\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\QQ{{\mathbb Q}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
Symplectic snippets
Hodge diamond and deformation
definition 0.0.1
This definition uses the Hodge decomposition.
The Hodge diamond of a compact Kähler manifold \(X\) of complex dimension \(n\) is the array of Hodge numbers \(h^{p,q}(X)=\dim_{\CC}H^{p,q}(X)\) arranged so that entries with fixed \(p+q\) lie on the same row and reflection across the vertical axis interchanges the two indices. Thus the row indexed by \(k\) contains the numbers corresponding to the summands in
\[
H^k(X;\CC)=\bigoplus_{p+q=k}H^{p,q}(X).
\]
For a threefold, the diamond has rows
\[
\begin{array}{ccccccc}
&&& h^{0,0} &&&\\
&& h^{1,0} && h^{0,1} &&\\
& h^{2,0} && h^{1,1} && h^{0,2} &\\
h^{3,0} && h^{2,1} && h^{1,2} && h^{0,3}\\
& h^{3,1} && h^{2,2} && h^{1,3} &\\
&& h^{3,2} && h^{2,3} &&\\
&&& h^{3,3} &&&
\end{array}
\]
The diamond is not extra structure beyond the Hodge decomposition; it is a compact way of displaying the dimensions of its summands.
Let \(X\) be a compact Kähler \(n\)-fold. Its Hodge numbers satisfy:
- complex conjugation gives
\[
h^{p,q}(X)=h^{q,p}(X);
\]
- Serre duality gives
\[
h^{p,q}(X)=h^{n-p,n-q}(X);
\]
- together, these equalities imply that the usual Hodge diamond is symmetric across both the vertical and horizontal axes.
If \(X\) is Calabi--Yau in the sense of (Calabi--Yau manifold), then \(K_X\simeq\mathcal O_X\), and Serre duality applied to \(\mathcal O_X\) gives
\[
h^{0,q}(X)=h^{0,n-q}(X).
\]
Thus a connected Calabi--Yau threefold has
\[
h^{0,0}=h^{3,0}=h^{0,3}=h^{3,3}=1.
\]
In complex dimension three, this implies that if \(h^{1,0}(X)=0\), then \(h^{2,0}(X)=0\) as well.
For a connected Calabi--Yau threefold satisfying the additional vanishing \(h^{1,0}=0\), the preceding symmetries force the Hodge diamond to have the form
After the standard symmetries and this vanishing assumption are accounted for, the Hodge diamond is controlled by two numbers: \(h^{1,1}\) and \(h^{2,1}\). These two numbers measure two different deformation directions.
1: \(h^{1,1}\) and the K\"ahler cone
definition 0.0.2
Let \(X\) be a compact Kähler manifold. Under the Hodge decomposition, the real \((1,1)\)-cohomology is
\[
H^{1,1}(X;\RR):=H^{1,1}(X;\CC)\cap H^2(X;\RR).
\]
The K\"ahler cone of \(X\) is
\[
\mathcal K_X
:=\{[\omega]_{\mathrm{dR}}\in H^{1,1}(X;\RR)\mid \omega \text{ is a Kähler form on }X\}.
\]
It is an open convex cone in \(H^{1,1}(X;\RR)\). A point of \(\mathcal K_X\) is the de Rham cohomology class of a compatible symplectic form on the fixed complex manifold \(X\).
In mirror symmetry one often uses the complexified K\"ahler parameter
\[
[B]+i[\omega]\in H^2(X;\CC),
\]
where \([B]\in H^2(X;\RR)\) is usually considered modulo integral classes.
The number \(h^{1,1}\) controls first-order deformations of the Kähler class while keeping the complex structure fixed. More precisely, the Kähler cone \(\mathcal K_X\) is an open subset of the real vector space
\[
H^{1,1}(X)\cap H^2(X,\RR).
\]
Therefore its real dimension is \(h^{1,1}(X)\).
An infinitesimal deformation of the Kähler form is represented by a real closed \((1,1)\)-form \(\eta\). Modding out by exact deformations leaves the cohomology class \([\eta]\). In Dolbeault notation, the corresponding complex vector space is
\[
H^1(X,\Omega_X^1)=
\frac{\ker(\bar\partial: \Omega^{1,1}(X)\to \Omega^{1,2}(X))}
{\operatorname{im}(\bar\partial: \Omega^{1,0}(X)\to \Omega^{1,1}(X))}.
\]
This is the Hodge-theoretic shadow of the symplectic fact that, on a compact manifold, deformations of symplectic forms in a fixed de Rham cohomology class are equivalent up to isotopy under the hypotheses of Moser's theorem.
The takeaway is that \(h^{1,1}\) measures first-order Kähler, hence symplectic, deformations.
2: \(h^{2,1}\) and complex deformation
definition 1.0.1
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.