TLDR: Basically with what you're asking, only ARP broadcasts, unless you subnet your wireless from your wired network.
Overall it depends. First, your computer on the ethernet may occasionally send out broadcasts to the whole network. For example, if it needs the MAC address of a machine on the local network, it will send out an ARP broadcast to all machines on the local network, including the attacker's. This doesn't tell the attacker much, however, other than that there is another computer on the network, and he might possibly deduce it's connected by ethernet since he can't see any other Wi-Fi traffic.
However, at this point he can do what is called ARP cache poisoning. His computer could respond to the broadcast sent by your computer looking for the machine on the local network, claiming to be the machine it's looking for. Then your computer will continue to send it data directly, because it thinks it's talking to the right host.
Of course, the attacker would be found out pretty quickly if his computer wasn't set up to reply correctly to the traffic you are sending it. It helps for the attacker to anticipate what kind of connection your computer will be trying to make. And if, say, your computer is trying to open an RDP connection to the attacker's computer, that's not very useful for the attacker anyway. The real danger of ARP cache poisoning is if the host your computer was trying to initiate a connection to was its default gateway to the Internet. The attacker could just forward all your traffic to the real default gateway while acting as a middle-man, sniffing all your traffic.
Fortunately, if the scenario you are referring to is a home Wi-fi network, the default gateway your computer was looking for was the Wi-fi router itself, and though I am not an expert on Wi-fi routers I would assume the Wi-fi router would be smart enough to not forward an ARP broadcast requesting its own MAC address. Plus the attacker would have to beat the router at replying to the broadcast, which seems unlikely, but I suppose could be possible if you had like the worst Wi-fi router ever made.
Anyway, once your computer has the correct MAC address for its default Internet gateway or whatever MAC address it was looking for, all communications will be sent to that MAC address directly. If you were connecting to another computer connected to Wi-fi, the attacker can still see that traffic though.
This explanation also assumes your Wi-fi router is otherwise secured from the attacker controlling it directly. If you, say, left the router admin password at default or something then he might be able to change the settings to have the router forward him all your traffic, or overwrite the router's firmware with one he supplies to have the router forward him all your traffic.
Finally, if you're really worried about these ARP broadcasts coming from your computer, however infrequent and mostly useless they are unless they are for the default gateway which they shouldn't be, then on a nicer router you might be able to subnet the local network to have the Wi-fi be one subnet and the ethernet be another. Then the router shouldn't even forward ARP requests across Wi-fi.