A brief survey on complex geometry

Abstract

We introduce some fundamental principles and classic theorems in the research of complex geometry, with emphasis on some major research fields and significant progresses in Kähler geometry in the past decades.

Keywords

Complex geometry / Kähler-Einstein manifolds / Hodge conjecture / Calabi-Yau manifolds / canonical metrics

1 Riemann surface

The origin of complex geometry can be traced back to the study of Riemann surfaces. The classical uniformization theorem of Riemann surfaces shows that a simply connected Riemann surface is holomorphically equivalent (or conformal) to one of the following three types of surfaces:

(1.1)

For planar domains D\subseteq C, the Riemann mapping theorem states that if D is simply-connected and there are has at least 2 points in its boundary, then D must be holomorphically equivalent to \mathrm{\Delta }. A complete proof of the uniformization of Riemann surfaces was first given independently by Kobe and Poincaré in 1907.

According to the uniformization theorem of Riemann surfaces, apart from the one-dimensional complex sphere, the classification of Riemann surfaces is transformed into the classification of the holomorphic covering transformation groups of \mathbb{C} or \mathrm{\Delta }. It is known that the universal complex space is the Riemann surface \mathbb{C}\mathrm{\setminus }\left\{0\right\} of \mathbb{C} or the quotient space \mathbb{C}/{\mathbb{Z}}^2. Here, the {\mathbb{Z}}^2 transformation group is expressed as

\left(l,m\right):z\to z+l+m\omega , \mathrm{\forall }\left(l,m\right)\in {\mathbb{Z}}^2,

where ω is a complex number with modulus not exceeding 1 and non-zero imaginary part Im(ω). For a Riemann surface M with genus g larger than 1, there is

M\cong \mathrm{\Delta }/\Gamma ,

where the holomorphic covering transformation group \Gamma is a discrete subgroup of Möbius transformation on \mathrm{\Delta }.

Let {\mathcal{M}}_g be the set of all complex structures on the surface M of genus g. For complex structures, by introducing the holomorphic equivalence relation \sim, the quotient space R_g={\mathcal{M}}_g/\sim is called the moduli space of complex structures on the surface M. It can be shown that in the case of a torus:

R_1=H_1/\mathrm{S}\mathrm{L}\left(2,\mathbb{Z}\right),

where H₁ refers to the upper-half plane.

There are multiple proofs of the Riemann surface uniformization theorem. In one of them, the Hodge decomposition theorem can be used to construct the existence of meromorphic 1-forms with singularities. The Riemann theorem (or the Riemann-Roch theorem) can be used to calculate when g > 1,

(1.2)\mathrm{d}\mathrm{i}\mathrm{m}\left(R_g\right)=3g-3.

In particular, R_g is an open (3g-3)-dimensional complex manifold. In fact, by the Kodaira-Kuranishi theory of complex structure deformation [14] and the Serre duality theorem, there is

\mathrm{d}\mathrm{i}\mathrm{m}\left(R_g\right)=\mathrm{d}\mathrm{i}\mathrm{m}\left(H^1({\mathcal{T}}_M)\right)=\mathrm{d}\mathrm{i}\mathrm{m}\left(H^0(({\mathrm{\Omega }}^1{)}^{\otimes 2})\right),

where {\mathcal{T}}_M and {\mathrm{\Omega }}^p are the sheaf of holomorphic tangent spaces and holomorphic p-forms, respectively. So, by the Riemann theorem, we obtain (1.2). Riemann first conjectured the result (1.2), which was later proven by Teichmüller in the 1930s and 1940s. One of Teichmüller's important contributions was the introduction of the Teichmüller space T_g related to the moduli space R_g. The holomorphic equivalence relation in the Teichmüller space is restricted within the same homotopy class.

2 Complex manifold

Many fundamental theorems of high-dimensional complex manifolds rely on the function theory of several complex variables. By introducing the \overline{\mathrm{\partial }}-exterior differential operator

\overline{\mathrm{\partial }}:{\left(T{M}^{\ast }\right)}^{\left(p,q\right)}\to {\left(T{M}^{\ast }\right)}^{\left(p,q+1\right)},

we can define the \overline{\mathrm{\partial }}-de Rham cohomology group:

H_{\overline{\mathrm{\partial }}}^{p,q}\left(M\right)=\{\alpha \in (T{M}^{\ast }{)}^{(p,q)}|\overline{\mathrm{\partial }}\alpha =0\}/\mathrm{I}\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{e}\left(\overline{\mathrm{\partial }}\right).

Corresponding to the \overline{\mathrm{\partial }}-de Rham cohomology group, the Zěch cohomology group can be defined on a complex manifold. Both the \overline{\mathrm{\partial }}-de Rham cohomology group and the Zěch cohomology group can be defined on the space F of holomorphic complex vector bundles. According to the Dolbeault cohomology theory, there are the following isomorphism relation:

(2.1)H^q\left({\mathrm{\Omega }}^p\otimes F\right)=H_{\overline{\mathrm{\partial }}}^{p,q}\left(F\right).

The well-known infinitesimal deformation space H^1\left({\mathcal{T}}_M\right) of the complex structure by Kodaira is generated by the first Zěch cohomology group on {\mathcal{T}}_M[14]. According to (2.1), there is

(2.2)H^1\left({\mathcal{T}}_M\right)=H_{\overline{\mathrm{\partial }}}^{0,1}\left(T_M^{1,0}\right).

2.1 Kähler geometry and the Hodge theorem

Let \left(z^1,z^2,\dots ,z^n\right) be a local holomorphic coordinate on the complex manifold M^n. A Hermitian metric h on M corresponds to a Hermitian positive-definite matrix in the local coordinate system:

h=\left(h_{i\overline{j}}\right)>0.

Then

{\omega }_h=\sqrt{-1}\sum h_{i\overline{j}}\mathrm{d}{z}^i\wedge \mathrm{d}{z}^j

is a real (1,1)-form. If {\omega }_h satisfies

\mathrm{d}{\omega }_h=0,

then h is called a Kähler metric. If a complex manifold admits a Kähler metric, we call it a Kähler manifold. The main examples of Kähler manifolds come from smooth algebraic varieties, that is, complex sub-manifolds in complex projective spaces. Kähler geometry is a major research area in complex geometry. The famous Dolbeault-Hodge theorem is established in the context of Kähler geometry.

Let δ and {\overline{\mathrm{\partial }}}^{\ast } be the codifferential operators with respect to the exterior differential d and \overline{\mathrm{\partial }} respectively under the Kähler metric h. Then

\Delta =\mathrm{d}\cdot \delta +\delta \cdot \mathrm{d}, {\Delta }_{\overline{\mathrm{\partial }}}=\overline{\mathrm{\partial }}\cdot {\overline{\mathrm{\partial }}}^{\ast }+{\overline{\mathrm{\partial }}}^{\ast }\cdot \overline{\mathrm{\partial }}

are respectively the Hodge-Laplace operator and the Hodge-Dolbeault-Laplace operator defined on the space of forms. And there is

\Delta =2{\Delta }_{\overline{\mathrm{\partial }}}.

The Dolbeault-Hodge theorem shows the following isomorphism relation:

(2.3)H_{\mathrm{\partial }}^{p,q}\left(M\right)\cong {\mathcal{H}}_{\mathrm{\partial }}^{p,q}\left(M\right)=\{\alpha \in \mathrm{\Gamma }(\left(T{M}^{\ast }{)}^{p,q}\right)|{\mathrm{\Delta }}_{\overline{\mathrm{\partial }}}\alpha =0\}.

In particular, there is the Dolbeault-Hodge decomposition theorem:

H^l\left(M\right)=\sum _{p+q=l}^{}{\mathcal{H}}_{\overline{\mathrm{\partial }}}^{p,q}\left(M\right),\mathrm{\forall }l\le n.

Here, H^l\left(M\right) is the lth de Rham cohomology group. So, studying the de Rham cohomology group essentially means studying the harmonic (p,q)-forms. {\omega }_h\in H_{\overline{\mathrm{\partial }}}^{1,1}\left(M\right), which is also harmonic. The conjugate-symmetric, positive-definite \overline{\mathrm{\partial }}-closed elements in H_{\overline{\mathrm{\partial }}}^{1,1}\left(M\right) form a Kähler cone.

2.2 Holomorphic sectional curvature and the Frankel conjecture

The component K_{i\overline{j}k\overline{l}} of the holomorphic curvature tensor K of the Kähler metric h has the following formula:

K_{i\overline{j}k\overline{l}}=-\frac{{\mathrm{\partial }}^2{h}_{k\overline{j}}}{\mathrm{\partial }{z}_i\overline{\mathrm{\partial }}{z}_l}+h^{\gamma \overline{\delta }}\frac{\mathrm{\partial }{h}_{\gamma \overline{j}}}{\overline{\mathrm{\partial }}{z}_l}\frac{\mathrm{\partial }{h}_{k\overline{\delta }}}{\mathrm{\partial }{z}_i}.

If X=X^i\frac{\mathrm{\partial }}{\mathrm{\partial }{z}_i} is a (1,0)-type component field, K\left(X,\overline{X},X,\overline{X}\right) is called the holomorphic sectional curvature in the direction of X. K\left(X,\overline{X},Y,\overline{Y}\right) is called the biholomorphic sectional curvature with respect to the two (1,0)-type vector fields X, Y.

On the three types of Riemann surfaces in (1.1), the standard metric on the unit sphere, the flat metric on the plane, and the Poincaré metric on the unit disk have constant holomorphic sectional curvatures of 2, 0, and -1, respectively. Therefore, the Riemann uniformization theorem can also be geometrically stated as: a simply-connected complete Riemann surface with constant holomorphic sectional curvature is holomorphically equivalent to one of the three types of surfaces in (1.1). For high-dimensional complex manifolds, there is the following well-known Frankel conjecture:

Conjecture 2.1　 A compact complex manifold with positive biholomorphic sectional curvature must be holomorphically equivalent to the complex projective space.

The Frankel conjecture was proven respectively by Mori [20] and Yau-Siu [22] in the 1970s and 1980s using the algebro-geometric and minimal surface methods. Regarding non-compact complete complex manifolds, there is the following Yau uniformization conjecture:

Conjecture 2.2 [30]　 A non-compact complete complex manifold with positive biholomorphic sectional curvature must be holomorphically equivalent to a complex linear space.

For research related to Yau’s conjecture, one can refer to the works of Shi [21], Zhu [3], Tan [2], Liu [18], and others.

Under the Kähler metric, the component R_{i\overline{j}} of the Ricci curvature tensor Ric is defined as

R_{i\overline{j}}=h^{k\overline{l}}{K}_{i\overline{j}k\overline{l}}.

There is the following simple calculation formula for R_{i\overline{j}}:

R_{i\overline{j}}=-\frac{{\mathrm{\partial }}^2\mathrm{l}\mathrm{o}\mathrm{g}\left(h_{p\overline{q}}\right)}{\mathrm{\partial }{z}_i\mathrm{\partial }{\overline{z}}_j}.

So, the Ricci form \mathrm{R}\mathrm{i}\mathrm{c}\left(M\right)=\sqrt{-1}{R}_{i\overline{j}}\mathrm{d}{z}^i\wedge \mathrm{d}{z}_{\overline{j}} is a closed real (1,1)-form. In particular, when the complex manifold is compact, Ric(M) falls into an element of H_{\overline{\theta }}^{1,1}\left(M\right). \frac{1}{2\pi }\mathrm{R}\mathrm{i}\mathrm{c}\left(M\right) is called the first Chern class.

2.3 The Kähler-Einstein manifold

If a complex manifold M admits a Kähler metric h with constant Ricci curvature, M is called a Kähler-Einstein manifold. In other words, h satisfies

\mathrm{R}\mathrm{i}\mathrm{c}\left(h\right)=ch.

Here, the constant c can be normalized to −1, 0, 1. So, a compact Kähler-Einstein manifold has a negative-definite, trivial, or positive-definite first Chern class. Whether a compact Kähler manifold admits a Kähler-Einstein metric is usually referred to as the Calabi problem. In the 1950s, Calabi proved the uniqueness of the Kähler-Einstein metric in cases wherein the first Chern class was negative-definite or trivial. A complex manifold with a positive-definite first Chern class is usually called a Fano manifold. The uniqueness of the Kähler-Einstein metric in this case was proven by Mabuchi in the 1980s.

In the 1970s, Aubin proved the existence of Kähler-Einstein metrics in the case where the first Chern class was negative-definite. Around the same time, Yau [29] solved the problem of the existence of Kähler-Einstein metrics for cases wherein the first Chern class was negative-definite or trivial. For Fano manifolds, generally speaking, Kähler-Einstein metrics do not exist because there are many obstruction conditions. In the 1980s, Futaki proved that a necessary condition for a Fano manifold to admit a Kähler-Einstein metric was the vanishing of the Futaki invariant. In the 1990s, Tian Gang [24] proved that a necessary and sufficient condition for a Fano surface to admit a Kähler-Einstein metric was the vanishing of the Futaki invariant, thus proving the existence of Kähler-Einstein metrics in the case of surfaces.

Regarding the existence of Kähler-Einstein metrics on high-dimensional Fano manifolds, there is the following well-known Yau-Tian-Donaldson conjecture:

Conjecture 2.3 [25]　 A necessary and sufficient condition for a Fano manifold to admit a Kähler-Einstein metric is that it is K-poly-stable.

K-stability (K-poly-stability) was introduced by Tian Gang [25] in the 1990s, which was based on the generalized Futaki invariant. The Futaki invariant is defined on the Lie algebra space of holomorphic vector fields on a complex manifold. If the manifold itself has no non-trivial holomorphic vector fields, Ding Weiyue and Tian Gang [9] introduced the generalized Futaki invariant in 1992 by deforming the complex manifold via a projective group. If the generalized Futaki invariant is strictly positive for a class of special degeneration, the complex manifold is called K-stable. Donaldson [11] gave an algebraic definition of the generalized Futaki invariant in 2002.

Conjecture 2.3 was recently proven by Tian [26]. Chen et al. [5] also provided a proof. Therefore, the problem of the existence of Kähler-Einstein metrics on high-dimensional Fano manifolds boils down to studying the K-poly-stability of complex manifolds. Since K-poly-stability is an infinite-dimensional condition, how to verify K-poly-stability through the finite verification method remains a very important research topic. For some special Fano manifolds, such as toric manifolds, compactification spaces of reductive Lie groups, there are clear descriptions. One can refer to the works of Wang et al. [27], Delcroix [7], Li et al. [17], and others.

3 Several research focuses

3.1 Hodge conjecture

Let M be a smooth n-dimensional algebraic manifold. An analytic cycle T of order p\left(p\le n\right) is a finite sum

T=\sum _i {\lambda }_i\left[Z_i\right],

where each Z_i is a p-dimensional algebraic variety, and the coefficient λ_i is a real number or rational number. By using integration, as the Poincaré dual, T can be defined to be a current.

T\left(\beta \right)={\int }_M T\wedge \beta =:\sum {\lambda }_i{\int }_{Z_t} \beta ,\mathrm{\forall }\beta \in \mathrm{\Gamma }\left(\right(T{M}^{\ast }{)}^{p,p}).

So, T can be regarded as a weakly \overline{\mathrm{\partial }} closed (n-p, n-p)-form, hence a cohomology element of degree 2(n-p). We denote the Hodge group of degree p over the rational number field by the following formula:

{\mathrm{H}\mathrm{d}\mathrm{g}}_{\mathbb{Q}}^p\left(M\right)=\left(H^{p,p}\left(M,\mathbb{R}\right)\cap H^{2p}\left(M,\mathbb{Z}\right)/\mathrm{T}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n},\mathbb{Q}\right).

Then the Hodge conjecture can be described as:

Conjecture 3.1 [8]　 For any integer 0 ≤ p < n, the Hodge group {Hdg}_{\mathbb{Q}}^p\left(M\right) is finitely generated by analytic cycles of codimension p.

When p = 0, n, Conjecture 3.1 is true. Moreover, it can be proven that the conjecture holds when p = 1. By using the Poincaré duality, the conjecture is also true whenp=n-1. However, for a general p, the proof of Conjecture 3.1 remains extremely difficult to this day.

3.2 The Calabi-Yau manifold and mirror symmetry

According to Yau's theorem [29], a trivial first Chern class indicates that for any Kähler class on a complex manifold M^n, there exists a Ricci-flat Kähler metric. We usually call a compact Kähler manifold with the first Betti number being zero and a trivial first Chern class a Calabi-Yau manifold. Thus, a Calabi-Yau manifold always admits a non-vanishing holomorphic n-form {\mathrm{\Omega }}_M everywhere. A two-dimensional Calabi-Yau manifold is also called a K₃-surface. A typical example in higher dimensions is a homogeneous smooth hypersurface of degree n + 2 in the (n + 1)-dimensional projective space.

Due to the non-degeneracy property of {\mathrm{\Omega }}_M, there is the following isomorphism:

{\mathcal{T}}_M\cong {\left({\mathrm{\Omega }}^1\right)}^{\ast }\cong {\mathrm{\Omega }}^{n-1}.

And thus, it can be obtained from the Dolbeault isomorphism theorem that

H^p\left({\mathcal{T}}_M\right)\cong H_{\overline{\mathrm{\partial }}}^{n-1,p}\left(M\right).

In particular, the space of infinitesimal deformation H^1\left({\mathcal{T}}_M\right) of a complex structure is isomorphic to the \overline{\mathrm{\partial }}-de Rham cohomology group H_{\overline{\mathrm{\partial }}}^{n-1,1}\left(M\right). Similarly, there is

H^2\left({\mathcal{T}}_M\right)\cong H_{\overline{\mathrm{\partial }}}^{n-1,2}\left(M\right).

H^2\left({\mathcal{T}}_M\right) is the obstruction group for the deformation of the complex structure and is generally trivial. In particular, for three-dimensional Calabi-Yau manifolds,

H^2\left({\mathcal{T}}_M\right)\cong H_{\overline{\mathrm{\partial }}}^{2,2}\left(M\right)\cong H_{\overline{\mathrm{\partial }}}^{1,1}\left(M\right)

contains at least the Kähler class.

Although H^2\left({\mathcal{T}}_M\right)\ne 0, the famous Bogomolov-Tian-Todorov theorem shows that the complex structure deformation of a Calabi-Yau manifold never has any obstructions. In other words, the space of complex structure deformations is a smooth manifold. The following mainly discusses the three-dimensional Calabi-Yau manifolds.

By using the Dolbeault isomorphism theorem and b_1\left(M\right)=0, there are

H_{\overline{\mathrm{\partial }}}^{3,0}\left(M\right)\cong H^0\left({\mathrm{\Omega }}^3\right)\cong \mathbb{R},

H_{\overline{\mathrm{\partial }}}^{1,0}\left(M\right)\cong H_{\overline{\mathrm{\partial }}}^{2,3}\left(M\right)=0.

So, for the de Rham cohomology groups of a three-dimensional Calabi-Yau manifold, only H_{\overline{\mathrm{\partial }}}^{1,1}\left(M\right)\left(H_{\overline{\mathrm{\partial }}}^{2,2}(M)\right) and H_{\overline{\mathrm{\partial }}}^{2,1}\left(M\right)\left(H_{\overline{\mathrm{\partial }}}^{1,2}(M)\right) are non-trivial.

The mirror symmetry principle originates from the superconformal field theory in theoretical physics. The latter can be described by the σ-model. The σ-model aims to construct a certain moduli space of maps from a Riemann surface to a three-dimensional Calabi-Yau manifold, such as the moduli space of holomorphic curves. According to superconformal field theory, there should be an A-model and a B-model corresponding to two three-dimensional Calabi-Yau manifolds M and \check{M} respectively. Physicists believe that the de Rham cohomology groups of these two Calabi-Yau manifolds must have the following relationships:

\mathrm{d}\mathrm{i}\mathrm{m}\left(H_{\overline{\mathrm{\partial }}}^{1,1}(M)\right)=\mathrm{d}\mathrm{i}\mathrm{m}\left(H_{\overline{\mathrm{\partial }}}^{1,2}(\check{M})\right),\mathrm{d}\mathrm{i}\mathrm{m}\left(H_{\overline{\mathrm{\partial }}}^{2,1}(M)\right)=\mathrm{d}\mathrm{i}\mathrm{m}\left(H_{\overline{\mathrm{\partial }}}^{1,1}(\check{M})\right).

This is the so-called mirror symmetry phenomenon. Moreover, the variation of the de Rham cohomology group of \check{M} is completely determined by the Gromov-Witten invariants on M.

Regarding the mirror symmetry principle, there is also a famous Strominger-Yau-Zaslow (SYZ) conjecture.

Conjecture 3.2 [23]　 Any Calabi-Yau manifold M has the structure of a toric fiber bundle of special Lagrangian submanifolds. Moreover, the Calabi-Yau manifold \check{M} in the B model has a fiber bundle structure corresponding to that of M.

3.3 Canonical metrics

In recent decades, with the development of geometric analysis, people have introduced other canonical metrics closely related to the Kähler-Einstein metric, such as Kähler-Ricci solitons, Kähler-Einstein metrics with conical singularities, and Calabi’s extremal metrics. For a Kähler metric h on a complex manifold, if it satisfies the equation

R_{i\overline{j}}-c{h}_{i\overline{j}}=L_{X}h,

then h is called a Kähler-Ricci soliton. Here, X is a holomorphic vector field on M, and L_X denotes the Lie derivative. Ricci solitons, first introduced by Hamilton when studying the Ricci flow, are a special type of solution to the Ricci flow. According to the sign of the constant c, Ricci solitons can be classified into three types: expanding, steady, and shrinking. However, in the compact case, there is only the shrinking type. Currently, there are few examples of Kähler-Ricci solitons. Typical examples include the solutions of ordinary differential equations constructed by Cao [1], Feldman et al. [13], Yang [28], as well as the Kähler-Ricci solitons on Fano cyclic manifolds proven by Wang and Zhu [27] using the solution of the Monge-Ampere equation.

The earliest research on Kähler-Einstein metrics with conical singularities can be traced back to the study of constant-curvature metrics with conical singularities on Riemann surfaces. A few years ago, Donaldson [12] proposed a high-dimensional model and used it to study the existence of Kähler-Einstein metrics on Fano manifolds. The flat Kähler metric with conical singularities on {\mathbb{C}}^n can be expressed as

(3.1)h=\frac{{\left|\mathrm{d}{z}_1\right|}^2}{{\left|z_1\right|}^{2\left(1-\beta \right)}}+\sum _{i=2}^n{\left|\mathrm{d}{z}_i\right|}^2,

where 2\pi \beta \in \left(0,2\pi \right] refers to the cone angle along the hypersurface \left\{z_1=0\right\}. Generally, a Kähler-Einstein metric h with conical singularities on a Fano manifold M satisfies the following equation:

(3.2)\mathrm{R}\mathrm{i}\mathrm{c}\left(h\right)=\beta {\omega }_h+\frac{\left(1-\beta \right)}{\lambda }\left[D\right],

where D is a smooth divisor of M, with a homology class \left[D\right]\in \lambda c_1\left(M\right), h is a smooth metric on M \ D, and for any point p of D, there exists a local holomorphic coordinate such that the relation (3.1) is satisfied at the point p. Eq. (3.2) is defined globally in the sense of current. For the research on (3.2), one can refer to the recent works of Rubinstein [15], Li et al. [16].

In the 1980s, Calabi proposed to study the extremal metrics on a general Kähler class. In a special case, there is the so-called constant scalar curvature metric, that is, the Kähler metric h satisfies the following equation:

\sum h^{i\overline{j}}(\mathrm{l}\mathrm{o}\mathrm{g}(h_{p\overline{q}}){)}_{i\overline{j}}\equiv \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}.

Just like the Kähler-Einstein metric, this type of metric is unique within the same Kähler class. One can refer to the works of Donaldson [10], Chen and Tian [6], Mabuchi [19], etc. Regarding the existence problem, similar to that of the Kähler-Einstein metric (see Conjecture 2.3) there is a popular conjecture: the necessary and sufficient condition is that the Kähler class is K-poly-stable. Recently, Chen et al. [4] proved that if the Mabuchi energy was proper on the Kähler class, then there existed a constant scalar curvature metric on this Kähler class. Therefore, a natural question is how to prove that the properness of the Mabuchi energy is equivalent to the K-poly-stability of the Kähler class.

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