Ok, here goes.
I cannot guarantee this is correct, but I will give you the process we use in BC, Canada. The process is like a decision tree that causes different actions depending on the amount of damage and if it is an insurable/not insurable. From your description, I will just describe the most likely events.
So you need to get your HOA bylaws and HOA insurance. Your HOA should carry insurance for this and you need to determine if the damage goes above the HOA's deductible. If it is, then the HOA manager should be sending out a restoration company to mitigate the damage. /Tarp/etc.
1) the HOA manager should send the HOA's insurance adjuster to confirm the damage was caused by an insurable event.
2) They should also send a restoration company to do an estimate of the repair cost to confirm it is above the deductible.
If It is over deductible and it is an insurable event, then the HOA's insurance will get involved. If you do not have a copy of it, ask your property manager for a copy of the HOA insurance and the HOA bylaws. The HOA will usually be responsible for the full repairs Except that they will send you a bill for the deductible amount. HOA deductible could be anywhere from 5-50k.
3) You should review your own personal Condo insurance to make sure you have HOA deductible insurance to cover that deductible amount. Your personal insurance should cover the HOA's deductible and you in turn will only be responsible for your personal insurance deductible amount.
Let us know what the damage estimate is, your HOA deductible, and if the adjuster confirms it is an insurable event.