arXiv subjects

Tobias Shin

Publications and source records attributed to Tobias Shin.

6 recordsLinked to original sources

arXiv2025

Motivic Steenrod problem away from the characteristic

In topology, the Steenrod problem asks whether every singular homology class is the pushforward of the fundamental class of a closed oriented manifold. Here, we introduce an analogous question in algebraic geometry: is every element on the Chow line of the motivic cohomology of $X$ the pushforward of a fundamental class along a projective derived-lci morphism? If $X$ is a smooth variety over a field of characteristic $p \geq 0$, then a positive answer to this question follows up to $p$-torsion from resolution of singularities by alterations. However, if $X$ is singular, then this is no longer necessarily so: we give examples of motivic cohomology classes of a singular scheme $X$ that are not $p$-torsion and are not expressible as such pushforwards. A consequence of our result is that the Chow ring of a singular variety cannot be expressed as a quotient of its algebraic cobordism ring, as suggested by the first-named-author in his thesis.

math.AG↗

arXiv2024

A priori obstructions to resolution of singularities

We present an argument due to Thom to formulate a priori cohomology obstructions for a projective variety to admit an embedded resolution of singularities, and generalize the argument to a field of characteristic $p > 0$. We show that these obstructions are defined for a general cohomology theory that satisfies localization, étale excision, and $\mathbb{A}^1$-invariance, and is equipped with a proper pushforward. As examples, we show that odd degree Steenrod homology operations on higher Chow groups mod $\ell$, defined by a suitable $E_{\infty}$-algebra structure, vanish on the fundamental class of a smooth variety, where $\ell$ is any prime. We extend this general vanishing result to arbitrary varieties for operations defined mod $\ell \neq p$. Along the way, we also obtain a Wu formula for the mod $p$ Steenrod operations in the case of closed embeddings of smooth varieties.

math.AG↗

arXiv2021

Almost complex manifolds with small Nijenhuis tensor

We give several explicit examples of compact manifolds with a $1$-parameter family of almost complex structures having arbitrarily small Nijenhuis tensor in the $C^0$-norm. The $4$-dimensional examples possess no complex structure, whereas the $6$-dimensional example does not possesses a left invariant complex structure, and whether it possesses a complex structure appears to be unknown.

math.DG↗

arXiv2021

Prismatic cohomology and $p$-adic homotopy theory

This short note regards an observation about the recent theory of prismatic cohomology developed by Bhatt and Scholze. In particular, by applying a functor of Mandell, we see that the étale comparison theorem in the prismatic theory reproduces the $p$-adic homotopy type for a smooth proper complex variety with good reduction mod $p$.

math.AG↗

arXiv2021

Directed immersions for complex structures

We analyze the differential relation corresponding to integrability of almost complex structures, reformulated as a directed immersion relation by Demailly and Gaussier. Combining results of Clemente [3], we show that applying h-principle techniques yields the following statement: for an almost complex manifold with arbitrary metric $(X, J, g)$, and for $ε> 0$, there exists a smooth function $f : X \rightarrow \mathbb{R}$ and almost complex structure $J'$ on $X$ such that $J$ and $J'$ are $C^0$-close on the graph of $f$ with respect to the extended metric on $X \times \mathbb{R}$, and such that the Nijenhuis tensor of $J'$ on the graph has pointwise sup norm less than $Cε$, where $C$ is a constant depending only on $J$ and $g$. This is an updated version of a previous preprint titled "Almost complex manifolds are (almost) complex".

math.DG↗

arXiv2020

Microflexiblity and local integrability of horizontal curves

Let $ξ$ be an analytic bracket-generating distribution. We show that the subspace of germs that are singular (in the sense of Control Theory) has infinite codimension within the space of germs of smooth curves tangent to $ξ$. We formalise this as an asymptotic statement about finite jets of tangent curves. This solves, in the analytic setting, a conjecture of Y. Eliashberg and N.M. Mishachev regarding an earlier claim by M. Gromov about the microflexibility of the tangency condition. From these statements it follows, by an argument due to M. Gromov, that the $h$-principle holds for maps and immersions transverse to $ξ$.

math.DG↗