Noether inequality for irregular threefolds of general type

Abstract

Let $X$ be a smooth irregular $3$-fold of general type over $\mathbb{C}$. We prove that the optimal Noether inequality $$\mathrm{vol}(X)\ge \frac{4}{3}{p}_g(X)$$ holds if $p_g(X)\ge 16$ or if $X$ has a Gorenstein minimal model. Moreover, when $X$ attains the equality and $p_g(X)\ge 16$, its canonical model can be explicitly described.

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