Mathematics Lecture Notes

DefinitionFirst-Order Deformation

Let $X_0$ be a scheme (or object) over a field $k$ . A first-order deformation of $X_0$ is a flat morphism $\mathcal{X} \to \mathrm{Spec}\, k[\varepsilon]/(\varepsilon^2)$ together with an isomorphism $\mathcal{X} \otimes_{k[\varepsilon]} k \cong X_0$ .

The set of first-order deformations (up to isomorphism) is denoted $\mathrm{Def}_{X_0}(k[\varepsilon]/(\varepsilon^2))$ and forms the tangent space to the deformation functor.

DefinitionDeformation Functor

The deformation functor of $X_0$ is the functor $\mathrm{Def}_{X_0} : \mathsf{Art}_k \to \mathsf{Sets}$ from the category of local Artinian $k$ -algebras $(A, \mathfrak{m}_A)$ with residue field $k$ to sets, defined by $A \mapsto \left\lbrace \text{flat deformations } \mathcal{X} \to \mathrm{Spec}\, A \text{ with } \mathcal{X} \otimes_A k \cong X_0 \right\rbrace / \cong$

ExampleDeformations of a Smooth Variety

Let $X$ be a smooth projective variety over $k$ . First-order deformations of $X$ are classified by $\mathrm{Def}_X(k[\varepsilon]/(\varepsilon^2)) \cong H^1(X, T_X)$ where $T_X = \mathcal{H}om(\Omega^1_X, \mathcal{O}_X)$ is the tangent sheaf. This comes from the exact sequence $0 \to \mathcal{O}_X \xrightarrow{\varepsilon} \mathcal{O}_{\mathcal{X}} \to \mathcal{O}_X \to 0$ and the identification of extensions of $X$ by $\mathcal{O}_X$ with elements of $H^1(X, T_X)$ .

ExampleDeformations of a Smooth Curve

For a smooth curve $C$ of genus $g$ : $\mathrm{Def}_C(k[\varepsilon]/(\varepsilon^2)) \cong H^1(C, T_C)$ By Serre duality, $H^1(C, T_C) \cong H^0(C, \omega_C^{\otimes 2})^{\vee}$ , which has dimension $3g - 3$ for $g \geq 2$ . This is the tangent space to $\mathcal{M}_g$ at $[C]$ .

ExampleDeformations of a Node

Consider the nodal singularity $\mathrm{Spec}\, k[x,y]/(xy)$ . Its first-order deformations are given by the family $xy = t\varepsilon$ over $k[\varepsilon]/(\varepsilon^2)$ . The deformation space is 1-dimensional, reflecting the smoothing parameter $t$ .

More precisely, $T^1 = \mathrm{Ext}^1(\Omega^1_{k[x,y]/(xy)}, k) \cong k$ , generated by the smoothing $xy = t$ .

DefinitionObstruction

Given a deformation $\mathcal{X}_A$ over $\mathrm{Spec}\, A$ and a small extension $0 \to I \to A' \to A \to 0$ (where $I$ is an ideal with $\mathfrak{m}_{A'} \cdot I = 0$ , so $I$ is a $k$ -vector space), the obstruction to lifting $\mathcal{X}_A$ to a deformation over $\mathrm{Spec}\, A'$ is an element $\mathrm{ob}(\mathcal{X}_A, A' \to A) \in \mathrm{Ob}(X_0) \otimes_k I$ where $\mathrm{Ob}(X_0)$ is the obstruction space. The lift exists if and only if this element vanishes.

ExampleObstruction Space for Smooth Varieties

For a smooth projective variety $X$ , the obstruction space is $\mathrm{Ob}(X) = H^2(X, T_X)$ If $H^2(X, T_X) = 0$ , then $X$ is unobstructed: every deformation extends to all orders, and the formal moduli space is smooth.

ExampleCurves are Unobstructed

For a smooth curve $C$ of genus $g \geq 2$ : $H^2(C, T_C) = 0$ since $\dim C = 1$ . Therefore curves are always unobstructed, and $\mathcal{M}_g$ is smooth. This is why $\mathcal{M}_g$ is a smooth DM stack of dimension $3g - 3$ .

ExampleAbelian Varieties are Unobstructed

For an abelian variety $A$ of dimension $g$ : $T_A \cong \mathcal{O}_A^g, \quad H^1(A, T_A) \cong H^1(A, \mathcal{O}_A)^g \cong k^{g^2}$ $H^2(A, T_A) \cong H^2(A, \mathcal{O}_A)^g \cong k^{g \binom{g}{2}}$ So the obstruction space is nonzero for $g \geq 2$ ! However, by a theorem of Grothendieck-Mumford, abelian varieties are still unobstructed (the obstructions vanish). The moduli space $\mathcal{A}_g$ of principally polarized abelian varieties has dimension $\binom{g+1}{2}$ , matching $\dim H^1(A, T_A)$ when restricted to polarization-preserving deformations.

ExampleK3 Surfaces

For a K3 surface $S$ : $H^1(S, T_S) \cong H^1(S, \Omega^1_S) \cong k^{20}$ $H^2(S, T_S) \cong H^2(S, \Omega^1_S) \cong k^0 = 0$ (using the triviality of $\omega_S$ and the Hodge diamond). So K3 surfaces are unobstructed with 20-dimensional moduli.

ExampleObstructed Example: Rigid Calabi-Yau

Consider a general Calabi-Yau threefold $X$ with $h^{2,1}(X) = 0$ (a "rigid" Calabi-Yau). Then: $H^1(X, T_X) = 0, \quad H^2(X, T_X) \neq 0$ The deformation space is a point (no nontrivial deformations), but the obstruction space is nonzero. The obstructions are "vacuous" since there are no deformations to obstruct.

DefinitionCotangent Complex

For a morphism $f: X \to Y$ of schemes, the cotangent complex $\mathbb{L}_{X/Y}$ is an object in the derived category $D^{\leq 0}(\mathsf{QCoh}(X))$ . For a smooth morphism, $\mathbb{L}_{X/Y} \simeq \Omega^1_{X/Y}[0]$ . In general, if $X = \mathrm{Spec}\, B$ and $Y = \mathrm{Spec}\, A$ with a presentation $B = P/I$ where $P = A[x_1, \ldots, x_n]$ is a polynomial algebra, then $\mathbb{L}_{B/A} \simeq [I/I^2 \to \Omega^1_{P/A} \otimes_P B]$ concentrated in degrees $[-1, 0]$ .

TheoremDeformation Theory via Cotangent Complex

For a scheme $X_0$ over $k$ and a small extension $0 \to I \to A' \to A \to 0$ :

The set of liftings of a deformation $\mathcal{X}_A$ to $A'$ , if nonempty, is a torsor under $\mathrm{Ext}^0(L_{X_0/k}, I \otimes_k \mathcal{O}_{X_0}) = \mathrm{Hom}(\Omega^1_{X_0}, \mathcal{O}_{X_0}) \otimes I$ .
The obstruction to the existence of a lifting lies in $\mathrm{Ext}^1(\mathbb{L}_{X_0/k}, I \otimes_k \mathcal{O}_{X_0})$ .
Automorphisms of a lifting are given by $\mathrm{Ext}^{-1}(\mathbb{L}_{X_0/k}, I \otimes_k \mathcal{O}_{X_0})$ .

ExampleCotangent Complex of a Complete Intersection

Let $X = V(f_1, \ldots, f_r) \subset \mathbb{A}^n$ be a complete intersection. Then $\mathbb{L}_{X/k} \simeq [\mathcal{O}_X^r \xrightarrow{(df_1, \ldots, df_r)} \Omega^1_{\mathbb{A}^n}|_X]$ The tangent space to deformations is $\mathrm{Ext}^1(\mathbb{L}_{X/k}, \mathcal{O}_X)$ and obstructions lie in $\mathrm{Ext}^2(\mathbb{L}_{X/k}, \mathcal{O}_X)$ .

DefinitionDeformations of a Coherent Sheaf

Let $\mathcal{F}_0$ be a coherent sheaf on a fixed scheme $X$ over $k$ . A deformation of $\mathcal{F}_0$ over $A \in \mathsf{Art}_k$ is a coherent sheaf $\mathcal{F}$ on $X_A = X \times \mathrm{Spec}\, A$ , flat over $A$ , with $\mathcal{F} \otimes_A k \cong \mathcal{F}_0$ .

TheoremDeformation Theory of Sheaves

For a coherent sheaf $\mathcal{F}_0$ on a smooth projective variety $X$ :

Tangent space: $\mathrm{Def}_{\mathcal{F}_0}(k[\varepsilon]) \cong \mathrm{Ext}^1(\mathcal{F}_0, \mathcal{F}_0)$ .
Obstruction space: Obstructions lie in $\mathrm{Ext}^2(\mathcal{F}_0, \mathcal{F}_0)$ .
Automorphisms: $\mathrm{Aut}_{\mathrm{inf}}(\mathcal{F}_0) \cong \mathrm{Ext}^0(\mathcal{F}_0, \mathcal{F}_0) = \mathrm{End}(\mathcal{F}_0)$ .

ExampleDeformations of a Line Bundle

For a line bundle $\mathcal{L}$ on a smooth variety $X$ : $\mathrm{Ext}^1(\mathcal{L}, \mathcal{L}) \cong H^1(X, \mathcal{O}_X)$ $\mathrm{Ext}^2(\mathcal{L}, \mathcal{L}) \cong H^2(X, \mathcal{O}_X)$ So the deformation space of a line bundle is the same as the tangent space to $\mathrm{Pic}(X)$ , which is $H^1(X, \mathcal{O}_X)$ , consistent with $\dim \mathrm{Pic}^0(X) = h^{0,1}$ .

ExampleDeformations of the Structure Sheaf of a Point

Let $\mathcal{F}_0 = k(p) = \mathcal{O}_X / \mathfrak{m}_p$ for a smooth point $p \in X$ with $\dim X = n$ . Then: $\mathrm{Ext}^i(k(p), k(p)) \cong \binom{n}{i} \cdot k$ So the tangent space to deformations has dimension $n$ and the obstruction space has dimension $\binom{n}{2}$ . For a surface ( $n = 2$ ), the obstruction space is 1-dimensional but obstructions vanish (Fogarty's theorem), and for $n \geq 3$ , genuine obstructions can appear.

DefinitionPro-Representable Functor

A deformation functor $F: \mathsf{Art}_k \to \mathsf{Sets}$ is pro-representable if there exists a complete local $k$ -algebra $R = \varprojlim R/\mathfrak{m}_R^n$ such that $F(A) \cong \mathrm{Hom}_{\text{cont}}(R, A)$ for all $A \in \mathsf{Art}_k$ . The ring $R$ is the versal deformation ring (or hull).

ExampleFormal Moduli of a Smooth Curve

For a smooth curve $C$ of genus $g \geq 2$ , the formal moduli is pro-represented by $R = k[[t_1, \ldots, t_{3g-3}]]$ (a power series ring, since $C$ is unobstructed). The formal universal deformation $\mathcal{C} \to \mathrm{Spf}\, R$ induces all formal deformations of $C$ .

ExampleFormal Moduli of a Node

The formal moduli of the node $\mathrm{Spec}\, k[[x,y]]/(xy)$ is pro-represented by $R = k[[t]]$ . The universal deformation is $\mathrm{Spec}\, k[[x,y,t]]/(xy - t)$ , which smooths the node when $t \neq 0$ .

ExampleFormal Moduli with Obstructions

Consider a smooth projective surface $S$ with $H^2(S, T_S) \neq 0$ . The formal moduli is pro-represented by $R = k[[t_1, \ldots, t_m]] / (f_1, \ldots, f_r)$ where $m = h^1(S, T_S)$ and the relations $f_i$ come from obstructions. The moduli space can be singular.

For instance, the Barth-Nieto quintic in $\mathbb{P}^4$ has $H^2(T) \neq 0$ and the deformation space has a non-reduced component.

Definition$T^i$ Functors of Lichtenbaum-Schlessinger

For a $k$ -algebra $B = P/I$ with $P$ smooth, the $T^i$ functors are: $T^0_{B/k} = \mathrm{Der}_k(B, B) \quad \text{(derivations, i.e., infinitesimal automorphisms)}$ $T^1_{B/k} = \mathrm{coker}(\mathrm{Der}_k(P, B) \to \mathrm{Hom}_B(I/I^2, B)) \quad \text{(first-order deformations)}$ $T^2_{B/k} \quad \text{(obstructions)}$ For a global variety $X$ , $T^1_X = \mathrm{Ext}^1(\Omega^1_X, \mathcal{O}_X)$ and $T^2_X = \mathrm{Ext}^2(\Omega^1_X, \mathcal{O}_X)$ .

Example$T^1$ of the Whitney Umbrella

Consider $B = k[x,y,z]/(x^2 - y^2 z)$ . This is the Whitney umbrella, which has a non-isolated singularity along the $z$ -axis. The local $T^1$ is supported along the singular locus, and computing it involves: $T^1 = \mathrm{Hom}(I/I^2, B) / \mathrm{Der}(P, B)|_I$ where $I = (x^2 - y^2 z)$ . A generator of $I/I^2$ maps to $B$ , and the cokernel by derivation images gives the deformation space.

RemarkDeformation Theory in the Stacky Setting

For an algebraic stack $\mathcal{X}$ , the deformation theory is governed by the cotangent complex $\mathbb{L}_{\mathcal{X}}$ in the derived category. For a point $x: \mathrm{Spec}\, k \to \mathcal{X}$ with automorphism group $G_x$ :

$\mathrm{Ext}^{-1}(\mathbb{L}_{\mathcal{X}}|_x, k)$ : Lie algebra of $G_x$ (infinitesimal automorphisms).
$\mathrm{Ext}^0(\mathbb{L}_{\mathcal{X}}|_x, k)$ : First-order deformations (tangent space to $\mathcal{X}$ ).
$\mathrm{Ext}^1(\mathbb{L}_{\mathcal{X}}|_x, k)$ : Obstruction space.

This "three-term" deformation theory is characteristic of (1-)stacks. For derived stacks, the cotangent complex can have cohomology in arbitrary non-positive degrees.

ExampleDeformation Theory of $\mathcal{M}_g$ at $[C]$

At a point $[C] \in \mathcal{M}_g$ corresponding to a curve with automorphism group $G = \mathrm{Aut}(C)$ :

$\mathrm{Ext}^{-1} = \mathrm{Lie}(G) = H^0(C, T_C) = 0$ for $g \geq 2$ (no global vector fields).
$\mathrm{Ext}^0 = H^1(C, T_C) \cong k^{3g-3}$ (first-order deformations).
$\mathrm{Ext}^1 = H^2(C, T_C) = 0$ (unobstructed).

The formal neighborhood of $[C]$ in $\mathcal{M}_g$ is $[\mathrm{Spf}\, k[[t_1, \ldots, t_{3g-3}]] / G]$ .

Math Notes

Deformation Theory

1. Infinitesimal Deformations

2. Higher-Order Deformations and Obstructions

3. The Tangent-Obstruction Complex

4. Deformations of Sheaves

5. Formal Moduli and Pro-Representability

6. The $T^i$ Functors

7. Deformation Theory for Stacks