The history of holomorphic Morse inequalities was initiated from the seminal work of

J.-P. Demailly, which was influenced by the Witten’s analytic proof of classical Morse inequalities and Siu’s solution of Grauert-Riemenschneider conjecture on Moishezon manifolds. These inequalities provide a flexible way to produce holomorphic sections of high tensor powers of line bundle under mild positivity assumption and have various generalizations in complex geometry.

We will give an introduction on this subject and our results jointly with Bingxiao Liu and George Marinescu in arXiv:2506.00879. We establish a general Nakano–Griffiths inequality with boundary conditions and apply it to derive holomorphic Morse inequalities for domains satisfying analytic convexity assumptions. As applications, we obtain a criterion for Moishezon q-concave manifolds and extension theorems for q-concave domains.