Kodaira–Spencer deformation problem

Let $X$ be a compact complex manifold of complex dimension $n$, with holomorphic tangent bundle $T_X^{1,0}$. A pointed complex-manifold germ $(S,0)$ is a complex manifold with distinguished point $0$, considered up to restriction to neighbourhoods of $0$; maps of such germs are holomorphic maps preserving the distinguished points, identified when they agree on a smaller neighbourhood. For each fixed $(S,0)$, the analytic deformation groupoid of $X$ over $(S,0)$ has representatives $(\pi :\mathcal{X}\to U,\iota )$, where $U$ represents the base germ, $\mathcal{X}$ is a complex manifold, $\pi$ is proper (inverse images of compact sets are compact) and a holomorphic submersion (its complex differential is surjective at every point), and $\iota :{\pi }^{-1}(0)\stackrel{\sim }{\to }X$ is biholomorphic. A representative may be restricted by replacing $U$ by a smaller neighbourhood and $\mathcal{X}$ by its inverse image. Arrows are germs of biholomorphisms of these total spaces over the fixed base germ, preserving $\iota$; two representatives of an arrow agree when they coincide after a common restriction. Identity and composition are those of maps after common restriction. There are no arrows between different fixed bases in this groupoid.

Let $A$ be a finite-dimensional commutative local $\mathbb{C}$-algebra with residue field $\mathbb{C}$ and nilpotent maximal ideal ${\mathfrak{m}}_A$. Define $\mathrm{Specan}A$ to be the one-point locally ringed space with structure ring $A$. The formal deformation groupoid over $A$ means here the finite Artin analytic deformation groupoid: its objects are sheaves ${\mathcal{O}}_A$ of local $A$-algebras on the underlying space $X$, with a marking $\iota :{\mathcal{O}}_A\mathrm{/}{\mathfrak{m}}_A{\mathcal{O}}_A\stackrel{\sim }{\to }{\mathcal{O}}_X$, such that every stalk of ${\mathcal{O}}_A$ is flat as an $A$-module and the following local analytic presentation condition holds. Near each point, $(X,{\mathcal{O}}_A)$ is isomorphic over $A$ to $(Z,(A{\otimes }_{\mathbb{C}}{\mathcal{O}}_U)\mathrm{/}\mathcal{I})$, where $U\subseteq {\mathbb{C}}^N$ is open, $\mathcal{I}$ is a locally finitely generated ideal sheaf, and $Z$ is the support of this quotient, with its analytic-subspace topology. Flat means that tensoring any injective map of $A$-modules with the stalk remains injective. The marking identifies the reduced-base ringed space (reduction modulo ${\mathfrak{m}}_A$, not removal of arbitrary nilpotents) with $X$. Arrows are isomorphisms of these $A$-ringed spaces preserving the marking; in the chosen underlying space they induce the identity on $X$. The structure map to $\mathrm{Specan}A$ is proper since $X$ is compact. This definition is finite-order and asserts no inverse-limit effectivity or convergence. For $A=\mathbb{C}$ it gives the marked central fibre. Here $A\{z\}$ denotes $A{\otimes }_{\mathbb{C}}\mathbb{C}\{z\}$, so the local presentation agrees with the stalk notation for convergent power series.

The Kodaira–Spencer differential graded Lie algebra is $L_X^q=A^{0,q}(X,T_X^{1,0})$, with differential $\bar{\mathrm{\partial }}$ and the graded Kodaira–Spencer bracket of vector-valued forms. The bracket convention is the one for which a degree-one element $\xi$ obeys the Maurer–Cartan equation $\bar{\mathrm{\partial }}\xi +\frac{1}{2}[\xi ,\xi ]=0,$ which is the involutivity equation for the graph deformation of $T_X^{0,1}$. The formal Maurer–Cartan action groupoid over $A$ has objects $\xi \in L_X^1\otimes {\mathfrak{m}}_A$ satisfying that equation and arrows given by the pushforward action of formal diffeomorphisms generated by $L_X^0\otimes {\mathfrak{m}}_A$. Their exponential series are finite because ${\mathfrak{m}}_A$ is nilpotent. This item defines the analytic deformation groupoid and the formal Maurer–Cartan action groupoid separately; it does not assert that they are equivalent, or that every formal deformation converges to an analytic family.

Complex manifolds, holomorphic vector bundles and Hermitian metrics

Deformations of schemes and the infinitesimal deformation functor

Kodaira–Spencer forms form a differential graded Lie algebra

The Axiom of Choice

Voisin, §§0.2 and 1.1

The analytic deformation groupoid, the formal deformation groupoid, and the Maurer–Cartan action groupoid are distinct interfaces in this item. Comparing the latter two requires a local deformation-theory argument not supplied here. The algebraic supplier in [F2] is used only for the scheme-theoretic square-zero convention when $X$ is algebraic; the analytic statement does not claim that every analytic deformation is algebraizable.