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Sanghoon Baek

Publications and source records attributed to Sanghoon Baek.

At least 19 recordsLinked to original sources
arXiv2026

The Chow Characteristic Image of \(\Spin(10)\) via the Affine Cone over the Spinor Variety

Let \(G=\Spin(10)\) be the split spin group over a field of characteristic different from \(2\), and let \(T\subset G\) be a split maximal torus. We determine the image of the integral Chow restriction map \(\CH(BG)\to \CH(BT)^W\), equivalently \(\CH(BG)\) modulo torsion. The main new geometric ingredient in the proof is a construction of the class \(c_2c_3c_5\), where the \(c_i\) are the elementary Chern classes after restriction to \(T\). This class is obtained from the proper equivariant push-forward associated with the affine cone over the spinor variety in its half-spin embedding for the special Clifford group \(Γ^+(10)\). Modulo two, the image is the subring generated, over the smallest Steenrod-stable subring of \(\F[c_2,c_3,c_4,c_5]\) containing \(c_2^2,c_3^2,c_4^2,c_5\) and \(c_2c_3c_5\), by the torus restriction of the top Chern class of a half-spin representation. The integral characteristic image is the full inverse image of this mod-two subring under reduction modulo \(2\).

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arXiv2026

Integral Weyl Invariants in Chow Characteristic Images of Spin and Special Clifford Groups

Let $G=\Spin(n)$ be the split spin group over an arbitrary field, with $n\ge7$. Extending a Steenrod-theoretic obstruction of Karpenko, we classify the recursively defined integral Weyl invariants $q_i$ in the Benson--Wood generating set that lie in the Chow characteristic image: the only such invariant is $q_3$ for $\Spin(10)$. We obtain the analogous classification for the recursive invariants $f_i$ of the special Clifford group $Γ^+(n)$: in their finite generating range, the only such invariant is $f_2$ for $Γ^+(7)$. Over $\mathbb C$, the class corresponding to $q_i$ in the torsion-free quotient of the integral cohomology of the classifying space $BG$ is algebraic precisely when $(n,i)=(10,3)$. For each $n$, a single smooth projective approximation simultaneously realizes all the corresponding classes in the finite range. Every nonexceptional class remains nonalgebraic after the addition of any torsion class, as detected by a Bockstein--Steenrod operation.

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arXiv2025

Upper Bounds on the Torsion Index of Half-Spin Groups

The torsion index of split simple groups has been extensively studied, notably by Totaro, who calculated the torsion indexes of the spin groups and $E_{8}$ in [5] and [6], respectively. The aim of this paper is to provide upper bounds for the torsion index of half-spin groups, the only remaining case in the calculation of torsion indexes for split simple groups. We present general upper bounds for the torsion index of half-spin groups, showing that, except for certain exceptional cases, it is at most twice that of the corresponding spin groups. For these exceptional cases, the torsion index is bounded above by at most $2^3$ times that of the spin groups. Our results also reveal that in many cases, the torsion index of half-spin groups coincides with that of the spin groups.

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arXiv2023

Essential dimension of reductive groups via generically free representations

We provide a simple method to compute upper bounds on the essential dimension of split reductive groups with finite or connected center by means of their generically free representations. Combining our upper bound with previously known lower bound, the exact value of the essential dimension is calculated for some types of reductive groups. As an application, we determine the essential dimension of a semisimple group of classical type or $E_{6}$, and its strict reductive envelope under certain conditions on its center. This extends previous works on simple simply connected groups of type $B$ or $D$ by Brosnan-Reichstein-Vistoli and Chernousov-Merkurjev, strict reductive envelopes of groups of type $A$ by Cernele-Reichstein, and semisimple groups of type $B$ by the authors to any classical type and type $E_{6}$ in a uniform way.

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arXiv2023

Counter-examples to a conjecture of Karpenko for spin groups

Consider the canonical morphism from the Chow ring of a smooth variety $X$ to the associated graded ring of the topological filtration on the Grothendieck ring of $X$. In general, this morphism is not injective. However, Nikita Karpenko conjectured that these two rings are isomorphic for a generically twisted flag variety $X$ of a semisimple group $G$. The conjecture was first disproved by Nobuaki Yagita for $G=\mathop{\mathrm{Spin}}(2n+1)$ with $n=8, 9$. Later, another counter-example to the conjecture was given by Karpenko and the first author for $n=10$. In this note, we provide an infinite family of counter-examples to Karpenko's conjecture for any $2$-power integer $n$ greater than $4$. This generalizes Yagita's counter-example and its modification due to Karpenko for $n=8$.

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arXiv2022

Maximal orthogonal grassmannians of quadratic forms of dimensions up to $22$

Let $X$ be a connected component of the maximal orthogonal grassmannian of a generic $n$-dimensional quadratic form $q$ with trivial Clifford invariant. Consider the canonical epimorphism $ϕ$ from the Chow ring of $X$ to the associated graded ring of the coniveau filtration on the Grothendieck ring of $X$. In \cite{Kar2018} Karpenko proved that $ϕ$ is an isomorphism for all $n\leq 12$ (conjecturally for all $n$). Recently, in \cite{Yagita} Yagita showed that $ϕ$ is not an isomorphism for $n=17, 18$. In the present paper, together with Yagita's results for $n=17,18$, we show that the map $ϕ$ is not an isomorphism for all $13\leq n\leq 22$. In particualr, the case $n=13$ gives the smallest dimensional maximal orthogonal grassmannian whose two graded rings are not isomorphic.

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arXiv2021

Essential dimension of semisimple groups of type $B$

We determine the essential dimension of an arbitrary semisimple group of type $B$ of the form \[G=\big(\operatorname{\mathbf{Spin}}(2n_{1}+1)\times\cdots \times \operatorname{\mathbf{Spin}}(2n_{m}+1)\big)/\boldsymbolμ\] over a field of characteristic $0$, for all $n_{1},\ldots, n_{m}\geq 7$, and a central subgroup $\boldsymbolμ$ of $\operatorname{\mathbf{Spin}}(2n_{1}+1)\times\cdots \times \operatorname{\mathbf{Spin}}(2n_{m}+1)$ not containing the center of $\operatorname{\mathbf{Spin}}(2n_i+1)$ as a direct factor. We also find the essential dimension of $G$ for each of the following cases, where either $n_{i}=1$ for all $i$ or $m=2$, $n_{1}=1$, $2\leq n_{2}\leq 3$, $\boldsymbolμ$ is the diagonal central subgroup for both cases.

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arXiv2019

Degree 3 unramified cohomology of classifying spaces for exceptional groups

Let $G$ be a reductive group defined over an algebraically closed field of characteristic $0$ such that the Dynkin diagram of $G$ is the disjoint union of diagrams of types $G_{2}, F_{4}, E_{6}, E_{7}, E_{8}$. We show that the degree $3$ unramified cohomology of the classifying space of $G$ is trivial. In particular, combined with articles by Merkurjev \cite{Mer17} and the author \cite{Baek}, this completes the computations of degree $3$ unramified cohomology and reductive invariants for all split semisimple groups of a homogeneous Dynkin type.

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arXiv2019

Degree three invariants for semisimple groups of types $B$, $C$, and $D$

We determine the group of reductive cohomological degree $3$ invariants of all split semisimple groups of types $B$, $C$, and $D$. We also present a complete description of the cohomological invariants. As an application, we show that the group of degree $3$ unramified cohomology of the classifying space $BG$ is trivial for all split semisimple groups $G$ of types $B$, $C$, and $D$.

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arXiv2016

The K-theory of versal flags and cohomological invariants of degree 3

Let $G$ be a split semisimple linear algebraic group over a field and let $X$ be a generic twisted flag variety of $G$. Extending the Hilbert basis techniques to Laurent polynomials over integers we give an explicit presentation of the Grothendieck ring $K_0(X)$ in terms of generators and relations in the case $G=G^{sc}/μ_2$ is of Dynkin type ${\rm A}$ or ${\rm C}$ (here $G^{sc}$ is the simply-connected cover of $G$); we compute various groups of (indecomposable, semi-decomposable) cohomological invariants of degree 3, hence, generalizing and extending previous results in this direction.

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arXiv2016

Commuting involution graphs of linear groups

In this paper, we determine the diameter of the commuting involution graphs of special and general linear groups over an arbitrary field. It turns out that our results also determine the diameter for certain projective special linear groups over finite fields. Moreover, we find the diameter of the commuting graphs of general linear groups on the set of all involutions over a field of characteristic 2, which completes the diameter of general linear groups on the set of all involutions. As an application, we classify the structure of the four-dimensional linear groups over finite fields according to the distance from a fixed involution.

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arXiv2015

Codimension 2 cycles on products of projective homogeneous surfaces

In the present paper, we provide general bounds for the torsion in the codimension 2 Chow groups of the products of projective homogeneous surfaces. In particular, we determine the torsion for the product of four Pfister quadric surfaces and the maximal torsion for the product of three Severi-Brauer surfaces. We also find an upper bound for the torsion of the product of three quadric surfaces with the same discriminant.

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arXiv2015

Chow groups of products of Severi-Brauer varieties and invariants of degree 3

We study the semi-decomposable invariants of a split semisimple group and their extension to a split reductive group by using the torsion in the codimension $2$ Chow groups of a product of Severi-Brauer varieties. In particular, for any $n\geq 2$ we completely determine the degree $3$ invariants of a split semisimple group, the quotient of $(\operatorname{\mathbf{SL}}_{2})^{n}$ by its maximal central subgroup, as well as of the corresponding split reductive group. We also provide an example illustrating that a modification of our method can be applied to find the semi-decomposable invariants of a split semisimple group of type A.

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arXiv2014

A lower bound on the essential dimension of $\operatorname{\mathbf{PGL}}_{4}$ in characteristic $2$

In the present paper, we provide a lower bound of the essential dimension over a field of positive characteristic via Kato's cohomology group, defined by cokernel of a general Artin-Schreier operator. Combining this with Tignol's result on the second trace form of simple algebras of degree $4$, we show that $\operatorname{ed}(\operatorname{\mathbf{PGL}}_{4})\geq 4$ over a field of characteristic $2$.

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arXiv2014

On the torsion of Chow groups of Severi-Brauer varieties

In this paper, we generalize a result of Karpenko on the torsion in the second quotient of the gamma filtration for Severi-Brauer varieties to higher degrees. As an application, we provide a nontrivial torsion in higher Chow groups and the topological filtration of the associated generic variety and obtain new upper bounds for the annihilators of the torsion subgroups in the Chow groups of a large class of Severi-Brauer varieties. In particular, using the torsion in higher degrees, we show indecomposability of certain algebras.

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arXiv2013

Erratum: On the torsion of Chow groups of twisted Spin-flags

In the erratum we correct a mistake (due to a wrong choice of basic polynomial invariants over Z[1/2]) in the original paper (v1). Using the correct basic polynomial invariants we improve our results and bounds on the annihilator. We also simplify some of the proofs.

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