Zhaobo Tom Han | Searcharxiv

arXiv2024

Characteristic Currents on Cohesive Modules

Let $\mathcal{F}$ be a coherent sheaf on a complex variety $X$ that has a locally free resolution $E^{\bullet}$. In [19], the authors constructed a pseudomeromorphic current whose support is contained in $supp(E^{\bullet})$ that represents products of Chern classes of $\mathcal{F}.$ In this paper, we show that their construction works for general de-Rham characteristic classes and then generalize it to represent products (in de-Rham cohomology) of characteristic forms of cohesive modules defined by Block. Finally, we state a corollary to a transgression result in [16] that show that it is sufficient to only use the degree-$0$ and degree-$1$ parts of the superconnection to construct currents that represent characteristic forms of cohesive modules in the Bott-Chern cohomology.

math.AG↗

arXiv2018

Variations on the Arkhipov-Karačuba Type Counterexamples to Artin's Conjecture

It was conjectured by Emil Artin in the 1930's that every $d$-form $F(x_1, x_2, $\ldots$, x_n)$ over the $p$-adic field in more than $d^2$ variables has a solution that is not $(0, 0, \cdots, 0)$ (non-trivial solution) over the $p$-adic field. This is true for $d=2$ and $d=3$. However, many counterexamples for $d \geq 4$ were later discovered. The major types of counterexamples are Terjanian Type and Arkhipov-Karačuba Type. The degrees of all known counterexamples, however, are divisible by $p-1$, which means that they are even for all odd primes. In this article we apply modifications to the known Arkhipov-Karačuba Type counterexamples to construct counterexamples with odd degrees that are divisible by $\frac{p-1}2$ for all primes greater than $3$ and congruent to $3$ modulo $4$ and then propose some ideas about increasing the number of variables in the counterexamples.

math.NT↗