The relative minimal model program for excellent algebraic spaces and analytic spaces in equal characteristic zero

TL;DR

This paper develops a comprehensive relative minimal model program for various classes of algebraic and analytic spaces in equal characteristic zero, extending existing theories and establishing new results in the field.

Contribution

It generalizes the relative minimal model program with scaling to a broad class of spaces, including algebraic, formal, and analytic spaces, using advanced techniques like finite generation and GAGA theorems.

Findings

Established the relative minimal model program with scaling for algebraic and analytic spaces in characteristic zero.

Proved finite generation of relative adjoint rings for these spaces.

Extended the program to certain spaces in positive and mixed characteristic, and to complex analytic spaces.

Abstract

We establish the relative minimal model program with scaling for locally projective morphisms of quasi-excellent algebraic spaces admitting dualizing complexes, quasi-excellent formal schemes admitting dualizing complexes, semianalytic germs of complex analytic spaces, rigid analytic spaces, Berkovich spaces, and adic spaces locally of weakly finite type over a field, all in equal characteristic zero. To do so, we prove finite generation of relative adjoint rings associated to projective morphisms of such spaces using the strategy of Cascini and Lazi\'c and the generalization of the Kawamata-Viehweg vanishing theorem to the scheme setting recently established by the second author. To prove these results uniformly, we prove GAGA theorems for Grothendieck duality and dualizing complexes to reduce to the algebraic case. In addition, we apply our methods to establish the relative minimal model program with scaling for spaces of the form above in dimensions $\le 3$ in positive and mixed characteristic, and to show that one can run the relative minimal model program with scaling for complex analytic spaces without shrinking the base at each step.