Registered users receive a variety of benefits including the ability to customize email alerts, create favorite journals list, and save searches.
Please note that a Project Euclid web account does not automatically grant access to full-text content. An institutional or society member subscription is required to view non-Open Access content.
Contact firstname.lastname@example.org with any questions.
Let be an irreducible complex analytic space with an immersion of a smooth Zariski-open subset, and let be a variation of Hodge structure of weight over . Assume that is compact Kähler. Then, provided that the local monodromy operators at infinity are quasi-unipotent, is known to carry a pure Hodge structure of weight , while carries a mixed Hodge structure of weight at least . In this note it is shown that the image of the natural map 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 is not a hypersurface.
Two vanishing theorems for harmonic map and harmonic -form on complete noncompact manifolds are proved under certain geometric assumptions, which generalize results of , , , , and . As applications, we improve some main results in , , , , , , , , and .
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 , 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 , 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 surfaces , that is, actions of finite groups on which act on trivially. We show that the action on the lattice induced by a symplectic action of on depends only on up to isomorphism, except for five groups.