There are 3 ways to publish variants from your inventory to your eCommerce store. These are:
To publish a variant individually first go to Inventory and click on the product variant.
On the right hand side of each variant field you will see a dropdown box marked with an arrow. Click it.
A dropdown listing all your available e-Commerce integrations will appear.
Click on an eCommerce integration to publish that variant.
Publish in Bulk
The process for bulk publishing is very similar to the process above. Go to the product details page.
However this time, check the box top right hand corner to select all your variants.
You can now publish all your variants to any of your integrated Sales Channels.
Publishing via CSV
If you have too many variants from different products that you need to publish, the best way for you to do so is by updating via csv.
First, go into the Inventory section and click on Bulk Manage > Update Existing Products
Choose the option "No, I need to download a list of my products".
Click on Sales Channels and choose Shopify.
Click "Build Product Sheet" and download the CSV generated. On the CSV, under the columns for the Sales Channels, mark the variants as TRUE to publish.
Reupload the csv file. Your variants have now been published.