We prove that a certain positivity condition, considerably more general than pseudoconvexity, enables one to conclude that the regular order of contact and singular order of contact agree when these numbers are $4$.
We give a proof of the hard Lefschetz theorem for orbifolds that does not involve intersection homology. This answers a question of Fulton. We use a foliated version of the hard Lefschetz theorem due to El Kacimi.
We show that for every smooth generic projective hypersurface $Xsubsetmathbb P^{n+1}$, there exists a proper subvariety $Ysubsetneq X$ such that $operatorname{codim}_X Yge 2$ and for every non constant holomorphic entire map $fcolonmathbb Cto X$ one has $f(mathbb C)subset Y$, provided $deg Xge 2^{n^5}$. In particular, we obtain an effective confirmation of the Kobayashi conjecture for threefolds in $mathbb P^4$.
From the viewpoint of quantum walks, the Ihara zeta function of a finite graph can be said to be closely related to its evolution matrix. In this note we introduce another kind of zeta function of a graph, which is closely related to, as to say, the square of the evolution matrix of a quantum walk. Then we give to such a function two types of determinant expressions and derive from it some geometric properties of a finite graph. As an application, we illustrate the distribution of poles of this function comparing with those of the usual Ihara zeta function.