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Is it possible to get the name of the google cloud project configured for the GCP provider? (In general terms: Can I get values from the provider?)

No. You have keep them in your own context / config object.

It should expose the `project` value when you create the Provider yourself. If you are using the default global provider it's properties are available on `gcp.config.*`

<https://github.com/pulumi/pulumi-gcp/blob/master/sdk/dotnet/Config/Config.cs#L561>

Not sure how it would appear in your code (don't know C#, I just went to the same path as the others sdks)