{R} R/(π^{n+1}) for n ≥ 0. The continuous K-groups of X are defined as K_{i}^{cont}(X) := invlim K_{i}(X_{n}) (i ∈ ℤ) where K_{i}(X_{n}) are the algebraic K-groups of X_{n}. Thanks to works of Bloch-Esnault-Kerz and Morrow, the Hodge conjecture for abelian varieties has been reduced to an algebrization problem for K_{0}^{cont}(X) (in case R = ℂ[[t]]).

In this talk I explain a joint work with Moritz Kerz and Georg Tamme on a newly developed theory of analytic K-theory KH_{i}^{an}(Y) for rigid spaces Y over K. The construction is done by "pro-homotopization" and "analytic Bass delooping" of BGL for affinoids, and its globalization using descent for admissible coverings. I will explain a relation of KH_{i}^{an}(Y) with K_{i}^{cont}(X) for a formal model Χ of Y over R. I will also explain a natural isomorphism K_{0}(Y) ≃ KH_{0}^{an}(Y) for a regular affinoid Y.

## Marco Schlichting: Homology stability and the K-theory of quadratic forms

When studying the K-theory of symmetric bilinear forms, it turns out that the broader picture of quadratic forms is useful. In this talk, we will define a variant of Quillen's Q-construction for quadratic forms and explain the "Q=+" theorem in this context. The proof is inspired by methods coming from homology stability.

## Peter Scholze: On topological cyclic homology

Topological cyclic homology is an approximation to algebraic K-theory that has been very useful for computations in algebraic K-theory. Recently, it has also inspired some work in integral p-adic Hodge theory. Its definition however requires delicate tools from genuine stable homotopy theory, and explicit point-set models. In joint work with Thomas Nikolaus, we revisit this theory, by giving a simplified definition of the ∞-category of cyclotomic spectra, and corresponding simplified formulas for topological cyclic homology.

## Gonçalo Tabuada: Additive invariants of orbifolds

Using the recent theory of noncommutative motives, I will compute the additive invariants of orbifolds (equipped with a sheaf of Azumaya algebras) using solely "fixed-point data". As an application, I will then explain how this computation yields a proof of Grothendieck's standard conjectures of type C and D, as well as of Voevodsky's smash-nilpotence conjecture, in the case of "low-dimensional" orbifolds.

This is a joint work with Michel Van den Bergh.