LECTURE 4. GROMOV COMPACTNESS AND STABLE MAPS 197The topology can be recovered from the collection C of convergent sequences, as thecollection U c 2X of all subsets U C X such that, for all x 0 E U and all sequences
Xn E X with (xo, (x n)n) E C, there exists an N E N such that n ~ N impliesXn E U. Starting with a collection of sequences C c X x XN one obtains a topology
without imposing any axioms. The "Subsequence" axiom guarantees that, for every
subset F C X, X - F E U if and only if x 0 E F whenever there exists a sequence
Xn E F such that (xo, (xn)n) E C. The "Constant" and "Diagonal" axioms are
needed to show that the closure of a subset A C X consists of all elements x 0 E Xfor which there exists a sequence Xn EA such that (x 0 , (xn)n) EC. All five axioms
are needed to prove that a sequence Xn E X converges to x 0 with respect to the
topology U if and only if (xo, (xn )n ) EC. Details are left as an exercise. DIt is obvious from the definitions that the collection of all Gromov convergent
sequences in Mo,k,A ( M, J) satisfies the "Constant" and "Subsequence" axioms.Exercise 4.7. Let (u,z) ({ua}aEr,{Zaf3}aEf3,{ai,zi}1s;is;k) be a stable
J-holomorphic curve of genus zero with k marked points and fix a suf-ficiently small constant c; > 0. For any other stable map ( u', z')
( { u~}a'ET', { z~'/3' }a' E'/3', a~, z~hs;is;k) with k marked points define the real numberec,u,z(u', z') inf inf {sup sup d (u/ (a) o I.Pa, ua)
f'T~T' {'Pa }aET a S2-B (Z )
f(a;)=a; ' "+ sup I Ea (u, Bc(za13)) - Ef(a) (u', I.Pa (Bc(za13)))I
aE/3+ sup sup d ( <p"{3^1 o I.Pa, Zf3a)
aE/3 S^2 -B,(za13)
f(a)=f(fJ)+ sup d (i.p"f3
1( zt(fJ)f(a)), Zf3a) + sup d (i.p-;;f (z~), zi)}.
aE/3 l<i<k
f(a)"f.f(/3) - -Here the infimum runs over all surjective tree homomorphisms f : T ---+ T' which
satisfy f(ai ) =a~ for all i and all tuples {i.pa }aET E er. The number c; > 0 is
chosen such that E( ua, B c (zaf3)) :::; !i/2 and Bc (za/3) n B c (za,,) = 0 whenever aE/3,
aE1 and /3 -=f. r· Note that E a (u, Bc:(za13)) = E(ua, Bc(za13)) + ma13(u).
Prove that a sequence ( u,,, z,,) Gromov converges to ( u , z) if and only if the
sequence of real numbers ev = ec,u,z ( u", z") converges to zero. Prove that the
function ec,u,z is lower semi-continuous with respect to the Gromov topology in the
domain ec,u,z < 8 for 8 > 0 sufficiently small. Deduce that Gromov convergence
satisfies the "Subsubsequence" and "Diagonal" axioms. D
Theorem 4.8 (Uniqueness of limits). Let (u",z") be a sequence of stable J-holo-
morphic curves of genus zero with k marked points which Gromov converges to two
stable maps (u,z) and (ii,z). Then (u,z) is equivalent to (ii,z).
Theorem 4.9 (Gromov compactness). Every sequence (u", z") of stable J-holo-
morphic curves of genus zero with k marked points with sup,,, E(u,,) < oo has a
Gromov convergent subsequence.Theorem 4.10 (Second countable). The topology of Mo,k,A(M, J) has a count-
able basis.