Relative Fourier-Mukai transforms for Weierstrass fibrations, abelian schemes and Fano fibrations

TL;DR

This paper investigates the structure of relative Fourier-Mukai transforms across various fibrations, providing complete descriptions for some and establishing conditions for equivalences and finiteness of partners in others.

Contribution

It offers a comprehensive analysis of the group of relative Fourier-Mukai transforms for Weierstrass, Fano, anti-Fano fibrations, and abelian schemes, including new criteria for equivalences and finiteness.

Findings

Complete description of the Fourier-Mukai transform group for Weierstrass and Fano fibrations.

Establishment of isometric isomorphisms between fiber products of abelian schemes and their duals.

Finiteness of the number of Fourier-Mukai partners for abelian schemes over normal bases.

Abstract

We study the group of relative Fourier-Mukai transforms for Weierstrass fibrations, abelian schemes and Fano or anti-Fano fibrations. For Weierstrass and Fano or anti-Fano fibrations we are able to describe this group completely. For abelian schemes over an arbitrary base we prove that if two of them are relative Fourier-Mukai partners then there is an isometric isomorphism between the fibre products of each of them and its dual abelian scheme. If the base is normal and the slope map is surjective we show that these two conditions are equivalent. Moreover in this situation we completely determine the group of relative Fourier-Mukai transforms and we prove that the number of relative Fourier-Mukai partners of a given abelian scheme over a normal base is finite.