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 RF RE RV

d−2 d−1

d−2(f ) = edge1(f ) − edge2(f ) + edge3(f ) d−1(e) = head(e) − tail(e).

RF RE RV

d−2 d−1

d−2(f ) = edge1(f ) − edge2(f ) + edge3(f ) d−1(e) = head(e) − tail(e). H0(S) := RV /im d−1 = R H1(S) := ker d−1/im d−2 = R2 H2(S) := ker d−2 = R

RF RE RV

d−2 d−1

H0(Bunny) = R H1(Bunny) = 0 H2(Bunny) = R

Complexes

Let R be a ring. A complex A• of R-modules is a sequence of R-modules ... di−2 − − − → Ai−1 di−1 − − − → Ai

di

− → Ai+1 di+1 − − → ... such that d2 = 0 i.e. di−1 ◦ di = 0 ∀i. The ith cohomology of a complex A• of R-modules is defined as the R-module Hi(A•) := ker di im di−1

Complexes

Let R be a ring. A complex A• of R-modules is a sequence of R-modules ... di−2 − − − → Ai−1 di−1 − − − → Ai

di

− → Ai+1 di+1 − − → ... such that d2 = 0 i.e. di−1 ◦ di = 0 ∀i. The ith cohomology of a complex A• of R-modules is defined as the R-module Hi(A•) := ker di im di−1

Quasi-isomorphisms

A morphism of complexes f : A• → B• is a commutative diagram ... Ai−1 Ai Ai+1 ... ... Bi−1 Bi Bi+1 ...

di

A

di−1

A

f i−1 di

A

f i di+1

A

f i+1 di

B

di−1

B

di

B

di+1

B

i.e. fd = df or more precisely f i ◦ di−1

A

= di−1

B

• f i−1.

Any morphism induces a map f∗ : Hi(A•) → Hi(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 ... Ai−1 Ai Ai+1 ... ... Bi−1 Bi Bi+1 ...

di

A

di−1

A

f i−1 di

A

f i di+1

A

f i+1 di

B

di−1

B

di

B

di+1

B

i.e. fd = df or more precisely f i ◦ di−1

A

= di−1

B

• f i−1.

Any morphism induces a map f∗ : Hi(A•) → Hi(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 • A• B•

∼ f g

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 f1, ..., ft ∈ C[x1, ..., xn] be polynomials. We define the zero

set to be X := {(v1, ..., vn) ∈ Cn | fi(v1, ..., vn) = 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[x1, ..., xn]/

• f1, ..., ft.

◮ Hence, to an affine algebraic variety X we can associate a

derived category D(X) := D(R).

Affine Algebraic Varieties

◮ Let f1, ..., ft ∈ C[x1, ..., xn] be polynomials. We define the zero

set to be X := {(v1, ..., vn) ∈ Cn | fi(v1, ..., vn) = 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[x1, ..., xn]/

• f1, ..., ft.

◮ Hence, to an affine algebraic variety X we can associate a

derived category D(X) := D(R).

Affine Algebraic Varieties

◮ Let f1, ..., ft ∈ C[x1, ..., xn] be polynomials. We define the zero

set to be X := {(v1, ..., vn) ∈ Cn | fi(v1, ..., vn) = 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[x1, ..., xn]/

• f1, ..., ft.

◮ 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 Pn := Cn+1\0/ ∼ where the equivalence relation is given by scaling the vectors v ∼ λv ∀λ ∈ C∗.

◮ Let f1, ..., ft ∈ C[x0, ..., xn] be homogeneous polynomials.

X := {(v0 : ... : vn) ∈ Pn | fi(v1, ..., vn) = 0 ∀i} ⊆ Pn

◮ X is called an projective algebraic variety. The coordinate ring

• f X is the N-graded ring,

R = C[x0, ..., xn]/

• f1, ..., ft.

◮ 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 Pn := Cn+1\0/ ∼ where the equivalence relation is given by scaling the vectors v ∼ λv ∀λ ∈ C∗.

◮ Let f1, ..., ft ∈ C[x0, ..., xn] be homogeneous polynomials.

X := {(v0 : ... : vn) ∈ Pn | fi(v1, ..., vn) = 0 ∀i} ⊆ Pn

◮ X is called an projective algebraic variety. The coordinate ring

• f X is the N-graded ring,

R = C[x0, ..., xn]/

• f1, ..., ft.

◮ 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 Pn := Cn+1\0/ ∼ where the equivalence relation is given by scaling the vectors v ∼ λv ∀λ ∈ C∗.

◮ Let f1, ..., ft ∈ C[x0, ..., xn] be homogeneous polynomials.

X := {(v0 : ... : vn) ∈ Pn | fi(v1, ..., vn) = 0 ∀i} ⊆ Pn

◮ X is called an projective algebraic variety. The coordinate ring

• f X is the N-graded ring,

R = C[x0, ..., xn]/

• f1, ..., ft.

◮ 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, Favero1 ’12)

◮ False in dimension 3 (Lesieutre ’13)

1Reconstruction 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, Favero1 ’12)

◮ False in dimension 3 (Lesieutre ’13)

1Reconstruction 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.

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.

So far we have introduced two invariants of an algebraic variety X

• 1. D(X)
• 2. The fraction field of X.

Question

What is the relationship between these two? For example, if X and Y are birational, do we have D(X) = D(Y )?

Answer

• No. There is a simple procedure for modifying an algebraic variety

called a blow up. The blow-up of P2 at a point is birational to P2 but it is easy to show that there derived categories cannot be equivalent.

So far we have introduced two invariants of an algebraic variety X

• 1. D(X)
• 2. The fraction field of X.

Question

What is the relationship between these two? For example, if X and Y are birational, do we have D(X) = D(Y )?

Answer

• No. There is a simple procedure for modifying an algebraic variety

called a blow up. The blow-up of P2 at a point is birational to P2 but it is easy to show that there derived categories cannot be equivalent.

So far we have introduced two invariants of an algebraic variety X

• 1. D(X)
• 2. The fraction field of X.

Question

What is the relationship between these two? For example, if X and Y are birational, do we have D(X) = D(Y )?

Answer

• No. There is a simple procedure for modifying an algebraic variety

called a blow up. The blow-up of P2 at a point is birational to P2 but it is easy to show that there derived categories cannot be equivalent.

Kawamata’s Second Conjecture Setup: Calabi-Yaus

We will say that a smooth algebraic variety or complex manifold is Calabi-Yau if it is simply-connected, compact, and admits a non-vanishing holomorphic n-form.

Example

Consider following smooth projective algebraic variety {(v0 : ... : v4) | v5

0 + ... + v5 4 = 0} ⊆ P4

Remark: The Calabi-Yau condition is that 5 = 4 + 1.

Kawamata’s Second Conjecture

Conjecture (Kawamata ’02)

If X and Y are Calabi-Yau and birational (isomorphic on a (dense)

• pen algebraic subset), their derived categories are equivalent.

Known for the following types of “algebraic surgeries”

◮ Standard Flops (Bondal-Orlov ’95) ◮ Toroidal Flops (Kawamata ’02) ◮ Flops in dimension 3 (Bridgeland ’02) ◮ Elementary wall-crossings from variation of Geometric

Invariant Theory Quotients (Halpern-Leistner ’12, Ballard-Favero-Katzarkov2 ’12)

2Variation of Geometric Invariant Theory Quotients and Derived

Categories(63 pages). To appear in Journal f¨ ur die reine und angewandte Mathematik (Crelle’s journal).

Homological Mirror Symmetry Setup: The Symplectic Side

◮ A symplectic manifold (M, ω) is a manifold M equipped with

a closed non-degenerate differential 2-form ω called the symplectic form i.e. dω = 0 and at a point p ∈ M, we have a map ωp : 2 TpM → R such that for any nonzero X ∈ TpM there exists Y ∈ TpM with ω(X ∧ Y ) = 0.

◮ A Lagrangian submanifold of (M, ω) is a submanifold L ⊆ M

such that ω restricts to zero on L.

◮ The Fukaya category of (M, ω) is a category Fuk(M) such

that

◮ Objects of Fuk(M) are Lagrangian submanifolds ◮ Morphisms are (very roughly!) intersection points.

Homological Mirror Symmetry Setup: The Symplectic Side

◮ A symplectic manifold (M, ω) is a manifold M equipped with

a closed non-degenerate differential 2-form ω called the symplectic form i.e. dω = 0 and at a point p ∈ M, we have a map ωp : 2 TpM → R such that for any nonzero X ∈ TpM there exists Y ∈ TpM with ω(X ∧ Y ) = 0.

◮ A Lagrangian submanifold of (M, ω) is a submanifold L ⊆ M

such that ω restricts to zero on L.

◮ The Fukaya category of (M, ω) is a category Fuk(M) such

that

◮ Objects of Fuk(M) are Lagrangian submanifolds ◮ Morphisms are (very roughly!) intersection points.

Homological Mirror Symmetry Setup: The Symplectic Side

◮ A symplectic manifold (M, ω) is a manifold M equipped with

a closed non-degenerate differential 2-form ω called the symplectic form i.e. dω = 0 and at a point p ∈ M, we have a map ωp : 2 TpM → R such that for any nonzero X ∈ TpM there exists Y ∈ TpM with ω(X ∧ Y ) = 0.

◮ A Lagrangian submanifold of (M, ω) is a submanifold L ⊆ M

such that ω restricts to zero on L.

◮ The Fukaya category of (M, ω) is a category Fuk(M) such

that

◮ Objects of Fuk(M) are Lagrangian submanifolds ◮ Morphisms are (very roughly!) intersection points.

Homological Mirror Symmetry Setup: The Symplectic Side

◮ A symplectic manifold (M, ω) is a manifold M equipped with

a closed non-degenerate differential 2-form ω called the symplectic form i.e. dω = 0 and at a point p ∈ M, we have a map ωp : 2 TpM → R such that for any nonzero X ∈ TpM there exists Y ∈ TpM with ω(X ∧ Y ) = 0.

◮ A Lagrangian submanifold of (M, ω) is a submanifold L ⊆ M

such that ω restricts to zero on L.

◮ The Fukaya category of (M, ω) is a category Fuk(M) such

that

◮ Objects of Fuk(M) are Lagrangian submanifolds ◮ Morphisms are (very roughly!) intersection points.

Conjecture (Homological Mirror Symmetry, Kontsevich)

Let X be a Calabi-Yau manifold. There exists a “mirror” Calabi-Yau manifold X such that the following categories are equivalent Fuk( X) ∼ = D(X) and Fuk(X) ∼ = D( X) Known for

◮ Dimension 1 (Polishchuk-Zaslow ’98) ◮ Dimension 2 (Seidel ’03) ◮ Hypersurfaces in Projective Space (Sheridan ’11) ◮ Hypersurfaces in Weighted Projective Space (Sheridan-Smith

’17)

Conjecture (Homological Mirror Symmetry, Kontsevich)

Let X be a Calabi-Yau manifold. There exists a “mirror” Calabi-Yau manifold X such that the following categories are equivalent Fuk( X) ∼ = D(X) and Fuk(X) ∼ = D( X) Known for

◮ Dimension 1 (Polishchuk-Zaslow ’98) ◮ Dimension 2 (Seidel ’03) ◮ Hypersurfaces in Projective Space (Sheridan ’11) ◮ Hypersurfaces in Weighted Projective Space (Sheridan-Smith

’17)

There is are also formulations of Homological Mirror Symmetry for non Calabi-Yau cases.

◮ Fano Toric Varieties (Abouzaid ’06) ◮ Del Pezzo Surfaces (Auroux-Katzarkov-Kontsevich ’05) ◮ Abelian Surfaces (Abouzaid-Smith ’10) ◮ some non-Fano toric varieties

(Ballard-Diemer-Favero-Kerr-Katzarkov ’15)

◮ Many non-compact cases

What is the mirror?

Fundamental question:

Given a Calabi-Yau variety X, what is its mirror? There are many constructions of mirrors, each having different

• contexts. They don’t always agree and have internal

inconsistencies! Examples:

◮ Greene-Plesser-Roan ’90 ◮ Berglund-H¨

ubsch ’93

◮ Batyrev-Borisov ’95 ◮ Strominger-Yau-Zaslow ’96 ◮ Hori-Vafa ’00 ◮ Gross-Siebert ’01 ◮ Clarke ’08

What is the mirror?

Fundamental question:

Given a Calabi-Yau variety X, what is its mirror? There are many constructions of mirrors, each having different

• contexts. They don’t always agree and have internal

inconsistencies! Examples:

◮ Greene-Plesser-Roan ’90 ◮ Berglund-H¨

ubsch ’93

◮ Batyrev-Borisov ’95 ◮ Strominger-Yau-Zaslow ’96 ◮ Hori-Vafa ’00 ◮ Gross-Siebert ’01 ◮ Clarke ’08

Invertible Polynomials

Start with an invertible matrix A = (aij)n

i,j=0 with all nonnegative

integer entries. Take the polynomial, FA :=

n

• i=0

n

• j=0

xaij

j

Assume that:

◮ FA : Cn+1 → C has a unique critical point at the origin. ◮ FA is quasihomogeneous of degree d: there exists d ∈ N and

(q0, ..., qn) ∈ Nn+1 such that FA(λq0x0, ..., λqnxn) = λdFA(x0, ..., xn) for all λ ∈ C∗

◮ Calabi-Yau condition: n i=0 qi = d ◮ G is a finite group of diagonal symmetries of FA in Sln+1(C) ◮ Berglund and H¨

ubsch proposed the following basic duality: (A, G) ← → (AT, G T

A ).

This basic duality (A, G) ← → (AT, G T

A )

can be viewed as a duality between Calabi-Yau manifolds XA := {(v0 : ... : vn) | FA(v0 : ...vn)} ⊆ P(q0, ..., qn) where P(q0, ..., qn) := Cn+1\0/ ∼ and (v0, ..., vn) ∼ (λq0v0, ..., λqnvn). We get a duality: XA/G ← → XAT /G T

A

Example

A1 : =   3 3 3   FA1 = x3 + y 3 + z3 FAT

1 = x3 + y 3 + z3

A2 : =   2 1 3 3   FA2 = x2y + y 3 + z3 FAT

2 = x2 + xy 3 + z3

Set G = 1. Notice that XA1 ∼ = XA2 as we have changed the complex structure but not the symplectic structure. Therefore, they should have the same mirror. However, G T

A1 = (Z/3Z)

XAT

1 /(Z/3Z) ⊆ P3/(Z/3Z)

G T

A2 = 1

XAT

2 ⊆ P(3 : 1 : 2)

Prediction Kontsevich

Suppose X is a Calabi-Yau manifold, and X1, X2 are mirrors, either from different constructions or from ambiguities in a particular construction.

Conjecture (Homological Mirror Symmetry, Kontsevich)

Let X be a Calabi-Yau manifold. There exists a “mirror” Calabi-Yau manifold X such that the following categories are equivalent Fuk( X) ∼ = D(X) and Fuk(X) ∼ = D( X)

Prediction

D( X1) D( X2) Fuk(X)

∼ ∼ ∼

Prediction Kawamata

Theorem (Shoemaker, Borisov, Kelly, Clarke, ’12-’13)

Given any two Calabi-Yau hypersurfaces in the same finite quotient

• f weighted projective space, their Berglund-H¨

ubsch mirrors are birational (isomorphic on a dense open subset).

Conjecture (Kawamata ’02)

If X and Y are Calabi-Yau and birational, then, their derived categories are equivalent.

Prediction

Given any two Calabi-Yau hypersurfaces in the same finite quotient

• f weighted projective space, their Berglund-H¨

ubsch mirrors have equivalent derived categories.

Theorem (Favero-Kelly ’14)

Given any two hypersurfaces in the same quotient of weighted projective space, their Berglund-H¨ ubsch mirrors have equivalent derived categories. Fuk(X1)Sheridan-Smith

• symplectomorphic
• D( ˆ

X1)

Favero-Kelly

• Fuk(X2)

Corollary

D( ˆ

X2)

Corollary

Homological Mirror Symmetry holds for all projective Calabi-Yau Gorenstein Berglund-H¨ ubsch mirror pairs.

Theorem (Favero-Kelly ’14)

Given any two hypersurfaces in the same quotient of weighted projective space, their Berglund-H¨ ubsch mirrors have equivalent derived categories. Fuk(X1)Sheridan-Smith

• symplectomorphic
• D( ˆ

X1)

Favero-Kelly

• Fuk(X2)

Corollary

D( ˆ

X2)

Corollary

Homological Mirror Symmetry holds for all projective Calabi-Yau Gorenstein Berglund-H¨ ubsch mirror pairs.

Similar theorems

Conjecture (Batyrev/Nill ’08)/Theorem(Favero-Kelly3 ’14)

Batyrev and Nill’s conjecture holds: multiple mirrors in the Batyrev-Borisov construction of mirror symmetry (for Calabi-Yau complete intersections in toric varieties) have equivalent derived categories.

Theorem (Doran-Favero-Kelly ’15)

Multiple mirrors in Clarke’s construction of mirror symmetry for hypersurfaces have equivalent derived categories.

3Proof of a Conjecture of Batyrev and Nill, (23 pages). To appear in

American Journal of Mathematics.

Invariants from the Derived Category

Many invariants descend from derived categories:

◮ Algebraic K-theory (Thomason-Trobaugh ’90) ◮ Cohomology (Swan ’96) ◮ Geometric motives for certain equivalences (Orlov ’05) ◮ Zeta Functions, dim 2,3, abelian varieties (Honigs ’17) ◮ Griffiths Groups (Favero-Iliev-Katzarkov4 ’14)

4Griffiths Groups for Derived Categories with applications to

Fano-Calabi-Yaus, Pure and Applied Mathematics Quarterly, V. 10 N. 1 pgs 1-55, 2014.

Invariants from the Derived Category

Many invariants descend from derived categories:

◮ Algebraic K-theory (Thomason-Trobaugh ’90) ◮ Cohomology (Swan ’96) ◮ Geometric motives for certain equivalences (Orlov ’05) ◮ Zeta Functions, dim 2,3, abelian varieties (Honigs ’17) ◮ Griffiths Groups (Favero-Iliev-Katzarkov4 ’14)

4Griffiths Groups for Derived Categories with applications to

Fano-Calabi-Yaus, Pure and Applied Mathematics Quarterly, V. 10 N. 1 pgs 1-55, 2014.

More Invariants of the Derived Category

◮ Rouquier Dimension/Orlov Spectra (Rouquier ’08, Orlov ’09)

◮ Related to relations in the symplectic mapping class group and

“algebraic surgeries” (birational geometry) (Ballard-Favero-Katzarkov5 ’12)

◮ Related to Algebraic Cycles (Ballard-Favero-Katzarkov6 ’14)

◮ Enumerative Invariants?

5Orlov Spectra: Gaps and Bounds Inventiones Mathematicae, V. 189 I. 2,

pgs 359-430, 2012.

6A Category of Kernels for Equivariant Factorizations, Publications

Math´ ematiques de l’IH´ ES, V. 120 I. 1, pgs 1-111, 2014.

Future Directions: Enumerative Mirror Symmetry and Homological Mirror Symmetry Meet

◮ Around 1990, physicists Philip Candelas, Xenia de la Ossa,

Paul Green, and Linda Parkes famously used mirror symmetry to count rational curves on Calabi-Yau varieties

◮ The difficult problem of counting these curves was simplified

to calculating certain integrals on a complex manifold / algebraic variety

◮ These curve counts are called Gromov-Witten invariants.

Future Directions: Enumerative Mirror Symmetry and Homological Mirror Symmetry Meet

◮ Around 1990, physicists Philip Candelas, Xenia de la Ossa,

Paul Green, and Linda Parkes famously used mirror symmetry to count rational curves on Calabi-Yau varieties

◮ The difficult problem of counting these curves was simplified

to calculating certain integrals on a complex manifold / algebraic variety

◮ These curve counts are called Gromov-Witten invariants.

Future Directions: Enumerative Mirror Symmetry and Homological Mirror Symmetry Meet

◮ Around 1990, physicists Philip Candelas, Xenia de la Ossa,

Paul Green, and Linda Parkes famously used mirror symmetry to count rational curves on Calabi-Yau varieties

◮ The difficult problem of counting these curves was simplified

to calculating certain integrals on a complex manifold / algebraic variety

◮ These curve counts are called Gromov-Witten invariants.

Future Directions: Enumerative Mirror Symmetry and Homological Mirror Symmetry Meet

In joint work with Ciocan-Fontanine, Guere, Kim, and Shoemaker, we have constructed enumerative invariants using derived categories.

Theorem (in progress)

Gromov-Witten theory and FJRW theory can be recovered as special cases of our enumerative invariants.

Future Directions: An approach to Kawamata’s conjecture

Conjecture (Kawamata ’02)

If X and Y are Calabi-Yau and birational (isomorphic on a (dense)

• pen algebraic subset), their derived categories are equivalent.

Future Directions: An approach to Kawamata’s conjecture

Theorem (Weak Factorization Theorem, Wlodarczyk ’03)

Suppose X

Y

• are birational projective algebraic varieties.

Then, there exists a diagram of morphisms: X

X1

• Xn
• Y
• such that each triangle is an elementary wall-crossing.

This means there exists spaces Ai equipped with a C∗-action such that Xi = (Ai)+/C∗, Xi+1 = A−

i /C∗.

Theorem (Ballard, Favero, Katzarkov)

Suppose X, Y are smooth Calabi-Yau varieties and A is smooth. Suppose we have an elementary wall-crossing X

Y

• then, D(X) = D(Y ).

Future Directions: An approach to Kawamata’s conjecture

Theorem (Weak Factorization Theorem, Wlodarczyk ’03)

Suppose X

Y

• are birational projective algebraic varieties.

Then, there exists a diagram of morphisms: X

X1

• Xn
• Y
• such that each triangle is an elementary wall-crossing.

This means there exists spaces Ai equipped with a C∗-action such that Xi = (Ai)+/C∗, Xi+1 = A−

i /C∗.

Theorem (Ballard, Favero, Katzarkov)

Suppose X, Y are smooth Calabi-Yau varieties and A is smooth. Suppose we have an elementary wall-crossing X

Y

• then, D(X) = D(Y ).

Future Directions: An approach to Kawamata’s conjecture

Theorem (Weak Factorization Theorem, Wlodarczyk ’03)

Suppose X

Y

• are birational projective algebraic varieties.

Then, there exists a diagram of morphisms: X

X1

• Xn
• Y
• such that each triangle is an elementary wall-crossing.

This means there exists spaces Ai equipped with a C∗-action such that Xi = (Ai)+/C∗, Xi+1 = A−

i /C∗.

Theorem (Ballard, Favero, Katzarkov)

Suppose X, Y are smooth Calabi-Yau varieties and A is smooth. Suppose we have an elementary wall-crossing X

Y

• then, D(X) = D(Y ).

Future Directions: An approach to Kawamata’s conjecture

Suppose now X, Y are smooth Calabi-Yau varieties but A has arbitrary singularities. In joint work with Ballard and Diemer, we use geometric and simplicial methods to construct a functor F : D(X) → D(Y ). We recover the theorem above in the smooth case and ask if it is an equivalence in general. Chidambaram has checked the functor is an equivalence for complete intersection singularities.