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We introduce a basis of the Orlik–Solomon algebra labeled by chambers, the so called chamber basis. We consider structure constants of the Orlik–Solomon algebra with respect to the chamber basis and prove that these structure constants recover D. Cohen’s minimal complex from the Aomoto complex.

We study the collapsing behavior of the Kähler–Ricci flow on a compact Kähler manifold X admitting a holomorphic submersion inherited from its canonical bundle. We show that the flow metric degenerates at exactly the rate of e^{-t} as predicted by the cohomology information, and so the fibres collapse at the optimal rate. Consequently, it leads to some analytic and geometric extensions to the regular case of works by J. Song and G. Tian. Its applicability to general Calabi–Yau fibrations will also be discussed in local settings.

Numerical modeling of wave propagation in heterogeneous media is important in many applications. Due to their complex nature, direct numerical simulations on the fine grid are prohibitively expensive. It is therefore important to develop efficient and accurate methods that allow the use of coarse grids. In this paper, we present a multiscale finite element method for wave propagation on a coarse grid. The proposed method is based on the generalized multiscale finite element method (GMsFEM) (see [Y. Efendiev, J. Galvis, and T. Hou, <i>J. Comput. Phys.</i>, 251 (2012), pp. 116--135]). To construct multiscale basis functions, we start with two snapshot spaces in each coarse-grid block, where one represents the degrees of freedom on the boundary and the other represents the degrees of freedom in the interior. We use local spectral problems to identify important modes in each snapshot space. These local spectral