What do arrows above notes mean on sheet music for piano?

The arrows you are referring to are called "articulation markings" and they are used to indicate how the notes should be played. Here are a few examples of the most common articulation markings and their meanings:

- Staccato ( ) - Staccato means "detached" and it indicates that the note should be played short and detached from the following notes.

- Tenuto (` ') - Tenuto means "held" and it indicates that the note should be held for its full value.

- Accent (`>`) - Accent means "emphasized" and it indicates that the note should be played louder than the surrounding notes.

- Marcato (_m_) - Marcato is a combination of staccato and tenuto, and it indicates that the note should be played with a strong emphasis and then detached.

There are many other articulation markings that can be used in sheet music, but these are a few of the most common. By following these markings, you can play the music as the composer intended.