Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 118 / 2

For the nonnegative Hodge Laplacian ${\Delta }_d^{\mathrm{H}}=d{d}^{\ast }+d^{\ast }d$, the Kähler Laplacian identity gives ${\Delta }_d^{\mathrm{H}}=2{\Delta }_{\bar{\partial }}$. If $u$ is a holomorphic function, then $\bar{\partial }u=0$, while ${\bar{\partial }}^{\ast }u=0$ by degree. ThereforeThe divergence-of-gradient convention printed in part (b) is ${\Delta }_d=-{\Delta }_d^{\mathrm{H}}$ on functions. The sign changes no kernel, so the requested harmonicity of holomorphic functions on a Kähler manifold isHolomorphic functions are harmonic without any compactness assumption. This is a local identity from the Kähler Laplacian identity, not an integration argument requiring a compact manifold.

The real coordinate functions are real and imaginary parts of the holomorphic coordinates $z_j$. Part (a), together with the real coefficients of the Laplace-Beltrami operator, therefore shows that every $x_l$ is a harmonic function. For the given divergence-of-gradient convention, applying the operator to $x_l$ givesThese are the harmonic coordinate identities. Expanding the operator on an arbitrary smooth function now givesThe bracketed coefficients vanish. Hence there is no first-derivative term in these coordinates:The reason is that the real components of holomorphic coordinates are harmonic coordinates for a Kähler metric; a general real coordinate change need not retain this simplification.

Let ${\pi }_i:{\mathbb{C}}^{g_i}\to X_i$ be the quotient covering maps. Since the source universal cover is simply connected, the lifting criterion for a covering space gives a lift $F:{\mathbb{C}}^{g_1}\to {\mathbb{C}}^{g_2}$ of $f\circ {\pi }_1$. The lift is a holomorphic map because ${\pi }_2$ is locally a biholomorphism.

For every $\lambda \in {\Lambda }_1$, the difference $F(z+\lambda )-F(z)$ belongs to the discrete Euclidean lattice ${\Lambda }_2$. It is a continuous function of $z$ on a connected space, hence is constant. Differentiating shows that each coefficient of $DF$ is ${\Lambda }_1$-periodic. Such a coefficient is bounded on a compact fundamental parallelepiped, and periodicity bounds it on all of ${\mathbb{C}}^{g_1}$. The several-variable Liouville theorem, obtained by applying the one-variable theorem on coordinate lines, makes every coefficient constant.

Consequently $DF=A$ for a constant complex-linear map $A$ and $F(z)=Az+x$. The period differences become $A\lambda \in {\Lambda }_2$. Thus the affine lift of a holomorphic map between complex tori givesEvery holomorphic map between complex tori is a homomorphism followed by a translation. The translation vector $x=F(0)$ is determined modulo ${\Lambda }_2$, while $A$ is the derivative of any lift and is unaffected by that ambiguity.

Choose a constant positive Hermitian form on $V$. Its underlying flat Kähler metric is translation-invariant and descends through the Euclidean lattice to the compact complex torus $X$. The Dolbeault Hodge decomposition gives a unique Dolbeault Laplacian-harmonic representative of every Dolbeault cohomology class. By the Kähler Laplacian identity, these are also Hodge Laplacian-harmonic.

Choose a constant coframe on $V$. A lifted harmonic form is a sum of constant wedge basis forms with smooth periodic coefficients. For the flat metric, the Hodge Laplacian acts coefficientwise by the scalar constant-coefficient Laplacian. Each coefficient is a harmonic function on a compact flat real torus, hence is constant: integration by parts gives $\int |Da{|}^2=0$. Conversely every constant-coefficient form is harmonic. Thus the harmonic representatives are exactly the translation-invariant differential forms of type (p, q).

The constant $(1,0)$ covectors form $V^{\ast }={\mathrm{Hom}}_{\mathbb{C}}(V,\mathbb{C})$; the constant $(0,1)$ covectors form the space of antilinear maps $\overline{V^{\ast }}={\mathrm{Hom}}_{\overline{\mathbb{C}}}(V,\mathbb{C})$. Sending a constant wedge form to its cohomology class givesThe map is intrinsic and does not depend on the auxiliary flat metric: that metric proves every class has exactly one constant representative, while the inclusion of constant forms is defined directly by the quotient. The Dolbeault cohomology of a complex torus consists of constant forms, and in particularFor indices outside $0,\dots ,g$, both the exterior-power expression and the cohomology group vanish.