Hodge theory for twisted differentials

Extension of twisted Hodge metrics for Kähler morphisms

The subject in this paper is the positivity of direct image sheaves of adjoint bundles Rf∗(KX/Y ⊗ E), for a Kähler morphism f : X −→ Y endowed with a Nakano semipositive holomorphic vector bundle (E, h) onX. In our previous paper [MT2], generalizing a result [B] in case q = 0, we obtained the Nakano semi-positivity of Rf∗(KX/Y ⊗ E) with respect to a canonically attached metric, the so-called Ho...

Hodge Theory for Combinatorial Geometries

The matroid is called loopless if the empty subset of E is closed, and is called a combinatorial geometry if in addition all single element subsets of E are closed. A closed subset of E is called a flat of M, and every subset of E has a well-defined rank and corank in the poset of all flats of M. The notion of matroid played a fundamental role in graph theory, coding theory, combinatorial optim...

Hodge Theory for R- Manifolds

Let X be an R-fold, and let π : E −→ X be a real vector bundle, of rank r, equipped with a positive definite symmetric bilinear form. If e1, . . . , er ∈ π −1(X) are orthonormal, then e1 ∧ · · · ∧ er is a non-trivial vector in ∧r E. Proposition: If f1, . . . , fr is any other orthonormal basis for π −1(X), then e1 ∧ · · · ∧ er = ±f1 ∧ · · · ∧ fr. Proof. Note that fi = g · ei for g ∈ O(r), so de...

TWISTED K - HOMOLOGY THEORY , TWISTED Ext - THEORY

These are notes on twisted K-homology theory and twisted Ext-theory from the C *-algebra viewpoint, part of a series of talks on " C *-algebras, noncommutative geometry and K-theory " , primarily for physicists.

Hodge Theory and Representation Theory