ok, so I had the same problem as many people on here with my new Arnova 7G3. I could not get calendar to sync with my google calendar account. Contact synched fine but not calendar.
Anyway, here's what is needed to fix it. Note: This may work on other tablets running ICS since the relevant files come from google's own AOSP ICS base.
Assumption: You have already created your google account (required if you're running Play Store) and have proved access is available to the google market.

Under 'Settings>Accounts&Sync>whatever your google account is',  you will see Sync com.android.calendar, this is the problem. Under these settings it should read 'Sync Calendar' NOT 'Sync com.android.calendar', clearly ARNOVA have compiled this firmware with a bad link.

1. Obtain the AOSP ICS proper files; Calendar.apk, CalendarProvider.apk and GoogleCalendarSyncAdapter.apk.
2. Copy somewhere on the SDCard; e.g. /sdcard/Calendar.apk etc..
2. Install Root Explorer (or any other file manager that allows you to write to the system partition and also change permissions).
3. Open Root Explorer.
4. Go into '/system/app' directory.
5. Rename the 3 above file by long pressing on each until you see the options, one of the options will be to rename. I suggest you give them the .org extension, so, for example, Calendar.apk would become Calendar.org. NOTE: before you can rename anything in the /system/app directory you need to press 'Mount R/W' tab at the top. When you press this it will change to 'Mount R/O', at this stage you have write access to the system partition. Be careful here cause overwriting the wrong apps can render them useless.
6. Go up the directory structure to your /sdcard directory, select the 3 ICS AOSP files you copied there (note, your do a multi-select in root explorer by pressing the options tab at the bottom) and copy these into your /system/app directory. Remember to press the 'Mount R/W' to allow writing into this directory.
7. At this stage in /system/app directory you should have the original 3 files with the .org extension and these new files you've just copied with the .apk extensions.
8. While still in the /system/app directory long press on each of these 3 new files and set your permissions the same as your other files. So first long press, for example, Contacts.apk to see what the permissions on that are like and then your set your 3 files the same as that.
9. Exit root explorer.
10.Reboot (or power off and power on) your tablet.
11.Once rebooted go back to 'Settings>Accounts&Sync>whatever your google account is', and now it should read 'Syn Calendar' and your should be able to sync.
12. If all is well you can now delete the .org files in the /system/app, but i would recommend you leave them just in case.

I've attached the 3 ICS AOSP files (stated above) as a zip file, you'll need to search for root explorer yourself.