Negative $K$-theory and Hodge theory
We study the negative $K$-groups of complex varieties from a mixed Hodge-theoretic perspective, making use of the theory of higher singularities, Chow groups, and the Minimal Model Program.
math.AG↗
arXiv subjects
Wanchun Shen
Publications and source records attributed to Wanchun Shen.
On higher Du Bois singularities and $K$-regularity
We study the relationship between higher Du Bois singularities and $K$-regularity, a notion that measures the $\mathbb{A}^1$-invariance of the algebraic $K$-groups. Building on this relationship, we establish a strengthened form of Vorst's conjecture for local complete intersections in characteristic zero. Our work also provides tools to construct new examples that illustrate various phenomena in the study of $K$-regularity. The main inputs for our results are vanishing theorems for the Du Bois complexes.
math.AG↗
Elimination Templates in Macaulay2
We introduce the package \texttt{EliminationTemplates} for the Macaulay2 computer algebra system, which provides tools for constructing automatic solvers for families of zero-dimensional radical ideals depending on algebraically independent parameters. This article provides a self-contained description of how elimination templates are constructed for such families and their specialization properties. Additionally, we describe the main functionality and datatypes provided by our package, and illustrate its usage on several examples, including applications from computer vision from which elimination templates originated.
math.AC↗
Injectivity and Vanishing for the Du Bois Complexes of Isolated Singularities
We prove an injectivity theorem for the cohomology of the Du Bois complexes of varieties with isolated singularities. We use this to deduce vanishing statements for the cohomologies of higher Du Bois complexes of such varieties. Besides some extensions and conjectures in the non-isolated case, we also provide analogues for intersection complexes.
math.AG↗
Du Bois complexes of cones over singular varieties, local cohomological dimension, and K-groups
We compute the Du Bois complexes of abstract cones over singular varieties, and use this to describe the local cohomological dimension and the non-positive K-groups of such cones.
math.AG↗
Local vanishing for toric varieties
Let $X$ be a toric variety. We establish vanishing (and non-vanishing) results for the sheaves $R^if_*Ω^p_{\tilde X}(\log E)$, where $f: \tilde{X} \to X$ is a strong log resolution of singularities with reduced exceptional divisor $E$. These extend the local vanishing theorem for toric varieties in [MOP20]. Our consideration of these sheaves is motivated by the notion of $k$-rational singularities introduced by Friedman and Laza [FL22b]. In particular, our results lead to criteria for toric varieties to have $k$-rational singularities, as defined in [SVV23].
math.AG↗
On $k$-Du Bois and $k$-rational singularities
We introduce new notions of $k$-Du Bois and $k$-rational singularities, extending the previous definitions in the case of local complete intersections (lci), to include natural examples outside of this setting. We study the stability of these notions under general hyperplane sections and show that varieties with $k$-rational singularities are $k$-Du Bois, extending previous results in [MP22b] and [FL22b] in the lci and the isolated singularities cases. In the process, we identify the aspects of the theory that depend only on the vanishing of higher cohomologies of Du Bois complexes (or related objects), and not on the behaviour of the Kähler differentials.
math.AG↗
Quasi-Assouad dimensions for random measures supported on $[0,1]^d$
We introduce a probability distribution on $\mathcal{P}([0,1]^d)$, the space of all Borel probability measures on $[0,1]^d$. Under this distribution, almost all measures are shown to have infinite upper quasi-Assouad dimension and zero lower quasi-Assouad dimension (hence the upper and lower Assouad dimensions are almost surely infinite or zero). We also indicate how the results extend to other Assouad-like dimensions.
math.MG↗
The $L^q$-spectrum for a class of self-similar measures with overlap
It is known that the heuristic principle, referred to as the multifractal formalism, need not hold for self-similar measures with overlap, such as the $3$-fold convolution of the Cantor measure and certain Bernoulli convolutions. In this paper we study an important function in the multifractal theory, the $L^{q}$-spectrum, $τ(q)$, for measures of finite type, a class of self-similar measures that includes these examples. Corresponding to each measure, we introduce finitely many variants on the $% L^{q}$-spectrum which arise naturally from the finite type structure and are often easier to understand than $τ$. We show that $τ$ is always bounded by the minimum of these variants and is equal to the minimum variant for $q\geq 0$. This particular variant coincides with the $L^{q}$-spectrum of the measure $μ$ restricted to appropriate subsets of its support. If the IFS satisfies particular structural properties, which do hold for the above examples, then $τ$ is shown to be the minimum of these variants for all $q$. Under certain assumptions on the local dimensions of $μ$, we prove that the minimum variant for $q \ll 0$ coincides with the straight line having slope equal to the maximum local dimension of $μ$. Again, this is the case with the examples above. More generally, bounds are given for $τ$ and its variants in terms of notions closely related to the local dimensions of $μ$.
math.CA↗
The Entropy of Cantor--like measures
By a Cantor-like measure we mean the unique self-similar probability measure $μ$ satisfying $μ=\sum_{i=0}^{m-1}p_{i}μ\circ S_{i}^{-1}$ where $% S_{i}(x)=\frac{x}{d}+\frac{i}{d}\cdot \frac{d-1}{m-1}$ for integers $2\leq d 0$, $\sum p_{i}=1$. In the uniform case ($p_{i}=1/m$ for all $i$) we show how one can compute the entropy and Hausdorff dimension to arbitrary precision. In the non-uniform case we find bounds on the entropy.
math.MG↗