A counterexample of the birational Torelli problem via Fourier--Mukai transforms

2009/04/02 by Hokuto Uehara, Uehara, Hokuto
Mathematics · #Algebraic and Geometric Analysis #Geometric Analysis and Curvature Flows #Mathematics and Applications #math.AG #msc:14C34 #msc:14J27 #msc:14J30 #msc:18E30

paper · pdf · doi:10.48550/arxiv.0904.0303

21 pages. Theorem 1.1 becomes weaker, since the proof of Lemma 2.5 in v1 is incorrect. Title has been changed. To appear in Journal of Algebraic Geometry

arxiv created 2009/11/13 · arxiv updated 2009/12/01

Abstract

We study the Fourier--Mukai numbers of rational elliptic surfaces. As its application, we give an example of a pair of minimal 3-folds with Kodaira dimensions 1, h¹(\mc O)=h²(\mc O)=0 such that they are mutually derived equivalent, deformation equivalent, but not birationally equivalent. It also supplies a counterexample of the birational Torelli problem.

Related

• Infinitesimal Derived Torelli Theorem for K3 surfaces
• Fourier-Mukai transform on abelian surfaces
• The Euclid-Fourier-Mukai algorithm for elliptic surfaces
• An example of Fourier--Mukai partners of minimal elliptic surfaces
• Perverse coherent sheaves and Fourier-Mukai transforms on surfaces