Cluster varieties, toric degenerations, and mirror symmetry

Cluster varieties, toric degenerations, and mirror symmetry Timothy Magee Instituto de Matemáticas de la Univesidad Nacional Autónoma de México Unidad Oaxaca Ongoing joint work with Lara Bossinger, Juan Bosco Frías Medina, and Alfredo Nájera Chávez

Outline 1 Cluster varieties and their compactifications 2 Toric degenerations of cluster varieties 3 Connections to Batyrev-Borisov mirror symmetry

Moral definitions and examples

Moral definitions and examples Calabi-Yau variety A complex projective variety with a notion of complex volume

Moral definitions and examples Calabi-Yau variety A complex projective variety with a notion of complex volume Example Compact torus C/Z 2, Ω = dz

Moral definitions and examples Log Calabi-Yau variety A smooth complex variety U with a unique volume form Ω having at worst a simple pole along any divisor in any compactification of U

Moral definitions and examples Log Calabi-Yau variety A smooth complex variety U with a unique volume form Ω having at worst a simple pole along any divisor in any compactification of U Example Algebraic torus T = (C ) n, Ω = dz 1 z 1 dzn z n

Moral definitions and examples Log Calabi-Yau variety A smooth complex variety U with a unique volume form Ω having at worst a simple pole along any divisor in any compactification of U Example Algebraic torus T = (C ) n, Ω = dz 1 z 1 dzn z n Example Carefully glued tori U = i T i / µ ij : T i T j, µ ij (Ω j ) = Ω i

Moral definitions and examples Log Calabi-Yau variety A smooth complex variety U with a unique volume form Ω having at worst a simple pole along any divisor in any compactification of U Example Algebraic torus T = (C ) n, Ω = dz 1 z 1 dzn z n Example Carefully glued tori U = i T i / µ ij : T i T j, µ ij (Ω j ) = Ω i Cluster variety

Cluster varieties

Cluster varieties Initial data Lattice N = Z n Skew-form {, } : N N Z Basis s = (e 1,...,e n ) of N

Cluster varieties Initial data Lattice N = Z n Skew-form {, } : N N Z Basis s = (e 1,...,e n ) of N Gives a pair of dual tori and bases for their character lattices: T M;s := Spec ( C [ z ±e 1,...,z ±en]) ( [ ]) T N;s := Spec C z ±e 1,...,z ±e n

Cluster varieties Initial data Lattice N = Z n Skew-form {, } : N N Z Basis s = (e 1,...,e n ) of N Gives a pair of dual tori and bases for their character lattices: T M;s := Spec ( C [ z ±e 1,...,z ±en]) ( [ ]) T N;s := Spec C z ±e 1,...,z ±e n Recursive rule for changing basis s s, together with birational maps µ s,s : T M;s T M;s and µ s,s : T N;s T N;s

Cluster varieties

Cluster varieties Two flavors of cluster varieties X = s T M;s, A = s T N;s

Cluster varieties Two flavors of cluster varieties X = s T M;s, A = s T N;s X has a Poisson structure: {z n 1,z n 2 } = {n 1,n 2 }z n 1+n 2 A has a degenerate symplectic structure, with 2-form: i,j dz e i z e i dze j z e j

(Partial) compactification of X

(Partial) compactification of X Cluster varieties can be encoded with scattering diagrams.

(Partial) compactification of X Cluster varieties can be encoded with scattering diagrams. Example (A 2 cluster varieties) C 2 C 1 C 3 C 4 C 5 M s R

(Partial) compactification of X Cluster varieties can be encoded with scattering diagrams. Example (A 2 cluster varieties) C 2 C 1 C 3 C 4 C 5 M s R In general the cluster varieties will be encoded in only part of the scattering diagram, called the cluster complex.

(Partial) compactification of X Cluster varieties can be encoded with scattering diagrams. Example (A 2 cluster varieties) C 2 C 1 C 3 C 4 C 5 M s R In general the cluster varieties will be encoded in only part of the scattering diagram, called the cluster complex. Theorem ([GHKK16]) The cluster complex + has a simplicial fan structure.

(Partial) compactification of X

(Partial) compactification of X Note: X = i Spec ( C [ C i N ] gp )

(Partial) compactification of X Note: X = i Spec ( C [ C i N ] gp ) Much more natural object from toric geometry perspective: Special completion of X [FG15] X = i Spec ( C [ C i N ])

(Partial) compactification of A

(Partial) compactification of A Often A varieties come frozen variables basis functions shared by each torus.

(Partial) compactification of A Often A varieties come frozen variables basis functions shared by each torus. In this case, allowing these variables to vanish gives a partial compactification.

(Partial) compactification of A Often A varieties come frozen variables basis functions shared by each torus. In this case, allowing these variables to vanish gives a partial compactification. Example (Affine cone over Gr k (C n )) For each cyclic degree k Plücker define D i := { p [i+1,i+k] = 0 }.

(Partial) compactification of A Often A varieties come frozen variables basis functions shared by each torus. In this case, allowing these variables to vanish gives a partial compactification. Example (Affine cone over Gr k (C n )) For each cyclic degree k Plücker define D i := { p [i+1,i+k] = 0 }. C(Gr k (C n ))\ i D i is a cluster A-variety.

(Partial) compactification of A Often A varieties come frozen variables basis functions shared by each torus. In this case, allowing these variables to vanish gives a partial compactification. Example (Affine cone over Gr k (C n )) For each cyclic degree k Plücker define D i := { p [i+1,i+k] = 0 }. C(Gr k (C n ))\ i D i is a cluster A-variety. Each p [i+1,i+k] is a frozen variable, so C(Gr k (C n )) is such a compactification.

Terminology

Terminology The generators of C s are called g-vectors for the A-variables A i;s on the torus T N;s. They are denoted g i;s.

Terminology The generators of C s are called g-vectors for the A-variables A i;s on the torus T N;s. They are denoted g i;s. The generators of C s are called c-vectors for the X-variables X i;s on the torus T M;s. They are denoted c i;s.

Terminology Note: The generators of C s are called g-vectors for the A-variables A i;s on the torus T N;s. They are denoted g i;s. The generators of C s are called c-vectors for the X-variables X i;s on the torus T M;s. They are denoted c i;s. If we restrict to the tori T N;s A and T M;s X, then A i;s = z e i;s and X i;s = z e i;s.

Terminology Note: The generators of C s are called g-vectors for the A-variables A i;s on the torus T N;s. They are denoted g i;s. The generators of C s are called c-vectors for the X-variables X i;s on the torus T M;s. They are denoted c i;s. If we restrict to the tori T N;s A and T M;s X, then A i;s = z e i;s and X i;s = z e i;s. On other tori T N;s and T M;s, µ s,s (A i;s) and µ s,s (X i;s) will be rational functions in the variables A j;s and X j;s.

Toric degenerations of X

Toric degenerations of X What should we demand of a decent toric degeneration of X?

Toric degenerations of X What should we demand of a decent toric degeneration of X? Central fiber is TV ( + ).

Toric degenerations of X What should we demand of a decent toric degeneration of X? Central fiber is TV ( + ). X strata go to toric strata of TV ( + ).

Toric degenerations of X What should we demand of a decent toric degeneration of X? Central fiber is TV ( + ). X strata go to toric strata of TV ( + ). What additional properties would be nice?

Toric degenerations of X What should we demand of a decent toric degeneration of X? Central fiber is TV ( + ). X strata go to toric strata of TV ( + ). What additional properties would be nice? X-variables extend canonically to family.

Toric degenerations of X What should we demand of a decent toric degeneration of X? Central fiber is TV ( + ). X strata go to toric strata of TV ( + ). What additional properties would be nice? X-variables extend canonically to family. Extension of X i;s homogeneous under some natural torus action, with weight c i;s.

Toric degenerations of X

Toric degenerations of X Basic idea: Add coefficients carefully to get X t Xs0 = s Spec(R[X 1;s,...,X n;s ]) A n s 0 = Spec(R) where R := C[t 1;s0,...,t n;s0 ].

Toric degenerations of X Basic idea: Add coefficients carefully to get X t Xs0 = s Spec(R[X 1;s,...,X n;s ]) A n s 0 = Spec(R) where R := C[t 1;s0,...,t n;s0 ]. The fiber over 1 will be X. The fiber over 0 will be TV( + ).

Toric degenerations of X

Toric degenerations of X Define ε ij := {e i,e j } and [a] + := max{a,0}.

Toric degenerations of X Define ε ij := {e i,e j } and [a] + := max{a,0}. X gluing defined by µ k ( X i ) = X 1 k X i (1+X sgnε ik k if i = k ) εik if i k

Toric degenerations of X Define ε ij := {e i,e j } and [a] + := max{a,0}. X gluing defined by µ k ( X i ) = X 1 k X i (1+X sgnε ik k if i = k ) εik if i k We define X s0 gluing by µ k ( X i ) = X 1 k X i (t [sgnε ijc k ] + +t [ sgnε ij c k ] +X sgnε ik k if i = k ) εik if i k

Toric degenerations of X Theorem (Bossinger, Frías Medina, Nájera Chávez, M.) With this gluing, Xs0 satisfies all of the properties we asked our toric degenerations to satisfy.

Toric degenerations of compactified A-varieties

Toric degenerations of compactified A-varieties In [GHKK16] an analogous family is defined for A-varieties compactified by letting frozen variables vanish. A t A prin,s0 A n s 0 = Spec(R) where R := C[t 1;s0,...,t n;s0 ].

Batyrev-Borisov mirror symmetry Goal of Batyrev-Borisov: Construct mirror families of Calabi-Yau varieties living in dual toric varieties.

Batyrev-Borisov mirror symmetry Goal of Batyrev-Borisov: Construct mirror families of Calabi-Yau varieties living in dual toric varieties.

Batyrev-Borisov mirror symmetry Goal of Batyrev-Borisov: Construct mirror families of Calabi-Yau varieties living in dual toric varieties.

Batyrev-Borisov picture for Grassmannians

Batyrev-Borisov picture for Grassmannians Example (Gr 2 (C 5 )) Start with A 2 scattering diagram: C 2 C 1 C 3 C 4 C 5 M s R

Batyrev-Borisov picture for Grassmannians Example (Gr 2 (C 5 )) Add on 5 dimensional linear space, one dimension for each consecutive Plücker: C 2 C 1 R C 5 C 5 3 C 4

Batyrev-Borisov picture for Grassmannians Example (Gr 2 (C 5 )) Cut out the cone Ξ generated by the columns of the following matrix: 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1

Batyrev-Borisov picture for Grassmannians Example (Gr 2 (C 5 )) Have a fan structure Σ on Ξ.

Batyrev-Borisov picture for Grassmannians Example (Gr 2 (C 5 )) Have a fan structure Σ on Ξ. Relevant slice of Ξ: P = {x Ξ (1 1 1 1 1 1 1),x = 5}. Unique interior point: v = (0 0 1 1 1 1 1) T.

Batyrev-Borisov picture for Grassmannians Example (Gr 2 (C 5 )) Have a fan structure Σ on Ξ. Relevant slice of Ξ: P = {x Ξ (1 1 1 1 1 1 1),x = 5}. Unique interior point: v = (0 0 1 1 1 1 1) T. Translate generators of Σ by v to get a new fan. Σ is cone over, and the rays of are generated by vertices of P.

Batyrev-Borisov picture for Grassmannians Expectations: is the fan for the Batyrev-Borisov dual Y to the Grassmannian.

Batyrev-Borisov picture for Grassmannians Expectations: is the fan for the Batyrev-Borisov dual Y to the Grassmannian. Σ is the fan for a line bundle over Y. (So X is this line bundle.)

Batyrev-Borisov picture for Grassmannians Expectations: is the fan for the Batyrev-Borisov dual Y to the Grassmannian. Σ is the fan for a line bundle over Y. (So X is this line bundle.) Remark: The construction works the same way for all Gr k (C n ).

Batyrev-Borisov picture for Grassmannians Expectations: Remark: is the fan for the Batyrev-Borisov dual Y to the Grassmannian. Σ is the fan for a line bundle over Y. (So X is this line bundle.) The construction works the same way for all Gr k (C n ). A generalization (higher dimensional version of cone over higher codimension CY subvarieties) works for complete flag varieties.

References [FG15] V. Fock and A. Goncharov, Cluster X-varieties at infinity, (2015), arxiv:1104.0407v2 [math.ag]. [GHKK16] M. Gross, P. Hacking, S. Keel and M. Kontsevich, Canonical bases for cluster algebras, preprint (2016), arxiv:1411.1394v2 [math.ag].