Kleinian singularities

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\underline{Derived categories of Kleinian singularities}

This talk examines another case where the Fukaya category is completely
understood
on the derived level. Prerequisites are a basic understanding of symplectic
geometry
and Floer cohomology, as well as some very basic homological algebra. The
main
reference is [Khovanov-Seidel, Quivers, Floer cohomology, and braid group
actions].

\underline{Plan}

{\it Kleinian singularities.} We will mainly look at the Milnor fibres
of the $(A_m)$ type singularities, which are the affine algebraic surfaces
$xy + f(z) = 0$
where $f$ is a generic polynomial of degree $m+1$. We equip them with the
restriction of the standard symplectic structure on $\C^3$. We approach
the geometry of these surfaces by projecting to the $z$-plane. In
particular,
this leads to a construction of Lagrangian spheres as in [Khovanov-Seidel,
op. cit.],
showing that the Milnor fibres contain an $(A_m)$ diagram of such spheres.
It is also interesting to see the braid group acting by symplectic
monodromy.

{\it The $(A_m)$ quiver.} Taking an $(A_m)$ chain of spheres, we want to
see their Floer cohomology and its multiplicative structure (at least on
the
cohomological level). This can be written down explicitly as a certain
algebra.
The theorem is that this algebra gives a derived description of the Fukaya
category. As motivation, one can see directly how the previously
constructed
Lagrangian spheres correspond to certain complexes of projective modules
[Khovanov-Seidel].

{\it Formality.}
The final ingredient in proving that this description of the Fukaya
category
is accurate is the formality of the underlying cochain level $(A_\infty)$
before,
but the basic references are [Seidel-Thomas, Braid group actions on derived
categories of coherent sheaves], [Seidel, Fukaya categories and
Picard-Lefschetz
theory, Section 20].

\underline{Notes}

The generalization of this picture to $D$ and $E$ type singularities is not
well explored. We know that their Milnor fibres contain suitable chains of
Lagrangian spheres, but formality of the associated $A_\infty$-structures
has not been shown, as far as I know. An interesting question is whether
the associated
monodromy representations of braid groups (considered as taking values in
the
groups of symplectic automorphisms mod isotopy) are faithful. This is most
difficult in the case of $E$ type singularities.

\underline{Dependencies}

This relies on: Lagrangian submanifolds, Floer cohomology,
$A_\infty$-structures,
and to some extent on the derived categories talk.

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