Are Complex Elliptic Curves Kähler Manifolds?

Question

Manifolds prof. wasn't sure, so I thought I'd ask the community. Are all elliptic curves defined over $\mathbb{C}$ Kähler manifolds when considered as compact, connected, complex manifolds? Or are there maybe particular conditions on the coefficients of the Weierstrass equation or perhaps some other conditions on elliptic curves over the complex numbers which ensure being Kähler?

Answer 1

Every (complex) smooth projective variety is Kähler. Indeed, $\mathbb{CP}^n$ is Kähler using the Fubini-Study metric, and this metric then restricts to a Kähler metric on any complex submanifold of $\mathbb{CP}^n$. An elliptic curve is a complex submanifold of $\mathbb{CP}^2$, and hence is Kähler.

Another way to get a Kähler metric on an elliptic curve is to use the fact that it is a complex torus. That is, if $X$ is an elliptic curve, then there is an isomorphism of complex manifolds $\mathbb{C}/\Lambda\to X$ where $\Lambda$ is a lattice in $\mathbb{C}$. The flat metric on $\mathbb{C}$ (which is Kähler) can then be pushed forward along the quotient map $\mathbb{C}\to\mathbb{C}/\Lambda$ to give a Kähler metric on $X$.

Yet another way: every Hermitian metric on a 1-dimensional complex manifold is Kähler. Indeed, the Kähler form $\omega$ is a 2-form, which is automatically closed since the manifold has real dimension 2 ($d\omega$ is a 3-form, but there are no nonzero 3-forms).

Answer 2

(The analytification of) any smooth projective variety over $\mathbb{C}$ is Kahler. The Fubini-Study metric restricts to a Kahler metric on any submanifold.