[Paper Review] Lectures on deformations of complex manifolds

Marco Manetti|ArXiv.org|Jul 14, 2005

Homotopy and Cohomology in Algebraic Topology38 references54 citations

TL;DR

This paper provides a comprehensive, self-contained introduction to deformation theory of compact complex manifolds using differential graded Lie algebras (DGLA) and $L_∞$-algebras. It establishes the foundational framework for understanding deformations via DGLA, proves the unobstructedness of Calabi-Yau manifolds via the Bogomolov-Tian-Todorov theorem, and gives an algebraic proof of the Clemens-Ran theorem using $L_∞$-morphisms.

ABSTRACT

This paper is based on a course given by the author at the University of Rome ``La Sapienza'' in the Academic year 2000/2001. The intended aim of the course was to rapidly introduce, although not in an exhaustive way, the non-expert PhD student to deformations of compact complex manifolds, from the very beginning to some recent (i.e. at that time not yet published) results. The goal of these lectures is to give a soft introduction to extended deformation theory. In view of the aim (and the hope) of keeping this paper selfcontained, user friendly and with a tolerating number of pages, we consider only deformations of compact complex manifolds. Anyhow, most part of the formalism and of the results that we prove here will apply to many other deformation problems.

Motivation & Objective

• To provide a soft, accessible introduction to extended deformation theory for non-expert PhD students.
• To unify deformation theories through the framework of differential graded Lie algebras (DGLA), emphasizing common structural features across different mathematical objects.
• To establish the connection between deformation functors and DGLA, and to extend this to $L_\infty$-algebras for more general deformation problems.
• To prove the unobstructedness of Calabi-Yau manifolds using the Tian-Todorov lemma and DGLA techniques.
• To give an algebraic proof of the Clemens-Ran theorem on obstructions annihilating ambient cohomology using $L_\infty$-morphisms.

Proposed method

• Uses differential graded vector spaces and dg-algebras as the foundational algebraic structures for deformation theory.
• Introduces deformation functors associated to a DGLA and proves their homotopy invariance via the exponential and Baker-Campbell-Hausdorff formulas.
• Applies the inverse function theorem in the context of DGLA to analyze deformation obstructions.
• Constructs $L_\infty$-morphisms from DGLA to $L_\infty$-algebras using symmetrized coderivations and the décalage operation.
• Employs the Gerstenhaber-Batalin-Vilkoviski (GBV) algebra structure on polyvector fields to analyze the deformation complex.
• Uses the contraction map and formality theorems to relate cohomology classes to deformation invariants, particularly in the Calabi-Yau case.

Experimental results

Research questions

• RQ1 How can deformation theory of compact complex manifolds be uniformly described using DGLA?
• RQ2 What is the role of $L_\infty$-algebras in generalizing and simplifying classical deformation functors?
• RQ3 Why are Calabi-Yau manifolds unobstructed in their deformation space?
• RQ4 How do obstructions to deformations interact with ambient cohomology, as in the Clemens-Ran theorem?
• RQ5 What is the significance of the $L_\infty$-morphism constructed from the Kodaira-Spencer complex to the polyvector field complex?

Key findings

• The unobstructedness of Calabi-Yau manifolds follows from the Tian-Todorov lemma and the vanishing of the second cohomology of the Kodaira-Spencer complex under the holomorphic volume form.
• An $L_\infty$-morphism $\Theta: (C(KS_X), \delta) \to (C(M_X[-1]), 0)$ is constructed with linear term $F_1$, proving that the deformation complex is formal in the $L_\infty$-sense.
• The composition of the $L_\infty$-morphism with evaluation at the holomorphic volume form $\Omega$ vanishes on $\bigodot^m\{a \in L \mid \partial(a \vdash \Omega) = 0\}$, confirming the Clemens-Ran obstruction condition.
• The proof of the unobstructedness of Calabi-Yau manifolds is algebraic and relies on the $d$-Gerstenhaber structure on polyvector fields and the formality theorem.
• The $L_\infty$-morphism $\Theta$ satisfies $F \circ \delta = 0$, confirming it as a valid $L_\infty$-morphism, with the key identity verified via symmetrization and coderivation identities.
• The construction of the $L_\infty$-morphism via $F_m$ and the use of the Koszul sign rule and unshuffle maps provide a systematic method for lifting DGLA structures to higher homotopy algebras.