Pluripotential Homotopy Theory

TL;DR

This paper develops a new algebraic framework for complex manifolds that refines homotopy invariants by incorporating complex structure sensitivity, leading to explicit minimal models and higher operations.

Contribution

It introduces bigraded bidifferential algebra models for complex manifolds that are compatible with conjugation and quasi-isomorphisms, answering Sullivan's question.

Findings

Construction of algebraic models for forms on complex manifolds.

Refinement of homotopy groups to be sensitive to complex structures.

Existence of unique minimal models under simple connectedness.

Abstract

We build free, bigraded bidifferential algebra models for the forms on a complex manifold, with respect to a strong notion of quasi-isomorphism and compatible with the conjugation symmetry. This answers a question of Sullivan. The resulting theory naturally accomodates higher operations involving double primitives. As applications, we obtain various refinements of the homotopy groups, sensitive to the complex structure. Under a simple connectedness assumption, one obtains minimal models which are unique up to isomorphism and allow for explicit computations of the new invariants.