Book Review: Differential forms orthogonal to holomorphic functions or forms, and their properties

From path functions to differential forms

Tensor Product Finite Element Differential Forms and Their Approximation Properties

We discuss the tensor product construction for complexes of differential forms and show how it can be applied to define shape functions and degrees of freedom for finite element differential forms on cubes in n dimensions. These may be extended to curvilinear cubic elements, obtained as images of a reference cube under diffeomorphisms, by using the pullback transformation for differential forms...

Holomorphic Almost Modular Forms

Holomorphic almost modular forms are holomorphic functions of the complex upper half plane which can be approximated arbitrarily well (in a suitable sense) by modular forms of congruence subgroups of large index in SL(2,Z). It is proved that such functions have a rotation-invariant limit distribution when the argument approaches the real axis. An example for a holomorphic almost modular form is...

Higgs Bundles and Holomorphic Forms

For a complex manifold X which has a holomorphic form ̟ of odd degree k, we endow E = ⊕ p≥a Λ (p,0)(X) with a Higgs bundle structure θ given by θ(Z)(φ) := {i(Z)̟} ∧ φ. The properties such as curvature and stability of these and other Higgs bundles are examined. We prove (Theorem 2, section 2, for k > 1) E and additional classes of Higgs subbundles of E do not admit Higgs-Hermitian-Yang-Mills metr...

L-FUNCTIONS FOR HOLOMORPHIC FORMS ON GSp(4)×GL(2) AND THEIR SPECIAL VALUES

We provide an explicit integral representation for L-functions of pairs (F, g) where F is a holomorphic genus 2 Siegel newform and g a holomorphic elliptic newform, both of squarefree levels and of equal weights. When F, g have level one, this was earlier known by the work of Furusawa. The extension is not straightforward. Our methods involve precise double-coset and volume computations as well...