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$dim = 1$: Riemann surface, super Riemann surface
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Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
Let $V$ be a finite-dimensional real vector space. Then a linear Kähler structure on $V$ is
a linear complex structure on $V$, namely a linear endomorphism
whose composition with itself is minus the identity morphism:
a skew-symmetric bilinear form
such that
$\omega(J(-),J(-)) = \omega(-,-)$;
$g(-,-) \coloneqq \omega(-,J(-))$ is a Riemannian metric, namely
a non-degenerate positive-definite bilinear form on $V$
(necessarily symmetric, due to the other properties: $g(w,v) = \omega(w,J(v)) = -\omega(J(v),w) = - \omega(J(J(v)), J(w)) = \omega(w,J(w)) = g(v,w)$).
(e.g. Boalch 09, p. 26-27)
Linear Kähler space structure may conveniently be encoded in terms of Hermitian space structure:
(Hermitian form and Hermitian space)
Let $V$ be a real vector space equipped with a complex structure $J\colon V \to V$. Then a Hermitian form on $V$ is
a complex-valued real-bilinear form
such that this is symmetric sesquilinear, in that:
$h$ is complex-linear in the first argument;
$h(w,v) = \left(h(v,w) \right)^\ast$ for all $v,w \in V$
where $(-)^\ast$ denotes complex conjugation.
A Hermitian form is positive definite (often assumed by default) if for all $v \in V$
$h(v,v) \geq 0$
$h(v,v) = 0 \phantom{AA} \Leftrightarrow \phantom{AA} v = 0$.
A complex vector space $(V,J)$ equipped with a (positive definite) Hermitian form $h$ is called a (positive definite) Hermitian space.
(basic properties of Hermitian forms)
Let $((V,J),h)$ be a positive definite Hermitian space (def. ). Then
the real part of the Hermitian form
is a Riemannian metric, hence a symmetric positive-definite real-bilinear form
the imaginary part of the Hermitian form
is a symplectic form, hence a non-degenerate skew-symmetric real-bilinear form
hence
The two components are related by
Finally
and so the Riemannian metrics $g$ on $V$ appearing from (and fully determining) Hermitian forms $h$ via $h = g - i \omega$ are precisely those for which
These are called the Hermitian metrics.
The positive-definiteness of $g$ is immediate from that of $h$. The symmetry of $g$ follows from the symmetric sesquilinearity of $h$:
That $h$ is invariant under $J$ follows from its sesquilinarity
and this immediately implies the corresponding invariance of $g$ and $\omega$.
Analogously it follows that $\omega$ is skew symmetric:
and the relation between the two components:
as well as
As a corollary:
(relation between Kähler vector spaces and Hermitian spaces)
Given a real vector space $V$ with a linear complex structure $J$, then the following are equivalent:
$\omega \in \wedge^2 V^\ast$ is a linear Kähler structure (def. );
$g \in V \otimes V \to \mathbb{R}$ is a positive definite Hermitian metric (2)
where $\omega$ and $g$ are related by (1)
Hence Kähler vector spaces are equivalently the finite dimensional complex Hilbert spaces.
The archetypical elementary example is the following:
(standard Kähler vector space)
Let $V \coloneqq \mathbb{R}^2$ be the 2-dimensional real vector space equipped with the complex structure $J$ which is given by the canonical identification $\mathbb{R}^2 \simeq \mathbb{C}$, hence, in terms of the canonical linear basis $(e_i)$ of $\mathbb{R}^2$, this is
Moreover let
and
Then $(V, J, \omega, g)$ is a Kähler vector space (def. ).
The corresponding Kähler manifold is $\mathbb{R}^2$ regarded as a smooth manifold in the standard way and equipped with the bilinear forms $J, \omega g$ extended as constant rank-2 tensors over this manifold.
If we write
for the standard coordinate functions on $\mathbb{R}^2$ with
and
for the corresponding complex coordinates, then this translates to
being the differential 2-form given by
and with Riemannian metric tensor given by
The Hermitian form is given by
This is elementary, but, for the record, here is one way to make it fully explicit (we use Einstein summation convention and “$\cdot$” denotes matrix multiplication):
and similarly
Lecture notes include
Last revised on June 11, 2022 at 11:10:11. See the history of this page for a list of all contributions to it.