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
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
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
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
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
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
Ehrhart polynomials Let P ⊂ R n be a lattice polytope. We can encode the lattice point count of dilates of P in the Ehrhart polynomial. Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 3 / 24
Ehrhart polynomials Let P ⊂ R n be a lattice polytope. We can encode the lattice point count of dilates of P in the Ehrhart polynomial. Theorem (Ehrhart) The function L P ( m ) = | mP ∩ Z n | is a polynomial in m . Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 3 / 24
Ehrhart polynomials Let P ⊂ R n be a lattice polytope. We can encode the lattice point count of dilates of P in the Ehrhart polynomial. Theorem (Ehrhart) The function L P ( m ) = | mP ∩ Z n | is a polynomial in m . Now, we can produce the h ∗ -polynomial by considering the Ehrhart series ∞ � | mP ∩ Z n | u m E ( u ) = m =0 Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 3 / 24
Ehrhart polynomials Let P ⊂ R n be a lattice polytope. We can encode the lattice point count of dilates of P in the Ehrhart polynomial. Theorem (Ehrhart) The function L P ( m ) = | mP ∩ Z n | is a polynomial in m . Now, we can produce the h ∗ -polynomial by considering the Ehrhart series ∞ � | mP ∩ Z n | u m E ( u ) = m =0 and as a consequence of Ehrhart’s theorem, it can be written as h ∗ ( u ) E ( u ) = (1 − u ) dimP +1 where h ∗ ( t ) is a polynomial. Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 3 / 24
Ehrhart polynomials Let P ⊂ R n be a lattice polytope. We can encode the lattice point count of dilates of P in the Ehrhart polynomial. Theorem (Ehrhart) The function L P ( m ) = | mP ∩ Z n | is a polynomial in m . Now, we can produce the h ∗ -polynomial by considering the Ehrhart series ∞ � | mP ∩ Z n | u m E ( u ) = m =0 and as a consequence of Ehrhart’s theorem, it can be written as h ∗ ( u ) E ( 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
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
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 L P ( m ) obeys L P ( − m ) = ( − 1) dim P L P ◦ ( m ) . Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 4 / 24
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 L P ( m ) obeys L P ( − m ) = ( − 1) dim P L P ◦ ( 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
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 L P ( m ) obeys L P ( − m ) = ( − 1) dim P L P ◦ ( 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 L P ◦ in terms of L Q 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
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
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
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
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 α u x u = 0 } ⊂ ( C ∗ ) n be a hypersurface. The Newton polytope P of X ◦ is the convex hull of { u ∈ Z n | α u � = 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
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 α u x u = 0 } ⊂ ( C ∗ ) n be a hypersurface. The Newton polytope P of X ◦ is the convex hull of { u ∈ Z n | α u � = 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 of the cohomology with compact support, H ∗ c ( X ◦ ). Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 6 / 24
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, H k ( Z ) = � H p , q ( Z ) . p + q = k Write h p , q = dim H p , q ( Z ) . Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 7 / 24
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, H k ( Z ) = � H p , q ( Z ) . p + q = k Write h p , q = dim H p , q ( Z ) . It is useful to phrase the decomposition in terms of a decreasing filtration F 0 = H k ⊃ F 1 ⊃ · · · ⊃ F k such that h p , q = dim Gr p F ( H k ) . Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 7 / 24
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
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 H k c such that the associated gradeds with respect to W have a pure Hodge structure induced by F . We define c ( X ◦ )) = dim Gr p c ( X ◦ )) . h p , q ( H k F Gr W p + q ( H k Eric Katz (Waterloo) Mixed Ehrhart Theory October 26, 2014 8 / 24
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