$\def\Q{\mathbb Q}$ On an $n$-dimensional locally reduced complex analytic space $X$ on which the shifted constant sheaf $\Q_X[n]$ is perverse, it is well-known that, locally, $\Q_X[n]$ underlies a mixed Hodge module of weight $<= n$ on $X$, with weight $n$ graded piece isomorphic to the intersection cohomology complex $IC_X$ with constant $\Q$ coefficients. In this paper, we identify the weight $(n-1)$ graded piece $Gr_{n-1}^W \Q_X[n]$ in the case where X is a “parameterized space", using the comparison complex, a perverse sheaf naturally defined on any space for which the shifted constant sheaf $\Q_X[n]$ is perverse. 

I will report on recent results which deal with the manner in which diophantine arithmetic measures of distance to divisors relate to concepts of stability for polarized projective varieties. These results build on many previous insights including those of Boucksom-Chen, Evertse-Ferretti, K. Fujita, A. Levin, C. Li, McKinnon-Roth, Ru-Vojta, Ru-Wang and J. Silverman. Some emphasis will be placed on the case of K-(in)stability for Fano varieties. At the same time, I will present motivational examples which arise within the context of toric varieties.

I hope to give an elementary survey of recent works on the geography problem of simply connected smooth 4-dimensional manifolds. I will focus mainly on the existence and uniqueness of symplectic 4-manifolds that satisfy certain topological conditions. One such condition that I wish to explore in detail is the signature being nonnegative.