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Reflexive polytopes, Gorenstein polytopes, and combinatorial mirror symmetry

Benjamin Nill

U Kentucky 10/04/10

Benjamin Nill (U Georgia) Gorenstein polytopes 1 / 46

Goals of this talk:

Convince you that (reflexive &) Gorenstein polytopes

1 turn up naturally

2 consist of interesting examples

3 have fascinating and not yet understood properties

4 have relations to

Combinatorics, Algebraic Geometry, Topology, Statistics

Benjamin Nill (U Georgia) Gorenstein polytopes 2 / 46

Goals of this talk:

Convince you that (reflexive &) Gorenstein polytopes

1 turn up naturally

2 consist of interesting examples

3 have fascinating and not yet understood properties

4 have relations to

Combinatorics,

Algebraic Geometry, Topology, Statistics

Benjamin Nill (U Georgia) Gorenstein polytopes 2 / 46

Goals of this talk:

Convince you that (reflexive &) Gorenstein polytopes

1 turn up naturally

2 consist of interesting examples

3 have fascinating and not yet understood properties

4 have relations to

Combinatorics, Algebraic Geometry,

Topology, Statistics

Benjamin Nill (U Georgia) Gorenstein polytopes 2 / 46

Goals of this talk:

Convince you that (reflexive &) Gorenstein polytopes

1 turn up naturally

2 consist of interesting examples

3 have fascinating and not yet understood properties

4 have relations to

Combinatorics, Algebraic Geometry, Topology,

Statistics

Benjamin Nill (U Georgia) Gorenstein polytopes 2 / 46

Goals of this talk:

Convince you that (reflexive &) Gorenstein polytopes

1 turn up naturally

2 consist of interesting examples

3 have fascinating and not yet understood properties

4 have relations to

Combinatorics, Algebraic Geometry, Topology, Statistics

Benjamin Nill (U Georgia) Gorenstein polytopes 2 / 46

I. Reflexive polytopes

Benjamin Nill (U Georgia) Gorenstein polytopes 3 / 46

Combinatorial polytopes and duality

Combinatorial types of polytopes Isomorphisms: combinatorially isomorphic face posets

Benjamin Nill (U Georgia) Gorenstein polytopes 4 / 46

Combinatorial polytopes and duality

Combinatorial types of polytopes Isomorphisms: combinatorially isomorphic face posets

Benjamin Nill (U Georgia) Gorenstein polytopes 5 / 46

Realized polytopes and duality

Embedded polytopes: P ⊂ Rd Isomorphisms: affine isomorphisms

P ⊂ Rd d-polytope with interior point 0 =⇒

P∗ := {y ∈ (Rd)∗ : 〈y , x〉 ≥ −1 ∀ x ∈ P}

Benjamin Nill (U Georgia) Gorenstein polytopes 6 / 46

Realized polytopes and duality

Embedded polytopes: P ⊂ Rd Isomorphisms: affine isomorphisms

P ⊂ Rd d-polytope with interior point 0 =⇒

P∗ := {y ∈ (Rd)∗ : 〈y , x〉 ≥ −1 ∀ x ∈ P}

Benjamin Nill (U Georgia) Gorenstein polytopes 6 / 46

Lattice polytopes and duality

Lattice polytopes: P = conv(m1, . . . ,mk) for mi ∈ Zd isomorphisms: affine lattice isomorphisms of Zd (unimodular equivalence)

Definition (Batyrev ’94)

A reflexive polytope is a lattice polytope P with 0 ∈ int(P) such that P∗ is also a lattice polytope.

origin only interior lattice point.

Benjamin Nill (U Georgia) Gorenstein polytopes 7 / 46

Lattice polytopes and duality

Lattice polytopes: P = conv(m1, . . . ,mk) for mi ∈ Zd isomorphisms: affine lattice isomorphisms of Zd (unimodular equivalence)

Definition (Batyrev ’94)

A reflexive polytope is a lattice polytope P with 0 ∈ int(P) such that P∗ is also a lattice polytope.

origin only interior lattice point.

Benjamin Nill (U Georgia) Gorenstein polytopes 7 / 46

Lattice polytopes and duality

Lattice polytopes: P = conv(m1, . . . ,mk) for mi ∈ Zd isomorphisms: affine lattice isomorphisms of Zd (unimodular equivalence)

Definition (Batyrev ’94)

A reflexive polytope is a lattice polytope P with 0 ∈ int(P) such that P∗ is also a lattice polytope.

origin only interior lattice point.

Benjamin Nill (U Georgia) Gorenstein polytopes 7 / 46

Reflexive polytopes

Facts 1 [Lagarias/Ziegler ’91]: In each dimension only finitely many reflexive

polytopes up to lattice isomorphisms.

2 [Haase/Melnikov ’06]: Any lattice polytope is a face of a (higher-dimensional) reflexive polytope.

3 [Kreuzer/Skarke ’98-00]: Tons of them: d 2 3 4

# 16 4,319 473,800,776

4 Even basic questions are open: maximal number of vertices? d 2 3 4

vertices ≤ 6 14 36

Benjamin Nill (U Georgia) Gorenstein polytopes 8 / 46

Reflexive polytopes

Facts 1 [Lagarias/Ziegler ’91]: In each dimension only finitely many reflexive

polytopes up to lattice isomorphisms.

2 [Haase/Melnikov ’06]: Any lattice polytope is a face of a (higher-dimensional) reflexive polytope.

3 [Kreuzer/Skarke ’98-00]: Tons of them: d 2 3 4

# 16 4,319 473,800,776

4 Even basic questions are open: maximal number of vertices? d 2 3 4

vertices ≤ 6 14 36

Benjamin Nill (U Georgia) Gorenstein polytopes 8 / 46

Reflexive polytopes

Facts 1 [Lagarias/Ziegler ’91]: In each dimension only finitely many reflexive

polytopes up to lattice isomorphisms.

2 [Haase/Melnikov ’06]: Any lattice polytope is a face of a (higher-dimensional) reflexive polytope.

3 [Kreuzer/Skarke ’98-00]: Tons of them: d 2 3 4

# 16 4,319 473,800,776

4 Even basic questions are open: maximal number of vertices? d 2 3 4

vertices ≤ 6 14 36

Benjamin Nill (U Georgia) Gorenstein polytopes 8 / 46

Reflexive polytopes

Facts 1 [Lagarias/Ziegler ’91]: In each dimension only finitely many reflexive

polytopes up to lattice isomorphisms.

2 [Haase/Melnikov ’06]: Any lattice polytope is a face of a (higher-dimensional) reflexive polytope.

3 [Kreuzer/Skarke ’98-00]: Tons of them: d 2 3 4

# 16 4,319 473,800,776

4 Even basic questions are open:

maximal number of vertices? d 2 3 4

vertices ≤ 6 14 36

Benjamin Nill (U Georgia) Gorenstein polytopes 8 / 46

Reflexive polytopes

Facts 1 [Lagarias/Ziegler ’91]: In each dimension only finitely many reflexive

polytopes up to lattice isomorphisms.

2 [Haase/Melnikov ’06]: Any lattice polytope is a face of a (higher-dimensional) reflexive polytope.

3 [Kreuzer/Skarke ’98-00]: Tons of them: d 2 3 4

# 16 4,319 473,800,776

4 Even basic questions are open: maximal number of vertices?

d 2 3 4

vertices ≤ 6 14 36

Benjamin Nill (U Georgia) Gorenstein polytopes 8 / 46

Reflexive polytopes

Facts 1 [Lagarias/Ziegler ’91]: In each dimension only finitely many reflexive

polytopes up to lattice isomorphisms.

2 [Haase/Melnikov ’06]: Any lattice polytope is a face of a (higher-dimensional) reflexive polytope.

3 [Kreuzer/Skarke ’98-00]: Tons of them: d 2 3 4

# 16 4,319 473,800,776

4 Even basic questions are open: maximal number of vertices? d 2 3 4

vertices ≤ 6 14 36

Benjamin Nill (U Georgia) Gorenstein polytopes 8 / 46

Reflexive polytopes

Let P be a lattice polytope with 0 in its interior.

Definition

P is reflexive if and only if

each vertex is a primitive lattice point.

`P∗ `-reflexive and P = `(`P∗)∗.

Duality of `-reflexive polytopes!

Benjamin Nill (U Georgia) Gorenstein polytopes 9 / 46

Reflexive polytopes

Let P be a lattice polytope with 0 in its interior.

Definition

P is reflexive if and only if

each facet F has lattice distance 1 from the origin,

each vertex is a primitive lattice point.

`P∗ `-reflexive and P = `(`P∗)∗.

Duality of `-reflexive polytopes!

Benjamin Nill (U Georgia) Gorenstein polytopes 9 / 46

Reflexive polytopes

Let P be a lattice polytope with 0 in its interior.

Definition

P is reflexive of Gorenstein index 1 if and only if

each facet F has lattice distance 1 from the origin,

each vertex is a primitive lattice point.

`P∗ `-reflexive and P = `(`P∗)∗.

Duality of `-reflexive polytopes!

Benjamin Nill (U Georgia) Gorenstein polytopes 9 / 46

Reflexive polytopes of higher index! (Joint work with A. Kasprzyk)

Let P be a lattice polytope with 0 in its interior.

Definition (Kasprzyk/N. ’10)

P is reflexive of Gorenstein index ` if and only if

each facet F has lattice distance ` from the origin,

each vertex is a primitive lattice point.

`P∗ `-reflexive and P = `(`P∗)∗.

Duality of `-reflexive polytopes!

Benjamin Nill (U Georgia) Gorenstein polytopes 9 / 46

Reflexive polytopes of higher index! (Joint work with A. Kasprzyk)

Let P be a lattice polytope with 0 in its interior.

Definition (Kasprzyk/N. ’10)