Derived categories of complete intersections, crepant categorical resolutions and homological projective duality
File(s)
Author(s)
Turcinovic, Zak
Type
Thesis
Abstract
Homological projective duality describes the bounded derived category D(X) of coherent sheaves on X when the latter is a complete intersection of hypersurfaces of the same degree. We give a procedure for constructing new HP duals from old and move beyond HPD by studying complete intersections cut out by hypersurfaces of varying degrees.
First, starting with a HP dual pair X, Y and smooth orthogonal linear sections X_{L^{\perp}} , Y_L, we prove that the blowup of X in X_{L^{\perp}} is naturally HP dual to Y_L. The result also holds true when Y is a noncommutative variety or just a category. We extend the result to the case where the base locus X_{L^{\perp}} is a multiple of a smooth variety and the universal hyperplane has rational singularities; here the HP dual is a weakly crepant categorical resolution of singularities of Y_L.
Second, for a complete intersection X ⊂ P^n, we show that the nontrivial part of D(X) is a weakly crepant categorical resolutions of a certain fractional Calabi–Yau category. This turns out to give examples of the previous story where, starting with a noncommutative HP dual, the blowing up process nevertheless gives geometric HP duals. This is also a generalisation of Kuznetsov’s story for nodal cubic fourfolds [32] and in particular we obtain new examples of Calabi–Yau categories that admit a geometric crepant resolution. Furthermore, we get a characterisation of complete intersections of Calabi–Yau Hodge type in terms of their derived category and infinitely many examples of rational varieties that are categorically representable in codimension 2.
First, starting with a HP dual pair X, Y and smooth orthogonal linear sections X_{L^{\perp}} , Y_L, we prove that the blowup of X in X_{L^{\perp}} is naturally HP dual to Y_L. The result also holds true when Y is a noncommutative variety or just a category. We extend the result to the case where the base locus X_{L^{\perp}} is a multiple of a smooth variety and the universal hyperplane has rational singularities; here the HP dual is a weakly crepant categorical resolution of singularities of Y_L.
Second, for a complete intersection X ⊂ P^n, we show that the nontrivial part of D(X) is a weakly crepant categorical resolutions of a certain fractional Calabi–Yau category. This turns out to give examples of the previous story where, starting with a noncommutative HP dual, the blowing up process nevertheless gives geometric HP duals. This is also a generalisation of Kuznetsov’s story for nodal cubic fourfolds [32] and in particular we obtain new examples of Calabi–Yau categories that admit a geometric crepant resolution. Furthermore, we get a characterisation of complete intersections of Calabi–Yau Hodge type in terms of their derived category and infinitely many examples of rational varieties that are categorically representable in codimension 2.
Version
Open Access
Date Issued
2018-10
Date Awarded
2019-01
Copyright Statement
Creative Commons Attribution NonCommercial NoDerivatives Licence
Advisor
Thomas, Richard
Sponsor
Engineering and Physical Sciences Research Council
Grant Number
EP/L015234/1
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)