elliptic chain complex
The notion of elliptic chain complex is the generalization of the notion of elliptic operator from single linear maps to chain complexes of linear maps.
For a smooth manifold and a collection of vector bundles over , a chain complex of differential operators between the spaces of sections of these bundles
is called an elliptic chain complex if the corresponding sequence of symbols
(where is the cotangent bundle) is an exact sequence.
For instance (Pati, def. 9.4.1).
For a single differential operator this says that is exact, which means that is an isomorphism, hence that is an elliptic operator.
If is an elliptic complex of smooth sections of a vector bundle overa compact closed manifold , then the inclusion
into the complex of distributional sections is a quasi-isomorphism, in fact a homotopy equivalence.
This is due to (Atiyah-Bott). A localized refinement (suitable for factorization algebras of local observables) appears as Gwilliam, lemma 5.2.13.
The classical examples of elliptic complexes are discussed also in (Gilkey section 3).
de Rham complex
Let be a compact smooth manifold. Then the de Rham complex is an ellptic complex. The corresponding index of an elliptic complex is the Euler characteristic
The Yang-Mills complex
The Dolbeault complex
(…) Dolbeault complex
The index of an elliptic complex of the Dolbeault complex is the arithmetic genus
(…) index is A-hat genus (…)
- V. Pati, Elliptic complexes and index theory (pdf)
- Michael Atiyah, Raoul Bott, A Lefschetz fixed point formula for elliptic complexes. I, Ann. of Math. (2) 86 (1967), 374–407. MR 0212836 (35 #3701)
Revised on February 19, 2013 15:14:58
by Urs Schreiber