Global Sections and Birational Geometry of Calabi-Yau Type Varieties

We investigate aspects of the relationship between global sections of divisors and birational geometry. We show that for varieties $X$ of klt Calabi-Yau type, global sections are governed by the same birational data that control the $D$-minimal model program ($D$-MMP). As a consequence, global sections of arbitrary big divisors can be systematically reduced to Euler characteristics $\chi (Y,{\mathcal{O}}_Y(D))$ on suitable birational models $Y$ using vanishing theorems, extending classical positivity-based methods beyond the movable cone. The reduction proceeds by removing fixed divisorial components, which do not contribute to the space of global sections, either directly by subtraction or birationally by contraction to a suitable $D$-minimal model. We obtain piecewise quasipolynomial formulae for global sections on the big cone, whose domains are the Mori chambers of the $D$-MMP. Restricting to the subclass of Fano type varieties extends such formulae to the entire effective cone. These formulae can be constructed from just the small birational class of $X$, yet encode properties of all birational contractions of $X$. Alongside the computation of global sections from birational geometry, we study the inverse process, wherein finite computations of $h^0(X,{\mathcal{O}}_X(D))$ determine aspects of the chamber decomposition and the associated birational models.