Using Bochner-Martinelli type residual currents we prove some generalizations of Jacobis Residue Formula, which allow proper polynomial maps to have `common zeroes at infinity, in projective or toric situations.
We give new contributions to the existence problem of canonical surfaces of high degree. We construct several families (indeed, connected components of the moduli space) of surfaces $S$ of general type with $p_g=5,6$ whose canonical map has image $Sigma$ of very high degree, $d=48$ for $p_g=5$, $d=56$ for $p_g=6$. And a connected component of the moduli space consisting of surfaces $S$ with $K^2_S = 40, p_g=4, q=0$ whose canonical map has always degree $geq 2$, and, for the general surface, of degree $2$ onto a canonical surface $Y$ with $K^2_Y = 12, p_g=4, q=0$. The surfaces we consider are SIP s, i.e. surfaces $S$ isogenous to a product of curves $(C_1 times C_2 )/ G$; in our examples the group $G$ is elementary abelian, $G = (mathbb{Z}/m)^k$. We also establish some basic results concerning the canonical maps of any surface isogenous to a product, basing on elementary representation theory.
Let $alpha$ be a big class on a compact Kahler manifold. We prove that a decomposition $alpha=alpha_1+alpha_2$ into the sum of a modified nef class $alpha_1$ and a pseudoeffective class $alpha_2$ is the divisorial Zariski decomposition of $alpha$ if and only if $operatorname{vol}(alpha)=operatorname{vol}(alpha_1)$. We deduce from this result some properties of full mass currents.
A simple proof of Ramanujans formula for the Fourier transform of the square of the modulus of the Gamma function restricted to a vertical line in the right half-plane is given. The result is extended to vertical lines in the left half-plane by solving an inhomogeneous ODE. We then use it to calculate the jump across the imaginary axis.
In this article, we prove that a general version of Alladis formula with Dirichlet convolution holds for arithmetical semigroups satisfying Axiom $A$ or Axiom $A^{#}$. As applications, we apply our main results to certain semigroups coming from algebraic number theory, arithmetical geometry and graph theory, particularly generalizing the results of Wang 2021, Kural et al. 2020 and Duan et al. 2020.
We introduce and study some generalizations of regular spaces, which were motivated by studying continuity properties of functions between (regular) topological spaces. In particular, we prove that a first-countable Hausdorff topological space is regular if and only if it does not contain a topological copy of the Gutik hedgehog.