LECTURE 1. SYMPLECTIC FIXED POINTS AND MORSE THEORY 1491.2. The monotonicity condition
An almost complex structure J on TM is called compatible with w if the formula
(3) (~, TJ) = w(~, JTJ)
defines a Riemannian metric on M. The space :1 ( M, w) of such almost com-plex structures is nonempty and contractible. Thus the first Chern class c 1 =
c1(TM,J) E H^2 (M,Z) is independent of the choice of J E .J(M,w). The goal in
these lectures is to outline the proof of the Arnold conjecture in the case where the
cohomology classes c 1 and [w] satisfy the condition(4) r v*c1 = T r v*w
ls2 ls2for every smooth map v : 82 -> M and some constant T E R It is important to
distinguish the three cases T > 0, T = 0, and T < 0. Geometrically, this corresponds
to the conditions of positive, zero, and negative curvature. The toy models for
these cases are the 2-sphere (positive curvature), the 2-torus (zero curvature), and
surfaces of higher genus (negative curvature).^1 As these simple examples already
indicate, the case T < 0 is by far the most general. On the other hand the proof of
the Arnold conjecture is easier in the case T > 0 and symplectic manifolds with this
property are called monotone. This is the case originally treated by Floer in [11]
and it led him to the definition of what is now called Floer homology. The case
c 1 = 0 was treated by Hofer-Salamon in [13, 19]. This is an extension of Floer's
work and requires the construction of Floer homology groups with coefficients in
a suitable Novikov ring. The case T < 0, and indeed that of general compact
symplectic manifolds was only recently resolved by Fukaya-Ono [14], Liu-Tian [28],
and Hofer-Salamon [20, 21, 22, 23].
Remark 1.4. In [35] Ohta and Ono proved that the only symplectic 4-manifolds,
in which c 1 (TX, J) is a positive multiple of some integral lift of the cohomology
class of w, are 82 x 82 and CP^2 with up to eight points blown up. Also it is a
well known fact in Kahler geometry that the only simply connected Kahler surfaces
with c 1 = 0 are the K3-surfaces (e.g. hypersurfaces of degree 4 in CP^3 ) and they
are all diffeomorphic. D
Exercise 1.5. Let Xd c e,pn be a hypersurface of degree d. Explicitly, one can
think of this as the submanifold cut out by the equation zod + z 1 d +·· ·+Zn d = 0.
The Lefschetz hyerplane theorem asserts that this manifold is simply connected.
Prove that its first Chern class is given by
c1(Xd) = (n+ 1-d)i*h
where h = PD([cpn-^1 ]) E H^2 (CPn;z) is the canonical generator, and i: Xd->
e,pn denotes the inclusion. Deduce that xd satisfies (4) with T > 0 for d :::; n,
with T = 0 for d = n + 1, and with T < 0 for d 2::: n + 2. Hint: The direct sum
of the tangent bundle rcpn with the trivial line bundle c is isomorphic to the
(n + 1)-fold direct sum of the canonical bundle H. The normal bundle of X d can
be identified with the restriction of the dth tensor power of H to Xd. D
(^1) These examples do not quite fit the definition since 7r2(I:) = 0 for Riemann surfaces of genus
g 2: 1. However, if we strengthen (4) to ci = r[w], then Riemannn surfaces satisfy the condition
with T = (Vol(I:))-^1 (2 -2g).