c#: static variable in a static method

No.

You can’t have local static variables.

From MSDN:

C# does not support static local variables (variables that are declared in method scope).

And here:

The static modifier can be used with classes, fields, methods, properties, operators, events, and constructors, but it cannot be used with indexers, destructors, or types other than classes.

As you can see, local variables are not mentioned.

You can, however use a static field:

public class MyClass
{
    private static int MyVariable = 10;

    public static void MyMethod()
    {
        MyVariable++;
    }
}