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22
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0answers
442 views

Software for rational homotopy theory

Does anybody know a software manipulating commutative differential graded algebras, and providing a computation of the minimal model? I tried to use the package DGAlgebras of Macaulay2, but I got ...
13
votes
0answers
740 views

Hodge star and harmonic simplicial differential forms

Is there a notion of harmonic forms and Hodge theory for Sullivan's piecewise smooth differential forms on a simplicial set? Let me recall some background. Hodge Theory on a Riemannian manifold A ...
13
votes
0answers
985 views

Mixed Hodge structure on the rational homotopy type

A mixed Hodge structure (mHs) on a commutative differential graded algebra (cgda) over $\mathbf{Q}$ is a mixed Hodge structure on the underlying vector space such that the product and the differential ...
6
votes
0answers
265 views

Formality of algebraic varieties via l-adic cohomology?

The cohomology with say real coefficients of any smooth projective algebraic variety is formal, by Deligne, Griffiths, Morgan, Sullivan: Real homotopy theory of Kähler manifolds, Inv. Math. 29, ...
5
votes
0answers
161 views

Reference request: splittings in rational homotopy theory

It is well known that for simply-connected rational spaces, every suspension splits as a wedge of rational spheres and every loop space splits as a product of rational Eilenberg-Mac Lane spaces. ...
4
votes
0answers
178 views

Is there a picture I should have in my head of rational homotopy equivalence?

My understanding is that one thinks to rational homotopy theory for computational advantage. However, thinking about things in terms of localizations still lacks some amount of intuition for me. In ...
4
votes
0answers
172 views

Leray degeneration for smooth projective morphisms and formality of families of compact Kähler manifolds

Let $\pi \colon X \to S$ be a smooth projective morphism of algebraic varieties, say over $\mathbf C$. By Deligne's argument ("Théorème de Lefschetz...", 1968), there is for each $i$ an injection $$ ...