Let $X$ be an irreducible complex analytic space with $j:U\hookrightarrow X$ an immersion of a smooth Zariski-open subset, and let $\mathbb{V}$ be a variation of Hodge structure of weight $n$ over $U$. Assume that $X$ is compact Kähler. Then, provided that the local monodromy operators at infinity are quasi-unipotent, $I{H}^k(X,\mathbb{V})$ is known to carry a pure Hodge structure of weight $k+n$, while $H^k(U,\mathbb{V})$ carries a mixed Hodge structure of weight at least $k+n$. In this note it is shown that the image of the natural map $I{H}^k(X,\mathbb{V})\to H^k(U,\mathbb{V})$ is the lowest-weight part of this mixed Hodge structure. In the algebraic case this easily follows from the formalism of mixed sheaves, but the analytic case is rather complicated, in particular when the complement $X-U$ is not a hypersurface.

Two vanishing theorems for harmonic map and $L^2$ harmonic $1$-form on complete noncompact manifolds are proved under certain geometric assumptions, which generalize results of [13], [15], [18], [19], and [20]. As applications, we improve some main results in [2], [4], [6], [9], [12], [20], [22], [24], and [25].

We prove a Noether-Lefschetz type theorem for varieties of $r$-planes in complete intersections. We then use it to study the Abel-Jacobi map of planes on a smooth cubic fivefold.

Every threefold divisorial contraction to a non-Gorenstein point is a weighted blowup.

The order of the Milnor fiber monodromy operator of a central hyperplane arrangement is shown to be combinatorially determined. In particular, a necessary and sufficient condition for the triviality of this monodromy operator is given.

It is known that the complement of a complex hyperplane arrangement is cohomologically Tate and, if the arrangement is defined over $\mathbb{Q}$, has polynomial count. We show that these properties hold for the corresponding Milnor fibers if the monodromy is trivial.

We construct a hyperplane arrangement defined over $\mathbb{Q}$, whose Milnor fiber has a nontrivial monodromy operator, is cohomologically Tate, and has no polynomial count. Such examples are shown not to exist in low dimensions.

In this paper, we study finite symplectic actions on $K3$ surfaces $X$, that is, actions of finite groups $G$ on $X$ which act on $H^{2,0}\left(X\right)$ trivially. We show that the action on the $K3$ lattice $H^2(X,\mathbb{Z})$ induced by a symplectic action of $G$ on $X$ depends only on $G$ up to isomorphism, except for five groups.