According to the PMBOK Guide, 7th edition, one of the principles of project management is to engage stakeholders proactively. This means that the project manager should communicate effectively with the project sponsor and other key stakeholders about the project status, issues, risks, and changes. When the project manager observes that the project is likely to miss a deadline due to a lack of resources, the best course of action is to escalate the issue to the sponsor and debrief them about the situation. This way, the project manager can seek the sponsor's support and guidance on how to resolve the resource problem and mitigate the impact on the project schedule. The sponsor may also have the authority and influence to allocate more resources or negotiate a revised deadline with the customer or other stakeholders. By escalating the issue to the sponsor, the project manager demonstrates transparency, accountability, and professionalism. References:
PMBOK Guide, 7th edition, page 9, Principle 3: Engage Stakeholders Proactively; page 32, Section 2.3.2.1, Project Sponsor; page 36, Section 2.4.2.1, Project Manager.