First some style/structure comments:

next unless  /Synchronizing started|Synchronizing finished|Summary/;
my @date= m|(\d+)/(\d+)/(\d+)|;
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I would make sure that you check that the date regex matched before using @date. Something like:

my @date;
if (@date = m{(\d+)/(\d+)/(\d+)}) {
  ...
}
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The danger is that if the date regex fails, you will have no way of knowing about it. (Update: I guess this is not entirely true, but it is still good coding practice to always test if regex matches succeed or not.)

The next issue is: never mind... I didn't see how @today was defined

Here's what I think you want:

my %times;
my $job;
while (<FH1>) {
  next unless (s{^\[(\d+/\d+/\d+ \d+:\d+:\d+)\]\s*}{}); # malformed li
+ne
  my $timestamp = $1;
  if (m/^(Synchonization|Analyzing) (started|finished), job: "(.*?)"/)
+ {
    $job = $3;
    $times{$job}->{$1}->{$2} = $timestamp;
  } elsif (m/^Summary:/) {
    $times{$job}->{summary} = $_;
  }
}
print Dumper(\%times); use Data::Dumper;
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One key element of this loop is to save the last parsed job so we know which job a "Summary" line refers to. The list of jobs is keys %times.

There are other ways of storing the timestamp data, and you may want to choose another data structure depending on how you plan to use the data later.