A Survey of Derived Categories in Algebraic Geometry David Favero - PowerPoint PPT Presentation

A Survey of Derived Categories in Algebraic Geometry David Favero favero@ualberta.ca University of Alberta December 2017 University of California, Santa Barbara

Consider a surface triangulation S (with consistently oriented triangles). V = { vertices } E = { edges } F = { faces } We get a sequence of vector spaces called a complex d − 2 d − 1 R F R E R V d − 2 ( f ) = edge 1 ( f ) − edge 2 ( f ) + edge 3 ( f ) d − 1 ( e ) = head( e ) − tail( e ) .

d − 2 d − 1 R F R E R V d − 2 ( f ) = edge 1 ( f ) − edge 2 ( f ) + edge 3 ( f ) d − 1 ( e ) = head( e ) − tail( e ) . H 0 ( S ) := R V / im d − 1 = R H 1 ( S ) := ker d − 1 / im d − 2 = R 2 H 2 ( S ) := ker d − 2 = R

d − 2 d − 1 R F R E R V H 0 (Bunny) = R H 1 (Bunny) = 0 H 2 (Bunny) = R

Complexes Let R be a ring. A complex A • of R -modules is a sequence of R -modules ... d i − 2 → A i − 1 d i − 1 d i → A i +1 d i +1 → A i − − − − − − − − − → ... such that d 2 = 0 i.e. d i − 1 ◦ d i = 0 ∀ i . The i th cohomology of a complex A • of R -modules is defined as the R -module H i ( A • ) := ker d i im d i − 1

Complexes Let R be a ring. A complex A • of R -modules is a sequence of R -modules ... d i − 2 → A i − 1 d i − 1 d i → A i +1 d i +1 → A i − − − − − − − − − → ... such that d 2 = 0 i.e. d i − 1 ◦ d i = 0 ∀ i . The i th cohomology of a complex A • of R -modules is defined as the R -module H i ( A • ) := ker d i im d i − 1

Quasi-isomorphisms A morphism of complexes f : A • → B • is a commutative diagram d i − 1 d i +1 d i d i A A i − 1 A A i A A i +1 A ... ... f i − 1 f i f i +1 d i − 1 d i +1 d i d i B B i − 1 B B i B B i +1 B ... ... i.e. fd = df or more precisely f i ◦ d i − 1 = d i − 1 ◦ f i − 1 . A B Any morphism induces a map f ∗ : H i ( A • ) → H i ( B • ) . We say that f is a quasi-isomorphism if f ∗ is an isomorphism ∀ i .

Quasi-isomorphisms A morphism of complexes f : A • → B • is a commutative diagram d i − 1 d i +1 d i d i A A i − 1 A A i A A i +1 A ... ... f i − 1 f i f i +1 d i − 1 d i +1 d i d i B B i − 1 B B i B B i +1 B ... ... i.e. fd = df or more precisely f i ◦ d i − 1 = d i − 1 ◦ f i − 1 . A B Any morphism induces a map f ∗ : H i ( A • ) → H i ( B • ) . We say that f is a quasi-isomorphism if f ∗ is an isomorphism ∀ i .

The Derived Category ◮ The derived category is roughly the category of complexes where all quasi-isomorphisms have been inverted i.e. are isomorphisms. ◮ Objects of D( R ) are still complexes of R modules ◮ Morphisms f : A • → B • are equivalences classes of diagrams C • g f ∼ A • B • where f is a quasi-isomorphism and g is any morphism of complexes. ◮ The equivalence relation is a bit intricate and I will skip it.

Affine Algebraic Varieties ◮ Let f 1 , ..., f t ∈ C [ x 1 , ..., x n ] be polynomials. We define the zero set to be X := { ( v 1 , ..., v n ) ∈ C n | f i ( v 1 , ..., v n ) = 0 ∀ i } ◮ X is called an affine algebraic variety. The ring of regular functions g : X → C is isomorphic to the quotient ring � R = C [ x 1 , ..., x n ] / � f 1 , ..., f t � . ◮ Hence, to an affine algebraic variety X we can associate a derived category D ( X ) := D ( R ).

Affine Algebraic Varieties ◮ Let f 1 , ..., f t ∈ C [ x 1 , ..., x n ] be polynomials. We define the zero set to be X := { ( v 1 , ..., v n ) ∈ C n | f i ( v 1 , ..., v n ) = 0 ∀ i } ◮ X is called an affine algebraic variety. The ring of regular functions g : X → C is isomorphic to the quotient ring � R = C [ x 1 , ..., x n ] / � f 1 , ..., f t � . ◮ Hence, to an affine algebraic variety X we can associate a derived category D ( X ) := D ( R ).

Affine Algebraic Varieties ◮ Let f 1 , ..., f t ∈ C [ x 1 , ..., x n ] be polynomials. We define the zero set to be X := { ( v 1 , ..., v n ) ∈ C n | f i ( v 1 , ..., v n ) = 0 ∀ i } ◮ X is called an affine algebraic variety. The ring of regular functions g : X → C is isomorphic to the quotient ring � R = C [ x 1 , ..., x n ] / � f 1 , ..., f t � . ◮ Hence, to an affine algebraic variety X we can associate a derived category D ( X ) := D ( R ).

Projective Algebraic Varieties ◮ Projective n -space is the set of lines in an n + 1 dimensional vector space P n := C n +1 \ 0 / ∼ where the equivalence relation is given by scaling the vectors v ∼ λ v ∀ λ ∈ C ∗ . ◮ Let f 1 , ..., f t ∈ C [ x 0 , ..., x n ] be homogeneous polynomials. X := { ( v 0 : ... : v n ) ∈ P n | f i ( v 1 , ..., v n ) = 0 ∀ i } ⊆ P n ◮ X is called an projective algebraic variety. The coordinate ring of X is the N -graded ring, � R = C [ x 0 , ..., x n ] / � f 1 , ..., f t � . ◮ Using graded- R -modules, we can almost define the D ( X ) , derived category of X , the same way except that morphisms are a bit different. We skip this detail.

Projective Algebraic Varieties ◮ Projective n -space is the set of lines in an n + 1 dimensional vector space P n := C n +1 \ 0 / ∼ where the equivalence relation is given by scaling the vectors v ∼ λ v ∀ λ ∈ C ∗ . ◮ Let f 1 , ..., f t ∈ C [ x 0 , ..., x n ] be homogeneous polynomials. X := { ( v 0 : ... : v n ) ∈ P n | f i ( v 1 , ..., v n ) = 0 ∀ i } ⊆ P n ◮ X is called an projective algebraic variety. The coordinate ring of X is the N -graded ring, � R = C [ x 0 , ..., x n ] / � f 1 , ..., f t � . ◮ Using graded- R -modules, we can almost define the D ( X ) , derived category of X , the same way except that morphisms are a bit different. We skip this detail.

Projective Algebraic Varieties ◮ Projective n -space is the set of lines in an n + 1 dimensional vector space P n := C n +1 \ 0 / ∼ where the equivalence relation is given by scaling the vectors v ∼ λ v ∀ λ ∈ C ∗ . ◮ Let f 1 , ..., f t ∈ C [ x 0 , ..., x n ] be homogeneous polynomials. X := { ( v 0 : ... : v n ) ∈ P n | f i ( v 1 , ..., v n ) = 0 ∀ i } ⊆ P n ◮ X is called an projective algebraic variety. The coordinate ring of X is the N -graded ring, � R = C [ x 0 , ..., x n ] / � f 1 , ..., f t � . ◮ Using graded- R -modules, we can almost define the D ( X ) , derived category of X , the same way except that morphisms are a bit different. We skip this detail.

Derived Categories ◮ To summarize, for an (affine, projective, or actually any) algebraic variety X , we can associate a derived category D ( X ). ◮ There are 3 major conjectures I wish to discuss today concerning D ( X ) ◮ Two are due to Kawamata and one is due to Kontsevich.

Derived Categories ◮ To summarize, for an (affine, projective, or actually any) algebraic variety X , we can associate a derived category D ( X ). ◮ There are 3 major conjectures I wish to discuss today concerning D ( X ) ◮ Two are due to Kawamata and one is due to Kontsevich.

Derived Categories ◮ To summarize, for an (affine, projective, or actually any) algebraic variety X , we can associate a derived category D ( X ). ◮ There are 3 major conjectures I wish to discuss today concerning D ( X ) ◮ Two are due to Kawamata and one is due to Kontsevich.

Kawamata’s First Conjecture Conjecture (Kawamata ’02) Suppose X is smooth and projective. The following set is finite: { Y | D ( Y ) = D ( X ) } . ◮ True in dimension 1 (easy) ◮ True in dimension 2 (Orlov ’96, Bridgeland-Macocia ’01, Kawamata ’02 ) ◮ True for varieties with positive or negative curvature (Bondal-Orlov ’97) ◮ True for complex n-dimensional tori (Huybrechts—Nieper–Wisskirchen ’11, Favero 1 ’12) ◮ False in dimension 3 (Lesieutre ’13) 1 Reconstruction and Finiteness Results for Fourier-Mukai Partners , Advances in Mathematics , V. 229, I. 1, pgs 1955-1971, 2012.

Kawamata’s First Conjecture Conjecture (Kawamata ’02) Suppose X is smooth and projective. The following set is finite: { Y | D ( Y ) = D ( X ) } . ◮ True in dimension 1 (easy) ◮ True in dimension 2 (Orlov ’96, Bridgeland-Macocia ’01, Kawamata ’02 ) ◮ True for varieties with positive or negative curvature (Bondal-Orlov ’97) ◮ True for complex n-dimensional tori (Huybrechts—Nieper–Wisskirchen ’11, Favero 1 ’12) ◮ False in dimension 3 (Lesieutre ’13) 1 Reconstruction and Finiteness Results for Fourier-Mukai Partners , Advances in Mathematics , V. 229, I. 1, pgs 1955-1971, 2012.

Kawamata’s Second Conjecture Setup: Birationality ◮ If X is an algebraic variety, choose any open affine subset U ⊆ X and consider its ring of regular functions R . Let Q ( R ) := { a b | a , b ∈ R } / ∼ where a b ∼ c d if ad = bc . ◮ Q ( R ) is called the fraction field of X . It is independent of the choice of U . ◮ Two algebraic varieties X , Y are called birational if they have the same fraction field. Equivalently, they are isomorphic on an open (dense) subset.

Kawamata’s Second Conjecture Setup: Birationality ◮ If X is an algebraic variety, choose any open affine subset U ⊆ X and consider its ring of regular functions R . Let Q ( R ) := { a b | a , b ∈ R } / ∼ where a b ∼ c d if ad = bc . ◮ Q ( R ) is called the fraction field of X . It is independent of the choice of U . ◮ Two algebraic varieties X , Y are called birational if they have the same fraction field. Equivalently, they are isomorphic on an open (dense) subset.