\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\QQ{{\mathbb Q}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
Symplectic snippets
\(h^{2,1}\) and complex deformation
definition 0.0.1
Let \(X\) be a compact complex manifold. A first-order deformation of the complex structure on \(X\) is a flat deformation of \(X\) over the dual numbers \(\CC[\epsilon]/(\epsilon^2)\), together with an identification of the special fiber with \(X\). Equivalence classes of first-order deformations are naturally identified with the Kodaira--Spencer group
\[
H^1(X,T_X),
\]
where \(T_X=T^{1,0}X\) is the holomorphic tangent bundle.
Analytically, after choosing a splitting, a nearby almost complex structure may be represented by a Beltrami differential
\[
s\in \Omega^{0,1}(X,T_X).
\]
The integrability condition is the Maurer--Cartan equation
\[
\bar\partial s+\frac{1}{2}[s,s]=0.
\]
To first order this reduces to \(\bar\partial s=0\), and infinitesimal changes of coordinates change \(s\) by elements of \(\bar\partial\Omega^0(X,T_X)\). Thus the Zariski tangent space to the deformation functor or Kuranishi space is \(H^1(X,T_X)\), while obstruction classes naturally take values in \(H^2(X,T_X)\).
The number \(h^{2,1}\) controls first-order deformations of complex structure on a Calabi--Yau threefold. Let \(J\) and \(J'\) be two nearby almost complex structures. A nearby complex structure can be described by its antiholomorphic tangent bundle \(T^{0,1}_{J'}X\). After choosing the splitting determined by \(J\), this is the graph of a map
\[
s:T^{0,1}_JX\to T^{1,0}_JX,
\]
which is the same as an element
\[
s\in \Omega^{0,1}(X,T_X^{1,0}).
\]
- The condition that the new almost complex structure be integrable is the Maurer--Cartan equation
\[
\bar\partial s+\frac{1}{2}[s,s]=0.
\]
To first order, this becomes \(\bar\partial s=0\).
- Some deformations of complex structure arise from pulling back by a diffeomorphism. Infinitesimally, these lie in the image of
\[
\bar\partial: \Omega^0(X,T_X^{1,0})\to \Omega^{0,1}(X,T_X^{1,0}).
\]
- Therefore, the space of first-order deformations of complex structure is
\[
\frac{\ker(\bar\partial: \Omega^{0,1}(X,T_X^{1,0})\to \Omega^{0,2}(X,T_X^{1,0}))}
{\operatorname{im}(\bar\partial: \Omega^{0}(X,T_X^{1,0})\to \Omega^{0,1}(X,T_X^{1,0}))}
=H^{0,1}(X,T_X^{1,0})=H^1(X,T_X).
\]
On a Calabi--Yau \(n\)-fold, contraction with a holomorphic volume form identifies \(T_X\) with \(\Omega_X^{n-1}\). Thus
\[
H^1(X,T_X)\simeq H^1(X,\Omega_X^{n-1})\simeq H^{n-1,1}(X).
\]
For a Calabi--Yau threefold, this is \(H^{2,1}(X)\).
The Kodaira--Spencer map identifies the tangent space to the moduli space of complex structures with \(H^1(X,T_X)\). On Calabi--Yau manifolds these first-order deformations are unobstructed by the Bogomolov--Tian--Todorov theorem, so locally the deformation space is smooth of dimension \(h^{n-1,1}(X)\).
The takeaway is that \(h^{2,1}\) measures first-order complex-structure deformations.