\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\QQ{{\mathbb Q}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\CritVal{\operatorname{CritVal}} \def\FS{\operatorname{FS}} \def\Sing{\operatorname{Sing}} \def\Coh{\operatorname{Coh}} \def\Vect{\operatorname{Vect}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \)

Symplectic snippets

why is it called mirror symmetry?

Mirror symmetry is a proposed duality between symplectic geometry and complex geometry. In one of its most familiar forms, it starts with a Calabi--Yau manifold \(X\) and predicts another Calabi--Yau manifold \(\check X\), called a mirror, whose complex geometry records symplectic information about \(X\) and whose symplectic geometry records complex information about \(X\). The mirror is not determined by the Hodge numbers alone; the Hodge-number symmetry is the first visible numerical shadow of a stronger correspondence. This slogan is a useful entry point into the more categorical form of mirror symmetry discussed in HMS for Fanos. In the Calabi--Yau setting, the mirror symmetry conjectures predict two related kinds of data:
  • to a suitable symplectic or Kähler Calabi--Yau manifold \(X\), a complex Calabi--Yau manifold \(\check X\) called a mirror of \(X\);
  • a dictionary translating symplectic invariants of \(X\) into complex invariants of \(\check X\), and conversely.
Some influential early papers include [CdlOGP91], [Kon94], [SYZ96], and [GS03]. The purpose of this note is to explain the elementary reason for the word ``mirror'', then to point toward the SYZ and enumerative pictures which motivate later formulations.

definition 0.0.1

In these notes, a Calabi--Yau \(n\)-fold is a compact connected Kähler manifold \(X\) of complex dimension \(n\) whose canonical bundle \[ K_X:=\Lambda^n(T^{1,0}X)^* \] is holomorphically trivial. Equivalently, \(X\) admits a nowhere-vanishing holomorphic volume form \[ \Omega\in H^0(X,K_X). \] Thus a Calabi--Yau manifold carries compatible complex and symplectic data:
  • an integrable complex structure \(J:TX\to TX\);
  • a Kähler form \(\omega\in \Omega^2(X)\), which in particular makes \(X\) a symplectic manifold;
  • a holomorphic volume form \(\Omega\) trivializing \(K_X\).
This convention does not impose the stronger condition \(h^{p,0}(X)=0\) for \(0<p<n\), nor does it require \(X\) to be simply connected. When those extra hypotheses are needed, they will be stated explicitly. Historically, physicists were interested in Calabi--Yau manifolds because of their role in superstring theory: the compact Calabi--Yau directions provide candidates for the hidden internal geometry of space-time. Early expectations suggested that only a small number of Calabi--Yau threefolds might occur up to deformation, but the search for string geometries produced many more examples; see, for example, [Hüb92]. Mathematically, the point is that Calabi--Yau manifolds sit at the meeting place of symplectic, complex, and Riemannian geometry.

theorem 0.0.2

Let \((X,J)\) be a compact Kähler manifold of complex dimension \(n\) with \(c_1(TX)=0\) in \(H^2(X;\RR)\). For every Kähler class \[ \alpha\in H^{1,1}(X;\RR) \] there is a unique Kähler form \(\omega\in\alpha\) whose Ricci form is zero. If \(K_X\) is holomorphically trivial, then with respect to this metric a nowhere-vanishing holomorphic \(n\)-form is parallel, and the holonomy is contained in \(SU(n)\). Conversely, a Kähler metric with holonomy contained in \(SU(n)\) is Ricci-flat and admits a parallel nowhere-vanishing holomorphic \(n\)-form. Yau proved the Calabi conjecture in [Yau78]. Morally, the Calabi--Yau condition is what lets the symplectic, complex, and Riemannian parts of the geometry interact with unusually little friction: the Kähler class determines a Ricci-flat metric, while the holomorphic volume form constrains the complex geometry.

1: Hodge diamond and deformation

definition 0.0.3

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 definition 0.0.1, 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 Calabi--Yau threefold Hodge diamondAfter 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.

0.1: \(h^{1,1}\) and the K\"ahler cone

definition 0.0.4

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.

0.2: \(h^{2,1}\) and complex deformation

definition 0.1.1

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.

2: the mirror flip

This is the elementary origin of the name ``mirror symmetry.'' If \(X\) and \(\check X\) are mirror Calabi--Yau \(n\)-folds, then the expected numerical relation is \[ h^{p,q}(X)=h^{n-p,q}(\check X). \] For Calabi--Yau threefolds satisfying the simplifying vanishing used above, this says in particular that variations of Kähler or symplectic structure on \(X\) should correspond to variations of complex structure on \(\check X\), and conversely. Since the dimensions of these two deformation spaces are governed by Hodge numbers, one expects \[ h^{1,1}(X)=h^{2,1}(\check X),\qquad h^{2,1}(X)=h^{1,1}(\check X). \] Pictorially, for threefolds this mirror relation exchanges the vertical \(h^{1,1}\) entries with the horizontal \(h^{2,1}\) entries: mirror flip of Hodge diamondsMany mirror pairs of Calabi--Yau manifolds were discovered by searching for this kind of numerical symmetry, and those examples provided early evidence for mirror symmetry. The equality of Hodge numbers is only a necessary compatibility check; the stronger mirror statement also identifies deformation theory, enumerative invariants, and eventually categories.

Connections

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References

[CdlOGP91]Philip Candelas, Xenia C. de la Ossa, Paul S. Green, and Linda Parkes. A pair of Calabi--Yau manifolds as an exactly soluble superconformal theory. Nuclear Physics B, 359(1):21--74, 1991.
[GS03]Mark Gross and Bernd Siebert. Affine manifolds, log structures, and mirror symmetry. Turkish Journal of Mathematics, 27(1):33--60, 2003.
[Hüb92]Tristan Hübsch. Calabi--Yau manifolds: A bestiary for physicists. World Scientific, 1992.
[Kon94]Maxim Kontsevich. Homological algebra of mirror symmetry. In Proceedings of the International Congress of Mathematicians, 1994.
[SYZ96]Andrew Strominger, Shing-Tung Yau, and Eric Zaslow. Mirror symmetry is T-duality. Nuclear Physics B, 479(1--2):243--259, 1996.
[Yau78]Shing-Tung Yau. On the ricci curvature of a compact Kähler manifold and the complex Monge--Ampere equation, i. Communications on Pure and Applied Mathematics, 31(3):339--411, 1978.