# Fano manifolds

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\underline{Floer cohomology and Fukaya categories for Fano varieties}

The purpose of this talk is to see some examples of Fukaya categories
which are less similar to objects from classical homological algebra, but
can still be explicitly understood. The literature is extensive but
fragmentary. The prerequisite for giving this talk is a basic knowledge of
Lagrangian Floer cohomology. The speaker does not necessarily have to be a
specialist, but interest in symplectic geometry is important.

\underline{Plan}

{\it Monotone Lagrangian submanifolds, moduli spaces of holomorphic
discs.} The point is just to recall the monotonicity condition, and what it
implies for the spaces of holomorphic discs (space of any given dimension
becomes compact in the Gromov compactification). Possible references: [Oh,
Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks],
[Cho, Holomorphic disc, spin structures and Floer cohomology of the
Clifford torus]. It should be emphasized that monotonicity is introduced
here to simplify both the conceptual and technical framework. There is a
more general picture without this condition, but it's somewhat more
complicated.

{\it The central charge.} We do want to consider motonone Lagrangian
submanifolds equipped with flat $U(1)$-bundles. The point here is that
(assuming the Lagrangians are $Pin$) counting Maslov index $2$ holomorphic
discs which go through a given point, with suitable holomony factors yields
a complex number associated to the Lagrangian submanifold (the charge).
Floer cohomology between two Lagrangian submanifolds $(L_0,L_1)$
is well-defined only if their charges agree, because $d^2$ is the
difference between those. See [Oh, op. cit. with Appendix]. You'll
have to figure out how to do Floer cohomology with twisted coefficients,
probably in one of the papers I'm quoting here (for $L_0 = L_1$, it's in
Cho's work, but we want the general case as well).

{\it Restrictions on the central charge.} So for any value of $\lambda \in \C$, we get a Fukaya category consisting of motonone Lagrangian
submanifolds with that charge. The point is that that category is
zero unless $\lambda$ is an eigenvalue of quantum multiplication with
$c_1(M)$. This is proved in [Auroux, Mirror symmetry and T-duality in the
complement of an anticanonical divisor]. Discussion needs to include some
preliminaries, presumably about the module structure of Floer cohomology
over quantum cohomology.

{\it Examples.} With that in mind, we can do either $\C P^2$ or $\C P^1 \times \C P^1$. For the first one, see [Cho, op. cit.]. The point is that
for each eigenvalue of quantum multiplication we get a single nontrivial
object, which is the Clifford torus with a prescribed $U(1)$-bundle. Its
Floer endomorphism ring $HF^*(L,L)$ is a Clifford algebra, as computed in
[Cho, Products of Floer cohomology of torus fibers in toric Fano
manifolds]. It's not important to do the computation (the baby case of $\C P^1$ probably shows all we need), but the result should be stated clearly.
A more or less known, but unfortunately not written down, result says that
these tori split-generated the relevant Fukaya categories. This can be
mentioned without further discussion.

\underline{Notes}

For $\C P^1 \times \C P^1$, there is actually another Lagrangian
submanifold, the antidiagonal sphere, and its relation to the tori in the
Fukaya category is interesting. However, it would require a specialist to
explain that thoroughly. For $\C P^3$ and $\C P^1 \times \C P^1 \times \C P^1$, or non-toric things like a variety of flags in $\C^3$, there are a
lot of open questions.

\underline{Dependencies}

This relies on: basic topology of Lagrangian submanifolds (monotonicity
uses the relative first Chern class); basic holomorphic disks; Floer
cohomology and quantum cohomology; and knowing the example of the Clifford
torus
in projective space. All of these should be covered in previous talks.

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