QAMISHLI, Syria (Reuters) - Kurdish-led authorities held local elections on Friday in areas they control in northern Syria, pushing ahead with autonomy plans opposed by both the government of President Bashar al-Assad and by Turkey. Kurdish forces and their political allies now hold the largest part of Syria outside the grip of Assad s government. They have captured vast territory from Islamic State with the support of U.S. arms, jets and ground advisers, although Washington opposes their autonomy drive. Kurdish leaders say their goal is to establish self-rule within Syria, not secession. But their influence has infuriated Ankara, which considers the Kurdish YPG militia to be an offshoot of the banned Kurdistan Workers  Party (PKK) that has run a decades-long insurgency in Turkey. Assad has vowed to recover every inch of the country, as his territorial grip expanded rapidly over the past two years with help from Russia and Iran. Damascus has more forcefully asserted its claim to territory held by Kurdish-led forces in recent months. The head of Syria s delegation to U.N.-backed peace talks in Geneva, Bashar al-Ja afari, rejected the election, repudiating  any unilateral act that happens without coordination  with Damascus.   Since Syria s conflict began more than six years ago, the dominant Kurdish parties have been left out of international diplomacy in line with Turkish wishes. They were excluded again from U.N.-led peace talks which reconvened in Geneva this week.  Hadiya Yousef, a senior Kurdish politician, said the Kurdish-led administration would not be bound by decisions taken in its absence.  We are not present in these meetings, and therefore we are developing the solution on the ground,  Yousef told Reuters. Peace talks would not  arrive at solutions  so long as they do not involve those running 30 percent of the country, she added. Voters are picking from close to 6,000 candidates for town and city councils on Friday, the second part of a three-stage process that will culminate in electing a parliament early next year. They chose representatives for smaller-scale district councils in September.  Everyone should take part (in the election) because this is the fate of the entire region,  said Sheikhmous Qamishlo, a 65-year-old Syrian Kurd at a polling station in Qamishli.  This is a new experience, we wish it success,  he said, and described casting his vote as a  national duty .    The election was being monitored by a small group of politicians from other countries in the Middle East, Yousef said, including a member of the Kurdistan Democratic Party which runs the autonomous Kurdish region in neighboring Iraq. Yousef said the Iraqi Kurdish official s presence was  a kind of recognition  of the Syrian Kurdish political project. The Iraqi Kurdish authorities, whose own plans for independence were met with a swift backlash from states in the region in the past two months, have previously been hostile to the Syrian Kurdish parties. 