The K3 category of cubic fourfolds and Gushel-Mukai fourfolds Laura - - PowerPoint PPT Presentation

The K3 category of cubic fourfolds and Gushel-Mukai fourfolds Laura Pertusi

Università di Roma Tor Vergata December 20, 2019

Introduction Derived category K3 categories

Complex Algebraic Geometry

Aim: to study smooth projective varieties over C. Defined as zero loci of systems of homogeneous polynomial equations with coefficients in C. Example (Cubic fourfolds) In P5

C, given f ∈ C[x0, x1, x2, x3, x4, x5] homogeneous of degree 3

with (∂f /∂xi) = 0, define Y := Z(f ) = {x ∈ P5

C : f (x) = 0}.

For example, consider x3

0 + x3 1 + · · · + x3 5 = 0 in P5 C.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Complex Algebraic Geometry

Aim: to study smooth projective varieties over C. Defined as zero loci of systems of homogeneous polynomial equations with coefficients in C. Example (Cubic fourfolds) In P5

C, given f ∈ C[x0, x1, x2, x3, x4, x5] homogeneous of degree 3

with (∂f /∂xi) = 0, define Y := Z(f ) = {x ∈ P5

C : f (x) = 0}.

For example, consider x3

0 + x3 1 + · · · + x3 5 = 0 in P5 C.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Complex Algebraic Geometry

Aim: to study smooth projective varieties over C. Defined as zero loci of systems of homogeneous polynomial equations with coefficients in C. Example (Cubic fourfolds) In P5

C, given f ∈ C[x0, x1, x2, x3, x4, x5] homogeneous of degree 3

with (∂f /∂xi) = 0, define Y := Z(f ) = {x ∈ P5

C : f (x) = 0}.

For example, consider x3

0 + x3 1 + · · · + x3 5 = 0 in P5 C.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Complex Algebraic Geometry

Our tool: vector bundles ↔ locally free sheaves. Example (Trivial vector bundle) Given a smooth complex projective variety X, consider X × Cr → X. If r = 1, this is the structure sheaf OX.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Complex Algebraic Geometry

Our tool: vector bundles ↔ locally free sheaves. Example (Trivial vector bundle) Given a smooth complex projective variety X, consider X × Cr → X. If r = 1, this is the structure sheaf OX.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Complex Algebraic Geometry

Our tool: vector bundles ↔ locally free sheaves. Example (Trivial vector bundle) Given a smooth complex projective variety X, consider X × Cr → X. If r = 1, this is the structure sheaf OX.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Complex Algebraic Geometry

Our tool: vector bundles ↔ locally free sheaves. Example (Trivial vector bundle) Given a smooth complex projective variety X, consider X × Cr → X. If r = 1, this is the structure sheaf OX.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Motivations

Vector bundles are classified by using numerical invariants, like the rank or the degree. Theorem (Grothendieck) Over P1

C vector bundles are direct sums of line bundles.

Consider vector bundles with some fixed numerical invariants as points of an algebraic space. Problem Is there a moduli space parametrizing vector bundles on X with fixed numerical invariants having the structure of a smooth projective variety? A notion of stability is required!

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Motivations

Vector bundles are classified by using numerical invariants, like the rank or the degree. Theorem (Grothendieck) Over P1

C vector bundles are direct sums of line bundles.

Consider vector bundles with some fixed numerical invariants as points of an algebraic space. Problem Is there a moduli space parametrizing vector bundles on X with fixed numerical invariants having the structure of a smooth projective variety? A notion of stability is required!

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Motivations

Vector bundles are classified by using numerical invariants, like the rank or the degree. Theorem (Grothendieck) Over P1

C vector bundles are direct sums of line bundles.

Consider vector bundles with some fixed numerical invariants as points of an algebraic space. Problem Is there a moduli space parametrizing vector bundles on X with fixed numerical invariants having the structure of a smooth projective variety? A notion of stability is required!

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Motivations

Vector bundles are classified by using numerical invariants, like the rank or the degree. Theorem (Grothendieck) Over P1

C vector bundles are direct sums of line bundles.

Consider vector bundles with some fixed numerical invariants as points of an algebraic space. Problem Is there a moduli space parametrizing vector bundles on X with fixed numerical invariants having the structure of a smooth projective variety? A notion of stability is required!

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Motivations

Vector bundles are classified by using numerical invariants, like the rank or the degree. Theorem (Grothendieck) Over P1

C vector bundles are direct sums of line bundles.

Consider vector bundles with some fixed numerical invariants as points of an algebraic space. Problem Is there a moduli space parametrizing vector bundles on X with fixed numerical invariants having the structure of a smooth projective variety? A notion of stability is required!

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Stability

Assume dim(X) = 1, i.e. X is a curve. For example, X is the planar curve x3

0 + x3 1 + x3 2 = 0 in P2 C.

Example (Slope stability) Define the function µ(−) := deg(−) rk(−) . A vector bundle E is slope (semi)stable if for every subbundle F ֒ → E, we have µ(F)(≤) < µ(E).

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Stability

Assume dim(X) = 1, i.e. X is a curve. For example, X is the planar curve x3

0 + x3 1 + x3 2 = 0 in P2 C.

Example (Slope stability) Define the function µ(−) := deg(−) rk(−) . A vector bundle E is slope (semi)stable if for every subbundle F ֒ → E, we have µ(F)(≤) < µ(E).

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Stability

Assume dim(X) = 1, i.e. X is a curve. For example, X is the planar curve x3

0 + x3 1 + x3 2 = 0 in P2 C.

Example (Slope stability) Define the function µ(−) := deg(−) rk(−) . A vector bundle E is slope (semi)stable if for every subbundle F ֒ → E, we have µ(F)(≤) < µ(E).

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Moduli spaces

Theorem (Mumford) If X is a smooth projective curve of genus g ≥ 2, for every pair (r, d) ∈ N∗ × Z, there is a projective variety M(r, d) parametrizing semistable vector bundles of rank r and degree d on

• X. Moreover, the open subset

Ms(r, d) ⊂ M(r, d)

• f stable bundles is non-empty and smooth.

Theorem (Atiyah) Case of vector bundles over curves of genus 1.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Coherent sheaves

Problem: The category Vect(X) of vector bundles over X is not abelian. Solution: consider the category of coherent sheaves Vect(X) ⊂ Coh(X).

1

Kernel, image and exact sequence are well-defined notions.

2

Locally of the form O⊕k1

X

→ O⊕k2

X

→ E → 0.

3

Set µ(E) = +∞ if rk(E) = 0 notion of stability for coherent sheaves.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Coherent sheaves

Problem: The category Vect(X) of vector bundles over X is not abelian. Solution: consider the category of coherent sheaves Vect(X) ⊂ Coh(X).

1

Kernel, image and exact sequence are well-defined notions.

2

Locally of the form O⊕k1

X

→ O⊕k2

X

→ E → 0.

3

Set µ(E) = +∞ if rk(E) = 0 notion of stability for coherent sheaves.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Coherent sheaves

Problem: The category Vect(X) of vector bundles over X is not abelian. Solution: consider the category of coherent sheaves Vect(X) ⊂ Coh(X).

1

Kernel, image and exact sequence are well-defined notions.

2

Locally of the form O⊕k1

X

→ O⊕k2

X

→ E → 0.

3

Set µ(E) = +∞ if rk(E) = 0 notion of stability for coherent sheaves.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Pass to complexes

Remark: if X is smooth projective of dimension n, then every F ∈ Coh(X) has a locally free resolution 0 → En → · · · → E1 → E0 → F → 0. introduce the derived category Db(X)!

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Pass to complexes

Remark: if X is smooth projective of dimension n, then every F ∈ Coh(X) has a locally free resolution 0 → En → · · · → E1 → E0 → F → 0. introduce the derived category Db(X)!

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Derived category of a variety

Construction of Db(X):

1

Consider the category of bounded complexes of coherent sheaves on X: A• : . . . Ai−1 di−1 − − − → Ai

di

− → Ai+1 di+1 − − → . . . with di ◦ di−1 = 0. f : A• → B• such that f i+1 ◦ di

A = di B ◦ f i.

2

Cohomology: Hi(A•) =

ker di Im di−1 ∈ Coh(X).

f : A• → B• is a quasi-isomorphism if Hi(f ) : Hi(A•) ∼ = Hi(B•) for every i ∈ Z.

3

Require that quasi-isomorphisms are “invertible” in Db(X).

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Derived category of a variety

Construction of Db(X):

1

Consider the category of bounded complexes of coherent sheaves on X: A• : . . . Ai−1 di−1 − − − → Ai

di

− → Ai+1 di+1 − − → . . . with di ◦ di−1 = 0. f : A• → B• such that f i+1 ◦ di

A = di B ◦ f i.

2

Cohomology: Hi(A•) =

ker di Im di−1 ∈ Coh(X).

f : A• → B• is a quasi-isomorphism if Hi(f ) : Hi(A•) ∼ = Hi(B•) for every i ∈ Z.

3

Require that quasi-isomorphisms are “invertible” in Db(X).

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Derived category of a variety

Construction of Db(X):

1

Consider the category of bounded complexes of coherent sheaves on X: A• : . . . Ai−1 di−1 − − − → Ai

di

− → Ai+1 di+1 − − → . . . with di ◦ di−1 = 0. f : A• → B• such that f i+1 ◦ di

A = di B ◦ f i.

2

Cohomology: Hi(A•) =

ker di Im di−1 ∈ Coh(X).

f : A• → B• is a quasi-isomorphism if Hi(f ) : Hi(A•) ∼ = Hi(B•) for every i ∈ Z.

3

Require that quasi-isomorphisms are “invertible” in Db(X).

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Derived category of a variety

Construction of Db(X):

1

Consider the category of bounded complexes of coherent sheaves on X: A• : . . . Ai−1 di−1 − − − → Ai

di

− → Ai+1 di+1 − − → . . . with di ◦ di−1 = 0. f : A• → B• such that f i+1 ◦ di

A = di B ◦ f i.

2

Cohomology: Hi(A•) =

ker di Im di−1 ∈ Coh(X).

f : A• → B• is a quasi-isomorphism if Hi(f ) : Hi(A•) ∼ = Hi(B•) for every i ∈ Z.

3

Require that quasi-isomorphisms are “invertible” in Db(X).

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Derived category of a variety

Construction of Db(X):

1

Consider the category of bounded complexes of coherent sheaves on X: A• : . . . Ai−1 di−1 − − − → Ai

di

− → Ai+1 di+1 − − → . . . with di ◦ di−1 = 0. f : A• → B• such that f i+1 ◦ di

A = di B ◦ f i.

2

Cohomology: Hi(A•) =

ker di Im di−1 ∈ Coh(X).

f : A• → B• is a quasi-isomorphism if Hi(f ) : Hi(A•) ∼ = Hi(B•) for every i ∈ Z.

3

Require that quasi-isomorphisms are “invertible” in Db(X).

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Derived category of a variety

Construction of Db(X):

1

Consider the category of bounded complexes of coherent sheaves on X: A• : . . . Ai−1 di−1 − − − → Ai

di

− → Ai+1 di+1 − − → . . . with di ◦ di−1 = 0. f : A• → B• such that f i+1 ◦ di

A = di B ◦ f i.

2

Cohomology: Hi(A•) =

ker di Im di−1 ∈ Coh(X).

f : A• → B• is a quasi-isomorphism if Hi(f ) : Hi(A•) ∼ = Hi(B•) for every i ∈ Z.

3

Require that quasi-isomorphisms are “invertible” in Db(X).

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Bondal-Orlov Theorem

To summarize:

1

Objects in Db(X) are bounded complexes of coherent sheaves.

2

Morphisms are more complicated. They are obtained by inverting quasi-isomorphisms in the category of bounded complexes. Question: How much of the geometry of X is encoded by Db(X)? Canonical bundle: ωX := n T ∗

X.

Theorem (Bondal-Orlov) Let X and X ′ be two smooth projective varieties with ample (anti)canonical bundle. Then X ∼ = X ′ if and only if Db(X) ∼ − → Db(X ′).

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Bondal-Orlov Theorem

To summarize:

1

Objects in Db(X) are bounded complexes of coherent sheaves.

2

Morphisms are more complicated. They are obtained by inverting quasi-isomorphisms in the category of bounded complexes. Question: How much of the geometry of X is encoded by Db(X)? Canonical bundle: ωX := n T ∗

X.

Theorem (Bondal-Orlov) Let X and X ′ be two smooth projective varieties with ample (anti)canonical bundle. Then X ∼ = X ′ if and only if Db(X) ∼ − → Db(X ′).

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Bondal-Orlov Theorem

To summarize:

1

Objects in Db(X) are bounded complexes of coherent sheaves.

2

Morphisms are more complicated. They are obtained by inverting quasi-isomorphisms in the category of bounded complexes. Question: How much of the geometry of X is encoded by Db(X)? Canonical bundle: ωX := n T ∗

X.

Theorem (Bondal-Orlov) Let X and X ′ be two smooth projective varieties with ample (anti)canonical bundle. Then X ∼ = X ′ if and only if Db(X) ∼ − → Db(X ′).

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Stability on Db(X) - case of K3 surfaces

Important property (Bridgeland) There is a notion of stability for complexes in Db(X), which generalizes slope stability for sheaves. Definition A K3 surface is a smooth projective simply connected surface X with trivial canonical bundle, i.e. ωX := 2 T ∗

X ∼

= OX.

1

Bridgeland proved that there are stability conditions on Db(X).

2

By the work of Mukai, Huybrechts, O’Grady, Yoshioka, Bayer, Macrì, moduli spaces of Bridgeland stable objects in Db(X) with Mukai vector v are smooth projective hyperkähler varieties, under some mild assumptions.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Stability on Db(X) - case of K3 surfaces

Important property (Bridgeland) There is a notion of stability for complexes in Db(X), which generalizes slope stability for sheaves. Definition A K3 surface is a smooth projective simply connected surface X with trivial canonical bundle, i.e. ωX := 2 T ∗

X ∼

= OX.

1

Bridgeland proved that there are stability conditions on Db(X).

2

By the work of Mukai, Huybrechts, O’Grady, Yoshioka, Bayer, Macrì, moduli spaces of Bridgeland stable objects in Db(X) with Mukai vector v are smooth projective hyperkähler varieties, under some mild assumptions.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Stability on Db(X) - case of K3 surfaces

Important property (Bridgeland) There is a notion of stability for complexes in Db(X), which generalizes slope stability for sheaves. Definition A K3 surface is a smooth projective simply connected surface X with trivial canonical bundle, i.e. ωX := 2 T ∗

X ∼

= OX.

1

Bridgeland proved that there are stability conditions on Db(X).

2

By the work of Mukai, Huybrechts, O’Grady, Yoshioka, Bayer, Macrì, moduli spaces of Bridgeland stable objects in Db(X) with Mukai vector v are smooth projective hyperkähler varieties, under some mild assumptions.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Hyperkähler geometry

Hyperkähler (HK) varieties are higher dimensional analogous of K3 surfaces. Theorem (Beauville-Bogomolov Decomposition Theorem) Let X be a compact Kähler manifold with trivial first Chern class. There exists an étale finite cover d

i=1 Mi → X, where each of the

factors Mi is either a compact complex torus, a Calabi-Yau variety

• r a HK variety.

Problem: Very hard to construct examples. Only four classes are known up to deformation equivalence.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Hyperkähler geometry

Hyperkähler (HK) varieties are higher dimensional analogous of K3 surfaces. Theorem (Beauville-Bogomolov Decomposition Theorem) Let X be a compact Kähler manifold with trivial first Chern class. There exists an étale finite cover d

i=1 Mi → X, where each of the

factors Mi is either a compact complex torus, a Calabi-Yau variety

• r a HK variety.

Problem: Very hard to construct examples. Only four classes are known up to deformation equivalence.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Hyperkähler geometry

Hyperkähler (HK) varieties are higher dimensional analogous of K3 surfaces. Theorem (Beauville-Bogomolov Decomposition Theorem) Let X be a compact Kähler manifold with trivial first Chern class. There exists an étale finite cover d

i=1 Mi → X, where each of the

factors Mi is either a compact complex torus, a Calabi-Yau variety

• r a HK variety.

Problem: Very hard to construct examples. Only four classes are known up to deformation equivalence.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Questions

1

How to construct Bridgeland stability conditions on Db(X) or, more generally, on a triangulated subcategory of Db(X)? Having a Bridgeland stability condition:

1

How is the geometry of a moduli space of (semistable) objects with a certain numerical invariant, e.g. whether it has the structure of a projective variety?

2

Describe the birational models of the moduli space. My research interests: to deal with these problems in the case of cubic fourfolds and Gushel-Mukai fourfolds.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Questions

1

How to construct Bridgeland stability conditions on Db(X) or, more generally, on a triangulated subcategory of Db(X)? Having a Bridgeland stability condition:

1

How is the geometry of a moduli space of (semistable) objects with a certain numerical invariant, e.g. whether it has the structure of a projective variety?

2

Describe the birational models of the moduli space. My research interests: to deal with these problems in the case of cubic fourfolds and Gushel-Mukai fourfolds.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Questions

1

How to construct Bridgeland stability conditions on Db(X) or, more generally, on a triangulated subcategory of Db(X)? Having a Bridgeland stability condition:

1

How is the geometry of a moduli space of (semistable) objects with a certain numerical invariant, e.g. whether it has the structure of a projective variety?

2

Describe the birational models of the moduli space. My research interests: to deal with these problems in the case of cubic fourfolds and Gushel-Mukai fourfolds.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories

Questions

1

How to construct Bridgeland stability conditions on Db(X) or, more generally, on a triangulated subcategory of Db(X)? Having a Bridgeland stability condition:

1

How is the geometry of a moduli space of (semistable) objects with a certain numerical invariant, e.g. whether it has the structure of a projective variety?

2

Describe the birational models of the moduli space. My research interests: to deal with these problems in the case of cubic fourfolds and Gushel-Mukai fourfolds.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Subcategories of K3 type

Idea: Generalize the definition of K3 surface to the noncommutative setting. Remark: If S is a K3 surface, the Serre functor of Db(S) is the autoequivalence SDb(S) : Db(S) ∼ − → Db(S) given by SDb(S)(−) = (−) ⊗ ωS[2] = (−)[2]. Definition A full subcategory T ⊂ Db(X) is of K3 type if T has the same Serre functor and same Hochschild homology of Db(S) where S is a K3 surface.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Subcategories of K3 type

Idea: Generalize the definition of K3 surface to the noncommutative setting. Remark: If S is a K3 surface, the Serre functor of Db(S) is the autoequivalence SDb(S) : Db(S) ∼ − → Db(S) given by SDb(S)(−) = (−) ⊗ ωS[2] = (−)[2]. Definition A full subcategory T ⊂ Db(X) is of K3 type if T has the same Serre functor and same Hochschild homology of Db(S) where S is a K3 surface.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Subcategories of K3 type

Idea: Generalize the definition of K3 surface to the noncommutative setting. Remark: If S is a K3 surface, the Serre functor of Db(S) is the autoequivalence SDb(S) : Db(S) ∼ − → Db(S) given by SDb(S)(−) = (−) ⊗ ωS[2] = (−)[2]. Definition A full subcategory T ⊂ Db(X) is of K3 type if T has the same Serre functor and same Hochschild homology of Db(S) where S is a K3 surface.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Semiorthogonal decompositions

Motivation: divide Db(X) into “easier” subcategories. Definition A semiorthogonal decomposition of Db(X) is a sequence T 1, . . . , T n of full triangulated subcategories of Db(X) such that:

1

HomDb(X)(T i, T j) = 0 for every n ≥ i > j ≥ 0;

2

For any 0 = E ∈ Db(X), there is 0 = En

fn

− → En−1

fn−1

− − → En−2 . . . E1

f1

− → E0 = E such that the cone C(fi) ∈ T i for every 1 ≤ i ≤ n. Notation: Db(X) = T 1, . . . , T n.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Semiorthogonal decompositions

Motivation: divide Db(X) into “easier” subcategories. Definition A semiorthogonal decomposition of Db(X) is a sequence T 1, . . . , T n of full triangulated subcategories of Db(X) such that:

1

HomDb(X)(T i, T j) = 0 for every n ≥ i > j ≥ 0;

2

For any 0 = E ∈ Db(X), there is 0 = En

fn

− → En−1

fn−1

− − → En−2 . . . E1

f1

− → E0 = E such that the cone C(fi) ∈ T i for every 1 ≤ i ≤ n. Notation: Db(X) = T 1, . . . , T n.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Semiorthogonal decompositions

Motivation: divide Db(X) into “easier” subcategories. Definition A semiorthogonal decomposition of Db(X) is a sequence T 1, . . . , T n of full triangulated subcategories of Db(X) such that:

1

HomDb(X)(T i, T j) = 0 for every n ≥ i > j ≥ 0;

2

For any 0 = E ∈ Db(X), there is 0 = En

fn

− → En−1

fn−1

− − → En−2 . . . E1

f1

− → E0 = E such that the cone C(fi) ∈ T i for every 1 ≤ i ≤ n. Notation: Db(X) = T 1, . . . , T n.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Example 1: Cubic fourfolds

Let Y be a cubic fourfold. Proposition (Kuznetsov) Db(Y ) = Ku(Y ), OY , OY (1), OY (2) The Kuznetsov component of Y is Ku(Y ) := {E ∈ Db(Y ) : HomDb(Y )(OY (i), E) = 0 ∀i = 0, 1, 2} and is a subcategory of K3 type. Conjecture Y is rational if and only if Ku(Y ) ∼ − → Db(S), where S is a K3 surface.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Example 1: Cubic fourfolds

Let Y be a cubic fourfold. Proposition (Kuznetsov) Db(Y ) = Ku(Y ), OY , OY (1), OY (2) The Kuznetsov component of Y is Ku(Y ) := {E ∈ Db(Y ) : HomDb(Y )(OY (i), E) = 0 ∀i = 0, 1, 2} and is a subcategory of K3 type. Conjecture Y is rational if and only if Ku(Y ) ∼ − → Db(S), where S is a K3 surface.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Example 1: Cubic fourfolds

Let Y be a cubic fourfold. Proposition (Kuznetsov) Db(Y ) = Ku(Y ), OY , OY (1), OY (2) The Kuznetsov component of Y is Ku(Y ) := {E ∈ Db(Y ) : HomDb(Y )(OY (i), E) = 0 ∀i = 0, 1, 2} and is a subcategory of K3 type. Conjecture Y is rational if and only if Ku(Y ) ∼ − → Db(S), where S is a K3 surface.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Example 1: Cubic fourfolds

Let Y be a cubic fourfold. Proposition (Kuznetsov) Db(Y ) = Ku(Y ), OY , OY (1), OY (2) The Kuznetsov component of Y is Ku(Y ) := {E ∈ Db(Y ) : HomDb(Y )(OY (i), E) = 0 ∀i = 0, 1, 2} and is a subcategory of K3 type. Conjecture Y is rational if and only if Ku(Y ) ∼ − → Db(S), where S is a K3 surface.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Ku(Y ) vs Y

General goal: Understand the geometry of Y by studying the properties of Ku(Y ) and moduli spaces of objects in Ku(Y ). Remark: BO Theorem applies consider Ku(Y ) instead. Question: Does Ku(Y ) determine the isomorphism class of Y ? Theorem (P.) In the moduli space of cubic fourfolds there is a countable union of divisors whose general element Y has at least one non-isomorphic Y ′ with Ku(Y ) ∼ − → Ku(Y ′).

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Ku(Y ) vs Y

General goal: Understand the geometry of Y by studying the properties of Ku(Y ) and moduli spaces of objects in Ku(Y ). Remark: BO Theorem applies consider Ku(Y ) instead. Question: Does Ku(Y ) determine the isomorphism class of Y ? Theorem (P.) In the moduli space of cubic fourfolds there is a countable union of divisors whose general element Y has at least one non-isomorphic Y ′ with Ku(Y ) ∼ − → Ku(Y ′).

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Ku(Y ) vs Y

General goal: Understand the geometry of Y by studying the properties of Ku(Y ) and moduli spaces of objects in Ku(Y ). Remark: BO Theorem applies consider Ku(Y ) instead. Question: Does Ku(Y ) determine the isomorphism class of Y ? Theorem (P.) In the moduli space of cubic fourfolds there is a countable union of divisors whose general element Y has at least one non-isomorphic Y ′ with Ku(Y ) ∼ − → Ku(Y ′).

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Ku(Y ) vs Y

General goal: Understand the geometry of Y by studying the properties of Ku(Y ) and moduli spaces of objects in Ku(Y ). Remark: BO Theorem applies consider Ku(Y ) instead. Question: Does Ku(Y ) determine the isomorphism class of Y ? Theorem (P.) In the moduli space of cubic fourfolds there is a countable union of divisors whose general element Y has at least one non-isomorphic Y ′ with Ku(Y ) ∼ − → Ku(Y ′).

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Ku(Y ) vs rational curves on Y

Classical hyperkähler manifolds starting from moduli spaces of rational curves on Y : The Fano variety FY , parametrizing lines in Y , is a smooth projective HK fourfold, with same period point as Y (Beauville, Donagi). The eightfold MY constructed out of twisted cubic curves, for Y non containing a plane (Lehn, Lehn, Sorger, van Straten). Both are deformation equivalent to Hilbert schemes of points on a K3 surface. New approach: describe as moduli spaces of stable objects in Ku(Y ).

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Ku(Y ) vs rational curves on Y

Classical hyperkähler manifolds starting from moduli spaces of rational curves on Y : The Fano variety FY , parametrizing lines in Y , is a smooth projective HK fourfold, with same period point as Y (Beauville, Donagi). The eightfold MY constructed out of twisted cubic curves, for Y non containing a plane (Lehn, Lehn, Sorger, van Straten). Both are deformation equivalent to Hilbert schemes of points on a K3 surface. New approach: describe as moduli spaces of stable objects in Ku(Y ).

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Ku(Y ) vs rational curves on Y

Classical hyperkähler manifolds starting from moduli spaces of rational curves on Y : The Fano variety FY , parametrizing lines in Y , is a smooth projective HK fourfold, with same period point as Y (Beauville, Donagi). The eightfold MY constructed out of twisted cubic curves, for Y non containing a plane (Lehn, Lehn, Sorger, van Straten). Both are deformation equivalent to Hilbert schemes of points on a K3 surface. New approach: describe as moduli spaces of stable objects in Ku(Y ).

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Ku(Y ) vs rational curves on Y

Classical hyperkähler manifolds starting from moduli spaces of rational curves on Y : The Fano variety FY , parametrizing lines in Y , is a smooth projective HK fourfold, with same period point as Y (Beauville, Donagi). The eightfold MY constructed out of twisted cubic curves, for Y non containing a plane (Lehn, Lehn, Sorger, van Straten). Both are deformation equivalent to Hilbert schemes of points on a K3 surface. New approach: describe as moduli spaces of stable objects in Ku(Y ).

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Ku(Y ) vs rational curves on Y

Classical hyperkähler manifolds starting from moduli spaces of rational curves on Y : The Fano variety FY , parametrizing lines in Y , is a smooth projective HK fourfold, with same period point as Y (Beauville, Donagi). The eightfold MY constructed out of twisted cubic curves, for Y non containing a plane (Lehn, Lehn, Sorger, van Straten). Both are deformation equivalent to Hilbert schemes of points on a K3 surface. New approach: describe as moduli spaces of stable objects in Ku(Y ).

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Key ingredient

Theorem (Bayer, Lahoz, Macrì, Nuer, Perry, Stellari)

1

There are Bridgeland stability conditions on Ku(Y ).

2

Under mild assumptions, moduli spaces of stable objects with primitive Mukai vector v with v2 ≥ −2 are smooth projective HK manifolds of dimension v2 + 2, deformation equivalent to a Hilbert scheme of points on a K3 surface.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Lines

Theorem (Li, P., Zhao) The Fano variety FY is a moduli space of Bridgeland stable objects in Ku(Y ) with Mukai vector λ1 + λ2. Application: easier proof of the Categorical Torelli Theorem. Corollary (Huybrecht, Rennemo, Appendix of BLMS) Y ∼ = Y ′ if and only if Ku(Y ) ∼ − → Ku(Y ′) and the induced isometry N(Ku(Y )) ∼ = N(Ku(Y ′)) commutes with the degree shift functor.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Lines

Theorem (Li, P., Zhao) The Fano variety FY is a moduli space of Bridgeland stable objects in Ku(Y ) with Mukai vector λ1 + λ2. Application: easier proof of the Categorical Torelli Theorem. Corollary (Huybrecht, Rennemo, Appendix of BLMS) Y ∼ = Y ′ if and only if Ku(Y ) ∼ − → Ku(Y ′) and the induced isometry N(Ku(Y )) ∼ = N(Ku(Y ′)) commutes with the degree shift functor.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Twisted cubics

Theorem (Li, P., Zhao) Assume Y does not contain a plane. The LLSvS eightfold MY is a moduli space of Bridgeland stable objects in Ku(Y ) with Mukai vector 2λ1 + λ2. Applications:

1

Recover the period point of MY : Proposition There is a (up to twist) Hodge isometry H2(MY , Z)prim ∼ = H4(Y , Z)prim.

2

Try to address the Derived Torelli Theorem.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Twisted cubics

Theorem (Li, P., Zhao) Assume Y does not contain a plane. The LLSvS eightfold MY is a moduli space of Bridgeland stable objects in Ku(Y ) with Mukai vector 2λ1 + λ2. Applications:

1

Recover the period point of MY : Proposition There is a (up to twist) Hodge isometry H2(MY , Z)prim ∼ = H4(Y , Z)prim.

2

Try to address the Derived Torelli Theorem.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Example 2: Gushel-Mukai fourfolds

A GM fourfold X is a smooth intersection X = CGr(2, V5) ∩ Q ⊂ P(

2

• V5 ⊕ C) ∼

= P10, where Q is a quadric hypersurface in P8 ∼ = P(W ) ⊂ P(2 V5 ⊕ C). Proposition (Kuznetsov, Perry) Db(X) = Ku(X), OX, U∗

X, OX(1), U∗ X(1)

and Ku(X) is a K3 category. General goal: try to address similar questions as those for cubic fourfolds, answered by Addington, Thomas, Huybrechts, BLMNPS.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Example 2: Gushel-Mukai fourfolds

A GM fourfold X is a smooth intersection X = CGr(2, V5) ∩ Q ⊂ P(

2

• V5 ⊕ C) ∼

= P10, where Q is a quadric hypersurface in P8 ∼ = P(W ) ⊂ P(2 V5 ⊕ C). Proposition (Kuznetsov, Perry) Db(X) = Ku(X), OX, U∗

X, OX(1), U∗ X(1)

and Ku(X) is a K3 category. General goal: try to address similar questions as those for cubic fourfolds, answered by Addington, Thomas, Huybrechts, BLMNPS.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Example 2: Gushel-Mukai fourfolds

A GM fourfold X is a smooth intersection X = CGr(2, V5) ∩ Q ⊂ P(

2

• V5 ⊕ C) ∼

= P10, where Q is a quadric hypersurface in P8 ∼ = P(W ) ⊂ P(2 V5 ⊕ C). Proposition (Kuznetsov, Perry) Db(X) = Ku(X), OX, U∗

X, OX(1), U∗ X(1)

and Ku(X) is a K3 category. General goal: try to address similar questions as those for cubic fourfolds, answered by Addington, Thomas, Huybrechts, BLMNPS.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Associated HK

Theorem (Debarre, Kuznetsov) Out of a divisor, there is a hyperkähler fourfold ˜ Y with the same period point as X. Theorem (P.) Characterized when ˜ Y is birational to a moduli space of (twisted) stable sheaves on a K3 surface S, resp. to the Hilbert scheme of points on a K3 surface.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Associated HK

Theorem (Debarre, Kuznetsov) Out of a divisor, there is a hyperkähler fourfold ˜ Y with the same period point as X. Theorem (P.) Characterized when ˜ Y is birational to a moduli space of (twisted) stable sheaves on a K3 surface S, resp. to the Hilbert scheme of points on a K3 surface.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Stability conditions and moduli spaces

Theorem (Perry, P., Zhao)

1

There are Bridgeland stability conditions on Ku(X).

2

Under mild assumptions, moduli spaces of stable objects with primitive Mukai vector v with v2 ≥ −2 are smooth projective HK manifolds of dimension v2 + 2, deformation equivalent to a Hilbert scheme of points on a K3 surface.

3

Characterized when Ku(X) ∼ − → Db(S) where S is a K3 surface. Interesting future applications:

1

Study moduli spaces of stable objects in Ku(Y ).

2

Describe the image of the period map of GM fourfolds.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Stability conditions and moduli spaces

Theorem (Perry, P., Zhao)

1

There are Bridgeland stability conditions on Ku(X).

2

Under mild assumptions, moduli spaces of stable objects with primitive Mukai vector v with v2 ≥ −2 are smooth projective HK manifolds of dimension v2 + 2, deformation equivalent to a Hilbert scheme of points on a K3 surface.

3

Characterized when Ku(X) ∼ − → Db(S) where S is a K3 surface. Interesting future applications:

1

Study moduli spaces of stable objects in Ku(Y ).

2

Describe the image of the period map of GM fourfolds.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Stability conditions and moduli spaces

Theorem (Perry, P., Zhao)

1

There are Bridgeland stability conditions on Ku(X).

2

Under mild assumptions, moduli spaces of stable objects with primitive Mukai vector v with v2 ≥ −2 are smooth projective HK manifolds of dimension v2 + 2, deformation equivalent to a Hilbert scheme of points on a K3 surface.

3

Characterized when Ku(X) ∼ − → Db(S) where S is a K3 surface. Interesting future applications:

1

Study moduli spaces of stable objects in Ku(Y ).

2

Describe the image of the period map of GM fourfolds.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

Stability conditions and moduli spaces

Theorem (Perry, P., Zhao)

1

There are Bridgeland stability conditions on Ku(X).

2

Under mild assumptions, moduli spaces of stable objects with primitive Mukai vector v with v2 ≥ −2 are smooth projective HK manifolds of dimension v2 + 2, deformation equivalent to a Hilbert scheme of points on a K3 surface.

3

Characterized when Ku(X) ∼ − → Db(S) where S is a K3 surface. Interesting future applications:

1

Study moduli spaces of stable objects in Ku(Y ).

2

Describe the image of the period map of GM fourfolds.

Laura Pertusi K3 category of cubic and GM fourfolds

Introduction Derived category K3 categories Cubic fourfolds Gushel-Mukai fourfolds

References

• A. Perry, L. Pertusi, X. Zhao, Stability conditions and moduli

spaces for Kuznetsov components of Gushel-Mukai varieties, arXiv:1912.06935.

• C. Li, L. Pertusi, X. Zhao, Twisted cubics on cubic fourfolds

and stability conditions, arXiv:1802.01134.

• L. Pertusi, Fourier-Mukai partners for very general special cubic

fourfolds, to appear in Math. Research Letters, arXiv:1611.06687.

• L. Pertusi, On the double EPW sextic associated to a

Gushel-Mukai fourfold, J. London Math. Soc. (2) 00 (2018), 1-24.

Laura Pertusi K3 category of cubic and GM fourfolds