Mr Kollek, a Labour Party member, is already in his 27th year as Jerusalem's mayor and is now running, at the age of 83, for a further term. To Jews abroad, and to many in Israel, he has become a symbol of Israel's success at 'reuniting' Jerusalem. He has sought tirelessly to present the acceptable face of Israel's annexation of the Arab east side, pressing always for 'peaceful coexistence' between Arab and Jew, while refusing to budge on Israel's sovereignty claim.
The Palestinians, who number 160,000 out of a total population in the city of 570,000, have, until now, been little impressed with Mr Kollek, or with the prospect of any Israeli mayor. When Israeli law was imposed on East Jersualem immediately after it was seized by the state in 1967, the Arab residents were given the right to vote in the municipal elections. Since then the vast majority have chosen not to do so, on the grounds that to vote would be to grant legitimacy to the occupier's law. Of the 90,000 eligible to vote in the municipal elections, only about 2,000 have exercised their right in recent years, and most of these have been employees of the municipality.
On the east side the Arabs have a different view of 'united Jerusalem' to the one presented by Mr Kollek. Under his reign the downtrodden Arab neighbourhoods have been engulfed by new Jewish settlements. The sense of a living, Arab community has been far more undermined by occupation in Jerusalem than in the West Bank, where Jewish settlement has hardly impinged on the big Arab urban centres. Mr Kollek has promoted Jewish settlement in east Jerusalem so that this year, for the first time, Jews on the east side outnumber Arabs.
Nevertheless, during this election campaign some Arabs have warmed to Mr Kollek - or started to view him as the lesser of two evils.
Mr Kollek's main opponent is Ehud Olmert, a member of the right-wing Likud Party. The Arabs fear that whatever Mr Kollek may have done, Mr Olmert will be far worse. That fear has grown as Mr Olmert's support has increased in the polls, which suggest that Jerusalem's Jewish population may vote out Mr Kollek - worried by his age, anxious for change, and concerned too, perhaps, that his talk of 'Arab-Jewish' coexistence might contain the seeds of future compromise.
So deep is the Arab fear of a Likud mayor that some leading Palestinians - most notably Faisal Husseini, the de facto local leader, and a respected Jerusalemite - have said that 'Olmert must be stopped', thereby hinting that Arabs should abandon their boycott, and shore up Mr Kollek's vote.
The mayoral election takes place at a pivotal moment in Israeli-Palestinian relations. Negotiations are now well under way to implement the September agreement between the Palestine Liberation Organisation and Israel to establish Palestinian self-rule. The future status of Jerusalem is firmly on the secret long-term political agenda, but Israel refuses to discuss Jerusalem publicly until talks begin on the final status of the occupied territories, three years into the period of self-rule. The mayor chosen today will have much influence over the Jerusalem debate.
The Arabs know that if they are to have any chance of winning back some control over Jerusalem, they must halt - or at least slow down - Jewish settlement. Those in favour of voting for Mr Kollek argue that he is less likely to seize new lands and build inside Arab areas than Mr Olmert. If Jerusalem goes to the Likud, the Israeli opposition to Labour's peace deal, which has so far been weak, would have a new focus.
The opponents of an Arab vote have appealed to Yasser Arafat, the chairman of the PLO, to make a clear 'no vote' ruling. Mr Arafat has refused to give one, remaining publicly silent on the question. It is a calculated Arafat fudge, which leaves it open to Arab Jerusalemites today to make up their own minds on the mayor.
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