ar X iv :m at h. A G /0 51 02 69 v 1 1 3 O ct 2 00 5 GEOMETRICITY OF THE HODGE FILTRATION ON THE ∞-STACK OF PERFECT COMPLEXES OVER XDR CARLOS SIMPSON Contents 1. Introduction 1 2. ?-connections and the Hodge filtration 3 3. Variation of cohomology—an example 5 4. Perfect complexes over XDR 6 5. Dold-Puppe of differential graded categories 10 5.1. Homotopy fibers 13 5.2. Maurer-Cartan stacks 15 6. Complexes over sheaves of rings of differential operators 17 7. The Hochschild complex and weak ?-module structures 21 8. Cˇech globalization 28 8.1. A finite-dimensional replacement 31 8.2. The proof of Theorem 6.7 32 References 34 1. Introduction We construct a locally geometric ∞-stack MHod(X,Perf) of perfect complexes on X with ?-connection structure (for a smooth projective variety X). This maps to A := A1/Gm, so it can be considered as a filtration. The stack underlying the filtration, fiber over 1, is MDR(X,Perf) which parametrizes complexes of D-modules which are perfect over OX . The associated-graded, or fiber over 0, is MDol(X,Perf) which parametrizes complexes of Higgs sheaves perfect over OX , whose cohomology is locally free, semistable with vanishing Chern classes.
- hochschild complex
- rham cohomology
- complex conjugate
- moduli stack
- identification between
- kontsevich-style hochschild
- twistor space
- fiber over
- xdr
- mhod