Hard Lefschetz theorem and Hodge-Riemann relations for combinatorial - - PowerPoint PPT Presentation

Hard Lefschetz theorem and Hodge-Riemann relations for combinatorial geometries

June Huh

Institute for Advanced Study and Princeton University

with Karim Adiprasito and Eric Katz

June Huh 1 / 26

A graph is a ✶-dimensional space, with vertices and edges. Graphs are the simplest combinatorial structures.

June Huh 2 / 26

Hassler Whitney (1932): The chromatic polynomial of a graph ● is the function ✤●✭q✮ ❂ ✭the number of proper colorings of vertices of ● with q colors✮✿ ✎ ✎ ✎ ✎ ✤●✭q✮ ❂ ✶q✹ ✺q✸ ✰ ✽q✷ ✹q❀ ✤●✭✷✮ ❂ ✵❀ ✤●✭✸✮ ❂ ✻❀ ✿ ✿ ✿ ✤●✭q✮

• ❛✷

✐ ✕ ❛✐✶❛✐✰✶

✐

June Huh 3 / 26

Hassler Whitney (1932): The chromatic polynomial of a graph ● is the function ✤●✭q✮ ❂ ✭the number of proper colorings of vertices of ● with q colors✮✿

Example

✎ ✎ ✎ ✎ ✤●✭q✮ ❂ ✶q✹ ✺q✸ ✰ ✽q✷ ✹q❀ ✤●✭✷✮ ❂ ✵❀ ✤●✭✸✮ ❂ ✻❀ ✿ ✿ ✿ What can be said about the chromatic polynomial in general? ✤●✭q✮

• ❛✷

✐ ✕ ❛✐✶❛✐✰✶

✐

June Huh 3 / 26

Hassler Whitney (1932): The chromatic polynomial of a graph ● is the function ✤●✭q✮ ❂ ✭the number of proper colorings of vertices of ● with q colors✮✿

Example

✎ ✎ ✎ ✎ ✤●✭q✮ ❂ ✶q✹ ✺q✸ ✰ ✽q✷ ✹q❀ ✤●✭✷✮ ❂ ✵❀ ✤●✭✸✮ ❂ ✻❀ ✿ ✿ ✿

Read-Hoggar conjecture (1968,1974)

The coefficients of the chromatic polynomial ✤●✭q✮ form a log-concave sequence for any graph ●, that is, ❛✷

✐ ✕ ❛✐✶❛✐✰✶ for all ✐.

June Huh 3 / 26

Example

How do we compute the chromatic polynomial? We write ✎ ✎ ✎ ✎ = ✎ ✎ ✎ ✎

• ✎

✎ ✎ and use ✤●✭q✮ ❂ ✤●♥❡✭q✮ ✤●❂❡✭q✮✿ ✤●♥❡✭q✮ ❂ ✶q✹ ✹q✸ ✰ ✻q✷ ✸q ✤●❂❡✭q✮ ❂ ✶q✸ ✷q✷ ✰ q❀ ✤●✭q✮ ❂ ✤●♥❡✭q✮ ✤●❂❡✭q✮ ❂ ✶q✹ ✺q✸ ✰ ✽q✷ ✹q✿ ✤●✭q✮

June Huh 4 / 26

Example

How do we compute the chromatic polynomial? We write ✎ ✎ ✎ ✎ = ✎ ✎ ✎ ✎

• ✎

✎ ✎ and use ✤●✭q✮ ❂ ✤●♥❡✭q✮ ✤●❂❡✭q✮✿ From the calculation ✤●♥❡✭q✮ ❂ ✶q✹ ✹q✸ ✰ ✻q✷ ✸q ✤●❂❡✭q✮ ❂ ✶q✸ ✷q✷ ✰ q❀ Therefore ✤●✭q✮ ❂ ✤●♥❡✭q✮ ✤●❂❡✭q✮ ❂ ✶q✹ ✺q✸ ✰ ✽q✷ ✹q✿ This algorithmic description of ✤●✭q✮ makes the prediction of the conjecture interesting.

June Huh 4 / 26

For any finite set of vectors ❆ in a vector space over a field, define ❢✐✭❆✮ ❂ ✭number of independent subsets of ❆ with size ✐✮✿

Example

If ❆ is the set of all nonzero vectors in F✸

✷, then

❢✵ ❂ ✶❀ ❢✶ ❂ ✼❀ ❢✷ ❂ ✷✶❀ ❢✸ ❂ ✷✽✿ ❢✐✭❆✮ ❢✐✭❆✮ ❂ ❢✐✭❆ ♥ ✈✮ ✰ ❢✐✶✭❆ ❂ ✈✮✿

June Huh 5 / 26

For any finite set of vectors ❆ in a vector space over a field, define ❢✐✭❆✮ ❂ ✭number of independent subsets of ❆ with size ✐✮✿

Example

If ❆ is the set of all nonzero vectors in F✸

✷, then

❢✵ ❂ ✶❀ ❢✶ ❂ ✼❀ ❢✷ ❂ ✷✶❀ ❢✸ ❂ ✷✽✿ How do we compute ❢✐✭❆✮? We use ❢✐✭❆✮ ❂ ❢✐✭❆ ♥ ✈✮ ✰ ❢✐✶✭❆ ❂ ✈✮✿

June Huh 5 / 26

Welsh-Mason conjecture (1971,1972)

The sequence ❢✐ form a log-concave sequence for any finite set of vectors ❆ in any vector space over any field, that is, ❢ ✷

✐ ✕ ❢✐✶ ❢✐✰✶ for all ✐.

June Huh 6 / 26

Definition of matroids

Whitney, Nakasawa, Birkhoff (1935), Mac Lane (1936), van der Waerden (1937) . . . Let ❊ be a finite set. ▼ ❊ ❊ ❋✶ ❋✷ ❋✶ ❭ ❋✷ ❋ ❋ ❊ ♥ ❋

June Huh 7 / 26

Definition of matroids

Whitney, Nakasawa, Birkhoff (1935), Mac Lane (1936), van der Waerden (1937) . . . Let ❊ be a finite set. A matroid ▼ on ❊ is a collection of subsets of ❊, called (proper) flats, satisfying: (1) If ❋✶ and ❋✷ are flats, then ❋✶ ❭ ❋✷ is a flat, (2) If ❋ is a flat, then the flats covering ❋ forms a partition of ❊ ♥ ❋.

June Huh 7 / 26

We write ♥ ✰ ✶ for the size of ▼, the cardinality of the ground set ❊. We write r ✰ ✶ for the rank of ▼, the height of the poset of flats of ▼. In all interesting cases (except one), r ❁ ♥.

June Huh 8 / 26

Example

A projective space is a set with distinguished subsets, called lines, satisfying: (1) Any two distinct points are in exactly one line. (2) Each line contains more than two points. (3) If ①❀ ②❀ ③❀ ✇ are distinct points, no three colinear, then ①② intersects ③✇ ❂✮ ①③ intersects ②✇✿ ❦ ❦

June Huh 9 / 26

Example

A projective space is a set with distinguished subsets, called lines, satisfying: (1) Any two distinct points are in exactly one line. (2) Each line contains more than two points. (3) If ①❀ ②❀ ③❀ ✇ are distinct points, no three colinear, then ①② intersects ③✇ ❂✮ ①③ intersects ②✇✿ A projective space has a structure of flats (subspaces), and this structure is inherited by any of its finite subset, defining a matroid. ❦ ❦

June Huh 9 / 26

Example

A projective space is a set with distinguished subsets, called lines, satisfying: (1) Any two distinct points are in exactly one line. (2) Each line contains more than two points. (3) If ①❀ ②❀ ③❀ ✇ are distinct points, no three colinear, then ①② intersects ③✇ ❂✮ ①③ intersects ②✇✿ A projective space has a structure of flats (subspaces), and this structure is inherited by any of its finite subset, defining a matroid. Matroids arising from the projective space over a field ❦ are said to be realizable over ❦ (the idea of “coordinates”).

June Huh 9 / 26

Matroids are determined by their independent sets (the idea of “general position”), and can be axiomatized in terms of independent sets.

• ❊

❊ ▼ ❆ ❦ ❆ ❦ ▼ ❦

June Huh 10 / 26

Matroids are determined by their independent sets (the idea of “general position”), and can be axiomatized in terms of independent sets.

• 1. Let ● be a finite graph, and ❊ the set of edges.

Call a subset of ❊ independent if it does not contain a circuit. This defines a graphic matroid ▼. ❆ ❦ ❆ ❦ ▼ ❦

June Huh 10 / 26

Matroids are determined by their independent sets (the idea of “general position”), and can be axiomatized in terms of independent sets.

• 1. Let ● be a finite graph, and ❊ the set of edges.

Call a subset of ❊ independent if it does not contain a circuit. This defines a graphic matroid ▼.

• 2. Let ❆ a finite subset of a field containing ❦.

Call a subset of ❆ independent if it is algebraically independent over ❦. This defines a matroid ▼ algebraic over ❦.

June Huh 10 / 26

The Fano matroid is realizable iff char✭❦✮ ❂ ✷. The non-Fano matroid is realizable iff char✭❦✮ ✷. The non-Pappus matroid is not realizable over any field, but. . . How many matroids are realizable over a field?

June Huh 11 / 26

0% of matroids are realizable over a field.

In other words, almost all matroids are (conjecturally) not realizable over any field. ❦ ❂ ❦ ❂ ❀ ❀

June Huh 12 / 26

0% of matroids are realizable over a field.

In other words, almost all matroids are (conjecturally) not realizable over any field. Testing the realizability of a matroid over a given field is not easy. When ❦ ❂ Q, this is equivalent to Hilbert’s tenth problem over Q. When ❦ ❂ R❀ C❀ etc, there are universality theorems on realization spaces.

June Huh 12 / 26

One can define the characteristic polynomial of a matroid by the recursion ✤▼✭q✮ ❂ ✤▼♥❡✭q✮ ✤▼❂❡✭q✮✿

Rota-Welsh conjecture (1970, 1976)

The coefficients of the characteristic polynomial ✤▼✭q✮ form a log-concave sequence for any matroid ▼, that is, ✖✷

✐ ✕ ✖✐✶✖✐✰✶ for all ✐.

This implies the conjecture on ● and the conjecture on ❆ (Brylawski).

June Huh 13 / 26

A tropical viewpoint

A matroid ▼ on ❊ can be viewed as an r-dimensional fan in an ♥-dimensional space ✝▼ ✒ R❊❂R whose maximal cones correspond to flags of flats ∅ ❋✶ ❋✷ ✁ ✁ ✁ ❋r ❊✿ More precisely, the maximal cones are of the form cone✭e❋✶❀ ✿ ✿ ✿ ❀ e❋r ✮❀ e❋ ❂

❳

✐✷❋

e✐ ✷ R❊❂R✿ ✝▼ ▼

✷

June Huh 14 / 26

A tropical viewpoint

A matroid ▼ on ❊ can be viewed as an r-dimensional fan in an ♥-dimensional space ✝▼ ✒ R❊❂R whose maximal cones correspond to flags of flats ∅ ❋✶ ❋✷ ✁ ✁ ✁ ❋r ❊✿ ✭

❋✶❀ ✿ ✿ ✿ ❀ ❋r ✮❀ ❋ ❂

❳

✐✷❋ ✐ ✷ ❊❂ ✿

The simplicial fan ✝▼ is the tropical linear space associated to ▼: =The tropical picture of “4 points of P✷ in general position”.

June Huh 14 / 26

The Boolean matroid on ❊, the matroid for which every subset of ❊ is a flat, produces the Coxeter complex of type ❆, the fan of the ♥-dimensional permutohedron: Note the extra symmetry not coming from the action of the Weyl group.

June Huh 15 / 26

A geometric digression

Theorem

There is a “Chow equivalence” from a smooth projective variety over ❦ to the toric variety ❳ ✭✝▼✮ if and only if ▼ is realizable over the field ❦. In other words, ▼ is realizable over ❦ if and only if “There is a map from a smooth projective variety over ❦ ❱ ✦ ❳ ✭✝▼✮ which induces an isomorphism between the cohomology rings ❆✄✭❳ ✭✝▼✮✮ ✦ ❆✄✭❱ ✮✿✧ ❆✄✭▼✮ ✿❂ ❆✄✭❳ ✭✝▼✮✮ r ♥

June Huh 16 / 26

A geometric digression

Theorem

There is a “Chow equivalence” from a smooth projective variety over ❦ to the toric variety ❳ ✭✝▼✮ if and only if ▼ is realizable over the field ❦. ▼ ❦ ❦ ❱ ✦ ❳ ✭✝▼✮ ❆✄✭❳ ✭✝▼✮✮ ✦ ❆✄✭❱ ✮✿✧ It is tempting to think this as a “Chow homotopy”, or a retraction. (When the base field is C, it is important not to think this as the usual homotopy.) ❆✄✭▼✮ ✿❂ ❆✄✭❳ ✭✝▼✮✮ r ♥

June Huh 16 / 26

A geometric digression

Theorem

There is a “Chow equivalence” from a smooth projective variety over ❦ to the toric variety ❳ ✭✝▼✮ if and only if ▼ is realizable over the field ❦. ▼ ❦ ❦ ❱ ✦ ❳ ✭✝▼✮ ❆✄✭❳ ✭✝▼✮✮ ✦ ❆✄✭❱ ✮✿✧ It is tempting to think this as a “Chow homotopy”, or a retraction. (When the base field is C, it is important not to think this as the usual homotopy.) We show that, even in the non-realizable case, ❆✄✭▼✮ ✿❂ ❆✄✭❳ ✭✝▼✮✮ satisfies the “standard conjectures” (of dimension r, not ♥).

June Huh 16 / 26

The Chow ring of ▼ can be described explicitly by generators and relations, which can be taken as a definition.

Definition

The Chow ring of ▼ is the quotient of the polynomial ring ❆✄✭▼✮ ✿❂ Z❬①❋❪❂✭■✶ ✰ ■✷✮❀ where the variables are indexed by nonempty proper flats of ▼, and ■✶ ✿❂ ideal

✥

①❋✶①❋✷ ❥ ❋✶ and ❋✷ are incomparable flats of ▼

✦

❀ ■✷ ✿❂ ideal

✥ ❳

✐✶✷❋

①❋

❳

✐✷✷❋

①❋ ❥ ✐✶ and ✐✷ are distinct elements of ❊

✦

✿

June Huh 17 / 26

Like complex manifolds and algebraic varieties, there is a natural orientation on ❆r✭▼✮:

Proposition

There is an isomorphism deg ✿ ❆r✭▼✮ ✬ Z determined by the property that deg✭①❋✶①❋✷ ✁ ✁ ✁ ①❋r ✮ ❂ ✶ for any flag of nonempty proper flats ∅ ❋✶ ❋✷ ✁ ✁ ✁ ❋r ❊✿

June Huh 18 / 26

How should we polarize the Chow ring of ▼?

Definition

A real valued function ■ ✼ ✦ ❝■ , where ■ is a nonempty proper subset of ❊, is said to be strictly submodular if ❝■✶ ✰ ❝■✷ ❃ ❝■✶❭■✷ ✰ ❝■✶❬■✷ for any two incomparable subsets ■✶❀ ■✷ ✒ ❊, where we replace ❝∅ and ❝❊ by zero whenever they appear in the above inequality. ❝ ❵✭❝✮ ✿❂

❳

❋

❝❋①❋ ✷ ❆✶✭▼✮ ❀ ▼

June Huh 19 / 26

How should we polarize the Chow ring of ▼?

Definition

A real valued function ■ ✼ ✦ ❝■ , where ■ is a nonempty proper subset of ❊, is said to be strictly submodular if ❝■✶ ✰ ❝■✷ ❃ ❝■✶❭■✷ ✰ ❝■✶❬■✷ for any two incomparable subsets ■✶❀ ■✷ ✒ ❊, where we replace ❝∅ and ❝❊ by zero whenever they appear in the above inequality. A strictly submodular function ❝ defines an element ❵✭❝✮ ✿❂

❳

❋

❝❋①❋ ✷ ❆✶✭▼✮R❀ where the sum is over all nonempty proper flats of ▼.

June Huh 19 / 26

Main Theorem

Let ❵ be an element associated to a strictly submodular function, and let q ✔ r❂✷. (1) Hard Lefschetz: The multiplication by ❵ defines an isomorphism ❆q✭▼✮R ✦ ❆rq✭▼✮R❀ ❤ ✼ ✦ ❵r✷q ✁ ❤✿ ❵ ❆q✭▼✮ ✂ ❆q✭▼✮

• ✦

❀ ✭❤✶❀ ❤✷✮ ✼ ✦ ✭✶✮q ✭❵r✷q ✁ ❤✶ ✁ ❤✷✮ ❵r✷q✰✶ ✁ ❜✷ ✕ ❛❝

☞ ☞ ☞ ☞

❛ ❜ ❜ ❝

☞ ☞ ☞ ☞ ✔ ✵✿

June Huh 20 / 26

Main Theorem

Let ❵ be an element associated to a strictly submodular function, and let q ✔ r❂✷. (1) Hard Lefschetz: The multiplication by ❵ defines an isomorphism ❆q✭▼✮R ✦ ❆rq✭▼✮R❀ ❤ ✼ ✦ ❵r✷q ✁ ❤✿ (2) Hodge-Riemann: The multiplication by ❵ defines a symmetric bilinear form ❆q✭▼✮R ✂ ❆q✭▼✮R ✦ R❀ ✭❤✶❀ ❤✷✮ ✼ ✦ ✭✶✮qdeg✭❵r✷q ✁ ❤✶ ✁ ❤✷✮ that is positive definite on the kernel of ❵r✷q✰✶ ✁ . ❜✷ ✕ ❛❝

☞ ☞ ☞ ☞

❛ ❜ ❜ ❝

☞ ☞ ☞ ☞ ✔ ✵✿

June Huh 20 / 26

Main Theorem

Let ❵ be an element associated to a strictly submodular function, and let q ✔ r❂✷. (1) Hard Lefschetz: The multiplication by ❵ defines an isomorphism ❆q✭▼✮R ✦ ❆rq✭▼✮R❀ ❤ ✼ ✦ ❵r✷q ✁ ❤✿ (2) Hodge-Riemann: The multiplication by ❵ defines a symmetric bilinear form ❆q✭▼✮R ✂ ❆q✭▼✮R ✦ R❀ ✭❤✶❀ ❤✷✮ ✼ ✦ ✭✶✮qdeg✭❵r✷q ✁ ❤✶ ✁ ❤✷✮ that is positive definite on the kernel of ❵r✷q✰✶ ✁ . Why does this imply the log-concavity conjectures? Essentially because ❜✷ ✕ ❛❝ if and only if

☞ ☞ ☞ ☞

❛ ❜ ❜ ❝

☞ ☞ ☞ ☞ ✔ ✵✿

June Huh 20 / 26

For any two matroids on ❊ with the same rank, there is a diagram ✝▼

❭✌✐♣✧ ✝✶ ❭✌✐♣✧ ✝✷ ❭✌✐♣✧ ✁ ✁ ✁ ❭✌✐♣✧ ✝▼✵ ❀

where “flip” is a local modification of ✝ that preserves the validity of the “standard conjectures” in their cohomology rings.

June Huh 21 / 26

For any two matroids on ❊ with the same rank, there is a diagram ✝▼

❭✌✐♣✧ ✝✶ ❭✌✐♣✧ ✝✷ ❭✌✐♣✧ ✁ ✁ ✁ ❭✌✐♣✧ ✝▼✵ ❀

where “flip” is a local modification of ✝ that preserves the validity of the “standard conjectures” in their cohomology rings. The intermediate objects are tropical varieties with good cohomology rings, but not in general associated to a matroid (unlike in McMullen’s case of polytopes).

June Huh 21 / 26

. . . there is a diagram ✝▼

❭✌✐♣✧ ✝✶ ❭✌✐♣✧ ✝✷ ❭✌✐♣✧ ✁ ✁ ✁ ❭✌✐♣✧

✝▼✵ ❀ where “flip” is a local modification of ✝ . . .

✝▼ ✝▼✵

June Huh 22 / 26

. . . there is a diagram ✝▼

❭✌✐♣✧ ✝✶ ❭✌✐♣✧ ✝✷ ❭✌✐♣✧ ✁ ✁ ✁ ❭✌✐♣✧

✝▼✵ ❀ where “flip” is a local modification of ✝ . . .

Another difference with the case of polytopes: In general, the underlying simplicial complexes of ✝▼ and ✝▼✵ are not homeomorphic, and not even homotopic.

June Huh 22 / 26

. . . there is a diagram ✝▼

❭✌✐♣✧ ✝✶ ❭✌✐♣✧ ✝✷ ❭✌✐♣✧ ✁ ✁ ✁ ❭✌✐♣✧

✝▼✵ ❀ where “flip” is a local modification of ✝ . . .

Another difference with the case of polytopes: In general, the underlying simplicial complexes of ✝▼ and ✝▼✵ are not homeomorphic, and not even homotopic. It turns out that there are at least two useful type of “flips” in the tropical context. One is analogous to the classical flip, and is a homotopy equivalence. The other is not a homotopy equivalence, but a “Chow equivalence”.

June Huh 22 / 26

What should play the role of K¨ ahler classes for the intermediate objects,

• r more generally for tropical varieties?

June Huh 23 / 26

What should play the role of K¨ ahler classes for the intermediate objects,

• r more generally for tropical varieties?

June Huh 23 / 26

Why does this imply the log-concavity conjecture? Let ✐ be an element of ❬♥❪, and consider the linear forms ☛✭✐✮ ✿❂

❳

✐✷❙

①❙❀ ☞✭✐✮ ✿❂

❳

✐❙

①❙✿ These elements are ‘almost’ ample: ❝❙✶ ✰ ❝❙✷ ✕ ❝❙✶❭❙✷ ✰ ❝❙✶❬❙✷ ✭❝❀ ❂ ❝❬♥❪ ❂ ✵✮✿ Their images in the cohomology ring ❆✄✭▼✮ does not depend on ✐; they will be denoted by ☛ and ☞ respectively.

June Huh 24 / 26

Proposition

Under the isomorphism deg ✿ ❆r✭▼✮ ✦ Z❀ we have ☛r❦☞❦ ✼ ✦ ✭❦-th coefficient of the reduced characteristic polynomial of ▼✮✿ Although neither ☛ nor ☞ are in the ample cone K▼, we may take the limit ❵✶ ✦ ☛❀ ❵✷ ✦ ☞❀ ❵✶❀ ❵✷ ✷ K▼✿ This may be one reason why direct combinatorial reasoning was not easy.

June Huh 25 / 26

Geometrically, ☛ and ☞ are the pullbacks of hyperplanes from the first and the last Grassmannians: ❳ ✭✝▼✮

• ❳❆♥
• ❂❇
• ❂P✶
• ❂P✷

✁ ✁ ✁

• ❂P♥✶
• ❂P♥

June Huh 26 / 26