Semiorthogonal decompositions in algebraic geometry

Abstract

In this review we discuss what is known about semiorthogonal decompositions of derived categories of algebraic varieties. We review existing constructions, especially the homological projective duality approach, and discuss some related issues such as categorical resolutions of singularities.

Contribution to the ICM 2014; v2: acknowledgements updated; v3: a reference added

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Cited by in corpus (47)

• Hilbert schemes of lines and conics and automorphism groups of Fano threefolds
• Stability conditions on Kuznetsov components
• Derived categories view on rationality problems
• On linear sections of the spinor tenfold, I
• Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields
• Exceptional collections, and the Neron-Severi lattice for surfaces
• Non-commutative resolutions of quotient singularities
• Conservative descent for semi-orthogonal decompositions
• Moduli spaces on the Kuznetsov component of Fano threefolds of index 2
• A note on the Kuznetsov component of the Veronese double cone
• Topological Strings on Non-Commutative Resolutions
• Cycles, derived categories, and rationality
• The fundamental theorem of homological projective duality via variation of GIT stability
• Non-existence of exceptional collections on twisted flags and categorical representability via noncommutative motives
• Geometry of symmetric determinantal loci
• On genus one fibered Calabi-Yau threefolds with 5-sections
• Moduli spaces of semiorthogonal decompositions in families
• Categorical cones and quadratic homological projective duality
• K3 categories, one-cycles on cubic fourfolds, and the Beauville-Voisin filtration
• Decompositions of moduli spaces of vector bundles and graph potentials
• No full exceptional collections on non-split Brauer--Severi varieties of dimension $\le 3$
• Categorical duality between joins and intersections
• Tilting objects on twisted forms of some relative flag varieties
• Mirror symmetry for perverse schobers from birational geometry
• On nonexistence of semi-orthogonal decompositions in algebraic geometry
• Paracanonical base locus, Albanese morphism, and semi-orthogonal indecomposability of derived categories
• On Clifford double mirrors of toric complete intersections
• Stringy $E$-functions of Pfaffian-Grassmannian double mirrors
• Gluing Approximable Triangulated Categories
• Noncommutative resolutions and intersection cohomology for quotient singularities
• Blowing up linear categories, refinements, and homological projective duality with base locus
• Mirror Symmetry and Projective Geometry of Fourier-Mukai Partners
• Characteristic classes and stability conditions for projective Kleinian orbisurfaces
• The (he)art of gluing
• HPD-invariance of the Tate, Beilinson and Parshin conjectures
• On quantum cohomology of Grassmannians of isotropic lines, unfoldings of $A_n$-singularities, and Lefschetz exceptional collections
• Towards Homological Projective duality for S^2 P^3 and S^2 P^4
• Derived categories of the Cayley plane and the coadjoint Grassmannian of type F
• Fano visitors, Fano dimension and orbifold Fano hosts
• On stringy Euler characteristics of Clifford non-commutative varieties
• Classification of full exceptional collections on smooth toric Fano varieties with Picard rank two
• HPD-invariance of the Tate conjecture
• Exotic t-structure for partial resolutions of the nilpotent cone
• Some remarks on rigid sheaves, helices and exceptional vector bundles on Fano varieties over arbitrary fields
• Equivariant derived category of flat families
• Indecomposability of the bounded derived categories of Brill-Noether varieties
• Note on Motivic Semiorthogonal Decompositions for Elementary Abelian 2-Group Actions