Abstract: Recent work of Kento Fujita, Yuji Odaka and Chen Jiang shows that among K-semistable Fano manifolds, the projective space can be characterized in terms of either the alpha invariant or the volume. In this talk, I will generalize the definition of alpha invariant to arbitrary codimension, and show that we can characterize projective spaces among all K-semistable Fano manifolds in terms of higher codimensional alpha invariants. This result also demonstrates the relation between alpha invariants and volumes in the characterization problem of projective spaces among K-semistable Fano manifolds.

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