[PPT] - Ehrhart theory of subdivisions and mixed Hodge theory Eric Katz PowerPoint Presentation

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Ehrhart theory of subdivisions and mixed Hodge theory

Eric Katz (University of Waterloo) joint with Alan Stapledon (University of Sydney) October 26, 2014

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 1 / 24

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A word on contents...

This talk is about Ehrhart theory. The machinery discussed here naturally belongs in the framework of Eulerian posets.

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 2 / 24

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A word on contents...

This talk is about Ehrhart theory. The machinery discussed here naturally belongs in the framework of Eulerian posets. And when I think about Eulerian posets, I like to think about:

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 2 / 24

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A word on contents...

This talk is about Ehrhart theory. The machinery discussed here naturally belongs in the framework of Eulerian posets. And when I think about Eulerian posets, I like to think about:

1 Motives Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 2 / 24

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A word on contents...

This talk is about Ehrhart theory. The machinery discussed here naturally belongs in the framework of Eulerian posets. And when I think about Eulerian posets, I like to think about:

1 Motives 2 Intersection cohomology Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 2 / 24

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A word on contents...

This talk is about Ehrhart theory. The machinery discussed here naturally belongs in the framework of Eulerian posets. And when I think about Eulerian posets, I like to think about:

1 Motives 2 Intersection cohomology 3 Mixed Hodge theory Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 2 / 24

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Ehrhart polynomials

Let P ⊂ Rn be a lattice polytope. We can encode the lattice point count

f dilates of P in the Ehrhart polynomial.

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 3 / 24

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Ehrhart polynomials

Let P ⊂ Rn be a lattice polytope. We can encode the lattice point count

f dilates of P in the Ehrhart polynomial.

Theorem (Ehrhart) The function LP(m) = |mP ∩ Zn| is a polynomial in m.

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 3 / 24

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Ehrhart polynomials

Let P ⊂ Rn be a lattice polytope. We can encode the lattice point count

f dilates of P in the Ehrhart polynomial.

Theorem (Ehrhart) The function LP(m) = |mP ∩ Zn| is a polynomial in m. Now, we can produce the h∗-polynomial by considering the Ehrhart series E(u) =

∞

m=0

|mP ∩ Zn|um

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 3 / 24

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Ehrhart polynomials

Let P ⊂ Rn be a lattice polytope. We can encode the lattice point count

f dilates of P in the Ehrhart polynomial.

Theorem (Ehrhart) The function LP(m) = |mP ∩ Zn| is a polynomial in m. Now, we can produce the h∗-polynomial by considering the Ehrhart series E(u) =

∞

m=0

|mP ∩ Zn|um and as a consequence of Ehrhart’s theorem, it can be written as E(u) = h∗(u) (1 − u)dimP+1 where h∗(t) is a polynomial.

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 3 / 24

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Ehrhart polynomials

Let P ⊂ Rn be a lattice polytope. We can encode the lattice point count

f dilates of P in the Ehrhart polynomial.

Theorem (Ehrhart) The function LP(m) = |mP ∩ Zn| is a polynomial in m. Now, we can produce the h∗-polynomial by considering the Ehrhart series E(u) =

∞

m=0

|mP ∩ Zn|um and as a consequence of Ehrhart’s theorem, it can be written as E(u) = h∗(u) (1 − u)dimP+1 where h∗(t) is a polynomial. The h∗-polynomial is a wonderful invariant. It takes simple values on simple things. Its value on the standard simplex is 1.

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 3 / 24

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Ehrhart reciprocity

A truly amazing theorem is Ehrhart reciprocity. It relates the generating function for a lattice polytope with that of its interior.

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 4 / 24

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Ehrhart reciprocity

A truly amazing theorem is Ehrhart reciprocity. It relates the generating function for a lattice polytope with that of its interior. Theorem (Ehrhart reciprocity) The function LP(m) obeys LP(−m) = (−1)dim PLP◦(m).

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 4 / 24

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Ehrhart reciprocity

A truly amazing theorem is Ehrhart reciprocity. It relates the generating function for a lattice polytope with that of its interior. Theorem (Ehrhart reciprocity) The function LP(m) obeys LP(−m) = (−1)dim PLP◦(m). Something to think about (when you get unhappy in this talk, and you will get unhappy):

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 4 / 24

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Ehrhart reciprocity

A truly amazing theorem is Ehrhart reciprocity. It relates the generating function for a lattice polytope with that of its interior. Theorem (Ehrhart reciprocity) The function LP(m) obeys LP(−m) = (−1)dim PLP◦(m). Something to think about (when you get unhappy in this talk, and you will get unhappy): What conditions does Ehrhart reciprocity put on h∗? You can express LP◦ in terms of LQ as Q ranges over faces of P and then use inclusion/exclusion. What does that say about h∗?

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 4 / 24

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Subdivsions

Motivating question: what if we have a lattice subdivision of a lattice polytope? We decompose P into smaller lattice polytopes. How can we enrich Ehrhart theory by incoporating the subdivision?

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 5 / 24

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Subdivsions

Motivating question: what if we have a lattice subdivision of a lattice polytope? We decompose P into smaller lattice polytopes. How can we enrich Ehrhart theory by incoporating the subdivision? Here’s a square being subdivided into two triangles.

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 5 / 24

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A geometric avatar for the Ehrhart polynomial

We will try to study subdivisions by relating them to geometry. Let us first study h∗ geometrically. By the work of Danilov-Khovanskii, there’s a geometric interpretation of the h∗-polynomial:

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 6 / 24

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A geometric avatar for the Ehrhart polynomial

We will try to study subdivisions by relating them to geometry. Let us first study h∗ geometrically. By the work of Danilov-Khovanskii, there’s a geometric interpretation of the h∗-polynomial: Let X ◦ = {

u∈M αuxu = 0} ⊂ (C∗)n be a hypersurface. The Newton

polytope P of X ◦ is the convex hull of {u ∈ Zn | αu = 0}. Let us suppose that P is full-dimensional and that X ◦ is non-degenerate with respect to its Newton polytope which means that all of its initial degenerations are smooth.

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 6 / 24

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A geometric avatar for the Ehrhart polynomial

We will try to study subdivisions by relating them to geometry. Let us first study h∗ geometrically. By the work of Danilov-Khovanskii, there’s a geometric interpretation of the h∗-polynomial: Let X ◦ = {

u∈M αuxu = 0} ⊂ (C∗)n be a hypersurface. The Newton

polytope P of X ◦ is the convex hull of {u ∈ Zn | αu = 0}. Let us suppose that P is full-dimensional and that X ◦ is non-degenerate with respect to its Newton polytope which means that all of its initial degenerations are smooth. The h∗-polynomial will arise from looking at dimensions of graded pieces

f the cohomology with compact support, H∗

c (X ◦).

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 6 / 24

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Hodge strucutre

If X ◦ were a compact smooth variety, then its cohomology would have a pure Hodge structure. This implies that there’s a decomposition, Hk(Z) =

p+q=k

Hp,q(Z). Write hp,q = dim Hp,q(Z).

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 7 / 24

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Hodge strucutre

If X ◦ were a compact smooth variety, then its cohomology would have a pure Hodge structure. This implies that there’s a decomposition, Hk(Z) =

p+q=k

Hp,q(Z). Write hp,q = dim Hp,q(Z). It is useful to phrase the decomposition in terms of a decreasing filtration F 0 = Hk ⊃ F1 ⊃ · · · ⊃ Fk such that hp,q = dim Grp

F(Hk).

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 7 / 24

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Mixed Hodge structure

The cohomology H∗

c (X ◦) has a mixed Hodge structure which is a technical

way of saying linear algebra is much much harder than you ever thought

possible. It arises from picking a smooth compactification X ◦.

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 8 / 24

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Mixed Hodge structure

The cohomology H∗

c (X ◦) has a mixed Hodge structure which is a technical

way of saying linear algebra is much much harder than you ever thought

possible. It arises from picking a smooth compactification X ◦.

This implies that there is an increasing filtration W and a decreasing filtration F on Hk

c such that the associated gradeds with respect to W

have a pure Hodge structure induced by F. We define hp,q(Hk

c (X ◦)) = dim Grp F GrW p+q(Hk c (X ◦)).

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 8 / 24

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Mixed Hodge structure

The cohomology H∗

c (X ◦) has a mixed Hodge structure which is a technical

way of saying linear algebra is much much harder than you ever thought

possible. It arises from picking a smooth compactification X ◦.

This implies that there is an increasing filtration W and a decreasing filtration F on Hk

c such that the associated gradeds with respect to W

have a pure Hodge structure induced by F. We define hp,q(Hk

c (X ◦)) = dim Grp F GrW p+q(Hk c (X ◦)).

Warning: Note that we may have hp,q(Hk(Z)) = 0 even though p + q = k. So there’s a lot more data.

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 8 / 24

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Danilov-Khovanskii’s approach

To throw out some of the excess data, we take the Hodge-Deligne numbers ep,q(X ◦) =

k

(−1)khp,q(Hk

c (X ◦)).

We want to find ep,q.

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 9 / 24

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Danilov-Khovanskii’s approach

To throw out some of the excess data, we take the Hodge-Deligne numbers ep,q(X ◦) =

k

(−1)khp,q(Hk

c (X ◦)).

We want to find ep,q. First, ep,q(X ◦) is motivic: if U is an open subset of Z then ep,q(X ◦) = ep,q(U) + ep,q(X ◦ \ U).

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 9 / 24

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Danilov-Khovanskii’s approach

To throw out some of the excess data, we take the Hodge-Deligne numbers ep,q(X ◦) =

k

(−1)khp,q(Hk

c (X ◦)).

We want to find ep,q. First, ep,q(X ◦) is motivic: if U is an open subset of Z then ep,q(X ◦) = ep,q(U) + ep,q(X ◦ \ U). Now, we can form the Hodge-Deligne polynomial, E(X ◦; u, v) =

p,q

ep,qupv q

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 9 / 24

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Danilov-Khovanskii’s approach

To throw out some of the excess data, we take the Hodge-Deligne numbers ep,q(X ◦) =

k

(−1)khp,q(Hk

c (X ◦)).

We want to find ep,q. First, ep,q(X ◦) is motivic: if U is an open subset of Z then ep,q(X ◦) = ep,q(U) + ep,q(X ◦ \ U). Now, we can form the Hodge-Deligne polynomial, E(X ◦; u, v) =

p,q

ep,qupv q It obeys Poincar´ e duality for closed smooth X, E(X; u, v) = (uv)n−1E(X; u−1, v −1).

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 9 / 24

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Danilov-Khovanskii’s approach (cont’d)

One has the Lefschetz hyperplane theorem: for k > n − 1, the natural inclusion i : X ◦ → (C∗)n induces an isomorphism of mixed Hodge structures: i∗ : Hk(X ◦) → Hk+1((C∗)n).

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 10 / 24

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Danilov-Khovanskii’s approach (cont’d)

One has the Lefschetz hyperplane theorem: for k > n − 1, the natural inclusion i : X ◦ → (C∗)n induces an isomorphism of mixed Hodge structures: i∗ : Hk(X ◦) → Hk+1((C∗)n). So the high degree terms of H∗(X ◦) are controlled by Lefschetz hyperplane and the low-degree terms are controlled (after compactification) by Poincar´ e-duality, so you should only care about the middle dimension.

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 10 / 24

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Danilov-Khovanskii’s approach (cont’d)

One has the Lefschetz hyperplane theorem: for k > n − 1, the natural inclusion i : X ◦ → (C∗)n induces an isomorphism of mixed Hodge structures: i∗ : Hk(X ◦) → Hk+1((C∗)n). So the high degree terms of H∗(X ◦) are controlled by Lefschetz hyperplane and the low-degree terms are controlled (after compactification) by Poincar´ e-duality, so you should only care about the middle dimension. We let the primitive cohomology of X ◦ be Hn−1

prim(X ◦) = ker(i∗ : Hn−1(X ◦) → Hn((C∗)n).

The primitive cohomology has a mixed Hodge structure and we can form the E-polynomial.

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 10 / 24

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Danilov-Khovanskii’s approach (cont’d)

One has the Lefschetz hyperplane theorem: for k > n − 1, the natural inclusion i : X ◦ → (C∗)n induces an isomorphism of mixed Hodge structures: i∗ : Hk(X ◦) → Hk+1((C∗)n). So the high degree terms of H∗(X ◦) are controlled by Lefschetz hyperplane and the low-degree terms are controlled (after compactification) by Poincar´ e-duality, so you should only care about the middle dimension. We let the primitive cohomology of X ◦ be Hn−1

prim(X ◦) = ker(i∗ : Hn−1(X ◦) → Hn((C∗)n).

The primitive cohomology has a mixed Hodge structure and we can form the E-polynomial. After hard work, we get a formula for χy of the primitive cohomlogy of X ◦: uEprim(X ◦; u, 1) = (−1)dim P+1h∗

P(u).

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 10 / 24

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Batyrev-Borisov’s approach

There’s a big downside to the above approach. It’s hard to get a handle

n the full E-polynomial. The combinatorics of picking a smooth

compactification is really unmanageable, so let’s follow Batyrev-Borisov and use intersection cohomology which is a technical way of saying no one expects you to read the papers. Less jocularly, it says that we can pretend things are smooth if we keep track of singularities. In the smooth case, it agrees with usual cohomology.

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 11 / 24

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Batyrev-Borisov’s approach

There’s a big downside to the above approach. It’s hard to get a handle

n the full E-polynomial. The combinatorics of picking a smooth

compactification is really unmanageable, so let’s follow Batyrev-Borisov and use intersection cohomology which is a technical way of saying no one expects you to read the papers. Less jocularly, it says that we can pretend things are smooth if we keep track of singularities. In the smooth case, it agrees with usual cohomology. We have X ◦ ⊂ (C∗)n. We can compactify (C∗)n to P, the toric variety associated to the Newton polytope P. Let X be the closure of X ◦ in P. It may not be smooth but by nondegeneracy, each open strata of X ◦

Q which

is associated to a face Q ⊆ P is smooth.

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 11 / 24

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Batyrev-Borisov’s approach

There’s a big downside to the above approach. It’s hard to get a handle

n the full E-polynomial. The combinatorics of picking a smooth

compactification is really unmanageable, so let’s follow Batyrev-Borisov and use intersection cohomology which is a technical way of saying no one expects you to read the papers. Less jocularly, it says that we can pretend things are smooth if we keep track of singularities. In the smooth case, it agrees with usual cohomology. We have X ◦ ⊂ (C∗)n. We can compactify (C∗)n to P, the toric variety associated to the Newton polytope P. Let X be the closure of X ◦ in P. It may not be smooth but by nondegeneracy, each open strata of X ◦

Q which

is associated to a face Q ⊆ P is smooth. So there’s a black box IH∗

c (X). It has a lot of good properties like the

Lefschetz hyperplane theorem and Poincar´ e duality, but it is not motivic.

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 11 / 24

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Sum-over-strata formula

What replaces motivicity is a sum-over-strata formula that incorporates singularities: Eint(X; u, v) =

Q⊆P

Q=∅

Eint(X ◦

Q)g([Q, P]∗; uv).

Here g([Q, P]∗) is the g-polynomial of the dual Eulerian poset [Q, P]∗. It is exactly the same term that enters into the toric h-vector, and it knows about the singularities along strata. We can invert this formula to solve for Eint(X ◦).

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 12 / 24

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Sum-over-strata formula

What replaces motivicity is a sum-over-strata formula that incorporates singularities: Eint(X; u, v) =

Q⊆P

Q=∅

Eint(X ◦

Q)g([Q, P]∗; uv).

Here g([Q, P]∗) is the g-polynomial of the dual Eulerian poset [Q, P]∗. It is exactly the same term that enters into the toric h-vector, and it knows about the singularities along strata. We can invert this formula to solve for Eint(X ◦). Oh yeah, Ehrhart reciprocity for X ◦ becomes Poincar´ e-duality for Eint,prim(X; u, 1)!

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 12 / 24

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Local h∗-polynomial

And Eint,prim(X; u, 1) is a cool invariant of the polytope P. It’s called the local h∗-polynomial and it is denoted l∗. Here local will always refer to invariants of closed varieties. That’s an unfortunate naming convention that predates the geometric picture. l∗(P; u) =

Q⊆P

(−1)dim P−dim Qh∗(Q; u)g([Q, P]∗; u). It is symmetric and its coefficients are manifestly non-negative. It’s also been called the ˜ S-polynomial of P.

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 13 / 24

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Local h∗-polynomial

And Eint,prim(X; u, 1) is a cool invariant of the polytope P. It’s called the local h∗-polynomial and it is denoted l∗. Here local will always refer to invariants of closed varieties. That’s an unfortunate naming convention that predates the geometric picture. l∗(P; u) =

Q⊆P

(−1)dim P−dim Qh∗(Q; u)g([Q, P]∗; u). It is symmetric and its coefficients are manifestly non-negative. It’s also been called the ˜ S-polynomial of P. To find E(X ◦; u, v), you use Danilov-Khovanskii to find E(X ◦; u, 1), sum-over-strata to find Eint(X; u, 1), use the pure Hodge structure to find E(X; u, v), and then invert the sum-over-strata formula.

Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 13 / 24

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