That won't work for several reasons.
- That LED has a forward voltage VF of 3.1V. This means you need to provide 3.1V to it. A single AA cell will only supply 1.5V. There are various ways around this problem, including using three 1.5V cells in series, and a current regulator.
- That LED requires 3A (three amps) of forward current to run at maximum brightness (10W; ten watts). An AA cell - even a C cell - will struggle to provide 3A for more than a few minutes; even an alkaline one will have trouble. And the terminal voltage, at a 3A load, will drop significantly - probably below 1V (that's just a "gut feeling" estimate; I haven't checked any data sheets to confirm this).
- High-power LEDs are best driven with a regulated current. This means circuitry in between your power source and the LED to provide a steady, constant current, regardless of variations in temperature, input voltage, or any other parameters.
You need to investigate a current regulator to drive the LED with the specified current, if you want to run it at its maximum rated brightness. These come in several types, but probably your best option would be a buck switching converter with regulated output current. These run from a higher voltage (between 10V and 30V would be appropriate) and step the voltage down, while boosting the current, and regulating the current. Google the keywords in this paragraph. You can probably get them through eBay, but that doesn't mean that you should
You also need to look for batteries that can supply 10W for a useful time period. Actually, because of the loss of power in the current regulator, the battery will need to supply at least 12W, maybe as much as 15W. The battery voltage needs to be matched to the input voltage requirements of the current regulator, as mentioned above. I imagine you will want to use a Nickel Metal Hydride (NiMH), Lithium Ion (Li-ion) or Lithium Polymer (Li-Pol) rechargeable battery.
P.S. Welcome to Electronics Point