Answer
No, ArcObjects SDK 10.1 does not officially support Microsoft Visual Studio 2012. However, it is supported by ArcObjects SDK 10.2.
Desktop application development with ArcObjects SDK 10.1 is supported by the following:
• Microsoft Visual Studio 2010 (C#, VB.NET) Professional, Premium, Ultimate Edition
• Microsoft Visual Studio 2010 Visual Basic Express
• Microsoft Visual Studio 2010 Visual C# Express
ArcGIS 10.1 products that support Visual Studio 2012 for desktop application development are:
• ArcGIS Runtime SDK for WPF 10.1.1 or later
• ArcGIS API for Silverlight 3.1 or later
• ArcGIS API for JavaScript 3.0 or later
Note:
Visual Studio 2010 and 2012 can run side-by-side on the same machine.