This article from the UK Defence National Association (implacably opposed to defence cuts) has claimed that if Argentina was determined to invade then the UK could not defend the islands:
It warned that the islands' coastline could not be defended from an invasion and that Argentina could use almost all of its armed forces to invade at a time when the British forces consisted of "just four Typhoons, a Type-45 Destroyer, and Rapier short range missiles around Mount Pleasant airfield".OK, so let's assume that Argentina manages to get the element of surprise, lands a planeload of commandos at Mount Pleasant airfield (by pretending to be a commercial flight from Uruguay or something), puts a serious amphibious force to sea and devotes all its active airforce to supporting the invasion. Let's assume that there is no intel warning of the attack, so the first the Falkland Islands military contingent knows about it is when the commando force spills out of a C-130 on Mount Pleasant airstrip and the radar simultaneously picks up the inbound 34 Skyhawks and 15 Mirages comprising the entire long-range offensive capability of the Fuerza Aerea Argentina. Would the attack succeed?
The Falkland Islands military force could scramble one (QRF) Typhoon before the commandos made things too interesting to take the other aircraft out of their hangars. That Typhoon could at best take out 8 Skyhawks with a combination of missiles and guns before they could reach Mount Pleasant, but also reduce the fuel of other Skyhawks by locking them up with radar and forcing them to take evasive action without knowing if a missile is on the way. Then, assuming their mission is to target Mount Pleasant airfield and eliminate concentrations of forces, the Skyhawks have to avoid Rapier 2000 batteries specifically placed to expect attack by low-flying Skyhawks. Again, 1980s planes face missiles developed 10-15 years later than them. If 20 aircraft approach the target, another 5-10 will take hits from Rapier and machine gun fire so only 10-15 will be able to deliver their (dumb) bombs.
The problem the Argentine aircraft now have is the Type 45 destroyer alerted to the invasion. Generously assuming that the inbound strike aircraft have somehow managed to slip by it (say, by flying very low and thus reducing their already-low maximum range), any survivors will now be exposed to an alerted and very pissed-off Type 45 firing salvoes of modern Mach 3+ Aster missiles at 1980s design aircraft. If 3 or 4 aircraft manage to return from the strike, they'll be doing well.
Let's look at the commandos. Assuming they use their Special Operations Forces Group they should be able to muster 64 armed commandos (max capacity of a C-130) for the attack on Mount Pleasant. The commandos have the advantage of surprise and being elite forces; against that, even if only 1/3 of the 1200 British soldiers on the islands are in the area, those are unfavourable odds of 7:1 against the attacking force. Bear in mind also that most British Army units have spent time in Afghanistan and Iraq, and are very familiar with being shot at and conducting small-unit action under live fire - an advantage not to be underestimated. At best, the commandos will be able to destroy a couple of Typhoons and aircraft facilities before being surrounded and immolated or forced to surrender. An airfield is a very large area, and takes many hundreds of troops to secure.
So what of the amphibious force? Let's be generous and assume that the commandos have knocked out Mount Pleasant for at least a couple of days, so the amphibs can sail the 400 miles and land unopposed by aircraft. We also assume that no Astute or Trafalgar sub is in the area launching Harpoon and Tomahawk missiles at them. They will have no effective air cover, and the British will have reasonable advance warning of their approach, so they will either have to mount a major landing against dug-in and pre-warned opposition near a port to secure it (good luck with that), or land light forces away from military concentrations and then hike over land to their target. They will not be able to transport over any significant helicopter force, so will be restricted to what they can carry on their backs and in light vehicles.
All the UK forces need to do is secure Mount Pleasant, fix any damage to the runway, then wait a few days for a squadron of Typhoons to deploy via Ascension. They can then proceed to bomb the living crap out of the landed Argentine military, deny them air or sea resupply, and wait for surrender.
The 1982 invasion of the Falklands pitted 600+ Argentine troops against 60 UK Marines and 10 sailors (8:1 odds), and was by no means a cakewalk - one of the senior Argentine Marine commanders was killed assaulting Government House. One of the deciding factors was the presence of Amtracs (armoured amphibious vehicles). I don't see either overwhelming odds or east amphibious assault being realistic this time around. The Argentines are not going to assault the Falklands unless they are crazy - and if you want to stop them, just bolster the AAA defences and ground security of Mount Pleasant.
The maxim that generals always fight the last war is a useful one, but in this case it's almost literally true - the Argentine forces are essentially unchanged from the end of the 1982 conflict due to an economy that is constantly circling the drain, and meanwhile the British have had 30 years to evolve their military in general and the Falklands defences in particular against a very specific threat. The only way Argentina could muddy the waters is to bring the Venezuelans into the conflict or something.