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Abstract
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For varieties of general type, it is natural to study the distribution of birational invariants and relations between invariants. We are interested in the relation between two fundamental birational invariants: the geometric genus and the canonical volume. For a minimal projective surface S, M. Noether proved that $K_S^2\geq 2p_g(S)-4,$ which is known as the Noether inequality. It is thus natural and important to study the higher dimensional analogue. In this talk, we will talk about our recent work on the Noether inequality for projective 3-folds. We will show that the inequality $\text{vol}(X)\geq \frac{4}{3}p_g(X)-{\frac{10}{3}}$ holds for all projective 3-folds X of general type with either $p_g(X)\leq 4$ or $p_g(X)\geq 27$, where $p_g(X)$ is the geometric genus and $\text{vol}(X)$ is the canonical volume. This inequality is optimal due to known examples found by M. Kobayashi in 1992. This proves that the optimal Noether inequality holds for all but finitely many families of projective 3-folds (up to deformation and birational equivalence). I will briefly recall the history on this problem and give some idea of the proof. This is a joint work with Jungkai A. Chen and Meng Chen.
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