Contents

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complex geometry

# Contents

## Idea

If the ground ring (ground field) is equipped with a star structure (an anti-involution, such as complex conjugation in the complex numbers), then the operation of forming dual linear spaces may be “twisted” by this involution to yield a notion of “anti-dual” spaces. These relate to Hermitian forms as ordinary dual spaces relate to ordinary inner products.

## Definition

Let $\mathbb{K}$ be a star-ring, hence a unital ring (often a field, whence our notation, and here often the complex numbers) equipped with an anti-involution (for instance complex conjugation):

$\overline{(-)} \,\colon\, \mathbb{K} \to \mathbb{K}$
$\overline{x \cdot y} \,=\, \overline{y} \cdot \overline{x} \,.$

Consider a $\mathbb{K}$-module $\mathscr{V} \,\in\, \mathbb{K}Mod$ (say right modules for possibly non-commutative $\mathbb{K}$, but being just $\mathbb{K}$-vector spaces if $\mathbb{K}$ is a field).

###### Definition

The anti-dual space $\overline{\mathscr{V}^\ast}$ of $\mathscr{V}$ is the space of anti-linear maps to the ground ring (ground field):

$\overline{\mathscr{V}^\ast} \;\coloneqq\; \Big\{ \phi \,\colon\, \mathscr{V} \to \mathbb{K} \,\Big\vert\, \underset{v, v' \in \mathscr{V}}{\forall} \, \phi(v + v') = \phi(v) + \phi(v') ,\; \underset{ { v \in \mathscr{V} } \atop { \lambda \in \mathbb{K} } }{\forall} \, \phi(v \cdot \lambda) \,=\, \overline{\lambda} \cdot v \Big\}$

equipped itself with the structure of a (right) module by

$\phi \in \overline{\mathscr{V}^\ast} ,\, \,\lambda \in \mathbb{K} \;\;\; \vdash \;\;\; (\phi \cdot \lambda)(c) \;=\; \phi(v) \cdot \lambda \,.$

(eg. Karoubi & Villamayor 1973, Ex. 1; Karoubi 2010, §1)

As the notation already indicates, an equivalent definition is:

###### Definition

The anti-dual $\overline{\mathscr{V}^\ast}$ has as underlying abelian group the ordinary linear dual $\mathscr{V}^\ast$ but equipped with its anti-linear structure, namely with the (right) module action twisted by the involution as:

$\phi \,\in\, Hom_{\mathbb{K}}(\mathscr{V}, \mathbb{K}) \;\;\; \vdash \;\;\; \underset{v \in \mathscr{V}}{\forall} \, (\phi \cdot \lambda)(v) \;=\; \overline{\lambda} \cdot \phi(v) \,.$

(eg. Mishchenko 1976 §1.1).

An isomorphism between the two definitions

$\array{ \mathscr{V}^\ast &\longrightarrow& \mathscr{V}^\ast \\ \phi &\mapsto& \overline{\phi} }$

is established by postcomposing linear forms with the star-involution

$\overline{\phi} \,\colon\, v \,\mapsto\, \overline{\phi(v)}$

which respects the above (right) $\mathbb{K}$-actions due to

$\array{ \big( v \,\mapsto\, \phi(v)\cdot \lambda \big) &\mapsto& \big( v \,\mapsto\, \overline{\phi(v) \cdot \lambda} \big) \\ & = & \big( v \,\mapsto\, \overline{\lambda} \cdot \overline{\phi(v)} \big) \mathrlap{\,.} }$

## References

Discussion in the context of Hermitian K-theory: