Hard Lefschetz theorems and Hodge-Riemann relations

Guest post by June Huh

I will write about a common algebraic structure hidden behind seemingly distant objects: convex polytopes, Kähler manifolds, projective varieties, and lastly, matroids. Let $n$ be a positive integer.

1. Polytopes

1.1

A polytope in $\mathbb{R}^n$ is the convex hull of a finite subset of $\mathbb{R}^n$. Let’s write $\Pi$ for the abelian group with generators $[P]$, one for each polytope $P \subseteq \mathbb{R}^n$, which satisfy the following relations:

  1. $[P_1 \cup P_2]+[P_1 \cap P_2]=[P_1]+[P_2]$ whenever $P_1 \cup P_2$ is a polytope,
  2. $[P+t]=[P]$ for every point $t$ in $\mathbb{R}^n$, and
  3. $[\varnothing]=0$.

This is the polytope algebra of McMullen [McM89]. The multiplication in $\Pi$ is defined by the Minkowski sum
\[
[P_1] \cdot [P_2]=[P_1+P_2],
\]
and this makes $\Pi$ a commutative ring with $1=[\text{point}]$ and $0=[\varnothing]$.

The structure of $\Pi$ can be glimpsed through some familiar translation invariant measures on the set of polytopes. For example, the Euler characteristic shows that there is a surjective ring homomorphism
\[
\chi:\Pi \longrightarrow \mathbb{Z}, \qquad [P] \longmapsto \chi(P),
\]
and the Lebesgue measure on $\mathbb{R}^n$ shows that there is a surjective group homomorphism
\[
\text{Vol}:\Pi \longrightarrow \mathbb{R}, \qquad [P] \longmapsto \text{Vol}(P).
\]
A fundamental observation is that some power of $[P]-1$ is zero in $\Pi$ for every nonempty polytope $P$. Since every polytope can be triangulated, it is enough to check this when the polytope is a simplex. In this case, a picture drawing for $n=0,1,2,$ and if necessary $3$, will convince the reader that
\[
([P]-1)^{n+1}=0.
\]
The kernel of the Euler characteristic $\chi$ turns out to be torsion free and divisible. Thus we may speak about the logarithm of a polytope in $\Pi$ which satisfies the usual rule
\[
\text{log}[P_1+P_2]=\text{log}[P_1]
+\text{log}[P_2].
\]
The notion of logarithm leads to a remarkable identity concerning volumes of convex polytopes.

Theorem
Writing $\texttt{p}$ for the logarithm of $[P]$, we have
\[
\text{Vol}(P)=\frac{1}{n!}\text{Vol} (\texttt{p}^n).
\]

This shows that, more generally, Minkowski’s mixed volume of polytopes $P_1,\ldots,P_n$ can be expressed in terms of the product of the corresponding logarithms $\texttt{p}_1,\ldots,\texttt{p}_n$:
\[
\text{Vol}(P_1,\ldots,P_n)=\frac{1}{n!} \text{Vol}(\texttt{p}_1 \cdots \texttt{p}_n).
\]

1.2

Let’s write $P_1 \preceq P_2$ to mean that $P_1$ is a Minkowski summand of some positive multiple of $P_2$. This relation is clearly transitive. We way that $P_1$ and $P_2$ are equivalent when
\[
P_1 \preceq P_2 \preceq P_1.
\]
Let $\mathscr{K}(P)$ be the set of all polytopes equivalent to a given polytope $P$. The collection $\mathscr{K}(P)$ is a convex cone in the sense that
\[
P_1, P_2 \in \mathscr{K}(P) \Longrightarrow \lambda_1 P_1 + \lambda_2 P_2 \in \mathscr{K}(P) \ \ \text{for positive real numbers $\lambda_1, \lambda_2$.}
\]
We will meet an analogue of this convex cone in each of the following sections.

Definition
For each positive integer $q$, let $\Pi^q(P) \subseteq \Pi$ be the subgroup generated by all elements of the form
\[
\texttt{p}_1\texttt{p}_2 \cdots \texttt{p}_q,
\]
where $\texttt{p}_i$ is the logarithm of a polytope in $\mathscr{K}(P)$.

Note that any two equivalent polytopes define the same set of subgroups of $\Pi$. These subgroups are related to each other in a surprising way when $P$ is an $n$-dimensional simple polytope; this means that every vertex of the polytope is contained in exactly $n$ edges.

Theorem [McM93]
Let $\texttt{p}$ be the logarithm of a simple polytope in $\mathscr{K}(P)$, and let $1 \le q \le \frac{n}{2}$.

  1. Hard Lefschetz theorem: The multiplication by $\texttt{p}^{n-2q}$ defines an isomorphism
    \[
    \Pi^{q}(P) \longrightarrow \Pi^{n-q}(P), \quad x \longmapsto \texttt{p}^{n-2q} x.
    \]
  2. Hodge-Riemann relations: The multiplication by $\texttt{p}^{n-2q}$ defines a symmetric bilinear form
    \[
    \Pi^{q}(P) \times \Pi^{q}(P) \longrightarrow \mathbb{R}, \quad (x_1,x_2) \longmapsto (-1)^q \ \text{Vol}\big(\texttt{p}^{n-2q} x_1 x_2\big)
    \]
    that is positive definite when restricted to the kernel of the multiplication by $\texttt{p}^{n-2q+1}$.

In fact, the group $\Pi^q(P)$ can be equipped with the structure of a finite dimensional real vector space in a certain natural way, and the isomorphism between groups in the first part of the theorem turns out to be an isomorphism between vector spaces.

I will mention two concrete implications of geometric-combinatorial nature, one for each of the above two statements.

  1. The first statement is the main ingredient in the proof of the $g$-conjecture for simple polytopes [Stan80]. This gives a numerical characterization of sequences of the form
    \[
    f_0(P),f_1(P),\ldots,f_n(P),
    \]
    where $f_i(P)$ is the number of $i$-dimensional faces of an $n$-dimensional simple polytope $P$.
  2. The second statement, in the special case $q=1$, is essentially equivalent to the Aleksandrov-Fenchel inequality on mixed volumes of convex bodies:
    \[
    \text{Vol}(\texttt{p}_1\texttt{p}_1 \texttt{p}_3 \cdots \texttt{p}_n) \text{Vol}(\texttt{p}_2\texttt{p}_2 \texttt{p}_3 \cdots \texttt{p}_n) \le \text{Vol}(\texttt{p}_1\texttt{p}_2 \texttt{p}_3 \cdots \texttt{p}_n)^2.
    \]
    The inequality played a central role in the proof of the van der Waerden conjecture that the permanent of any doubly stochastic $n \times n$ nonnegative matrix is at least $n!/n^n$. An interesting account on the formulation and the solution of the conjecture can be found in [vLin82].

With suitable modifications, the hard Lefschetz theorem and the Hodge-Riemann relations can be extended to polytopes that are not necessarily simple [Karu04].

2. Kähler manifolds

2.1

Let $\omega$ be a Kähler form on an $n$-dimensional compact complex manifold $M$. This means that $\omega$ is a smooth differential $2$-form on $M$ that can be written locally in coordinate charts as
\[
i \partial \overline{\partial} f
\]
for some smooth real functions $f$ whose complex Hessian matrix $\Big[\frac{\partial^2 f}{\partial z_i\partial \overline{z}_j}\Big]$ is positive definite; here $z_1,\ldots,z_n$ are holomorphic coordinates and $\partial$, $\overline{\partial}$ are the differential operators
\[
\partial=\sum_{k=1}^n \frac{\partial}{\partial z_k} dz_k, \qquad \overline{\partial}=\sum_{k=1}^n \frac{\partial}{\partial \overline{z}_k} d\overline{z}_k.
\]
Like all other good definitions, the Kähler condition has many other equivalent characterizations, and we have chosen the one that emphasizes the analogy with the notion of convexity.

To a Kähler form $\omega$ on $M$, we can associate a Riemannian metric $g$ on $M$ by setting
\[
g(u,v)=w(u,Iv),
\]
where $I$ is the operator on tangent vectors of $M$ that corresponds to the multiplication by $i$. Thus we may speak of the length, area, etc., on $M$ with respect to $\omega$.

Theorem
The volume of $M$ is given by the integral
\[
\text{Vol}(M)=\frac{1}{n!} \int_M w^n.
\]
More generally, the volume of a $d$-dimensional complex submanifold $N \subseteq M$ is given by
\[
\text{Vol}(N)=\frac{1}{d!} \int_N w^d.
\]

Compare the corresponding statement of the previous section that $\text{Vol}(P)=\frac{1}{n!}\text{Vol} (\texttt{p}^n)$.

2.2

Let $\mathscr{K}(M)$ be the set of all Kähler forms on $M$. The collection $\mathscr{K}(M)$ is a convex cone in the sense that
\[
\omega_1, \omega_2 \in \mathscr{K}(M) \Longrightarrow \lambda_1 \omega_1 + \lambda_2 \omega_2 \in \mathscr{K}(M) \ \ \text{for positive real numbers $\lambda_1, \lambda_2$.}
\]
This follows from the fact that the sum of two positive definite matrices is positive definite.

Definition
For each nonnegative integer $q$, let $H^{q,q}(M) \subseteq H^{2q}(M,\mathbb{C})$ be the subset of all the cohomology classes of closed differential forms that can be written in local coordinate charts as
\[
\sum f_{k_1,\ldots,k_q,l_1,\ldots,l_q} dz_{k_1} \wedge \cdots \wedge dz_{k_q} \wedge d\overline{z}_{l_1} \wedge \cdots \wedge d\overline{z}_{l_q}.
\]

Note that the cohomology class of a Kähler form $\omega$ is in $H^{1,1}(M)$, and that
\[
[\varphi] \in H^{q,q}(M) \Longrightarrow [\omega \wedge \varphi] \in H^{q+1,q+1}(M).
\]

Theorem (Classical)
Let $\omega$ be an element of $\mathscr{K}(M)$, and let $q$ be a nonnegative integer $\le \frac{n}{2}$.

  1. Hard Lefschetz theorem: The wedge product with $\omega^{n-2q}$ defines an isomorphism
    \[
    H^{q,q}(M) \longrightarrow H^{n-q,n-q}(M), \quad [\varphi] \longmapsto [\omega^{n-2q} \wedge \varphi].
    \]
  2. Hodge-Riemann relations: The wedge product with $\omega^{n-2q}$ defines a Hermitian form
    \[
    H^{q,q}(M) \times H^{q,q}(M) \longrightarrow \mathbb{C}, \quad (\varphi_1,\varphi_2) \longmapsto (-1)^q \int_M \omega^{n-2q} \wedge \varphi_1 \wedge \overline{\varphi_2}
    \]
    that is positive definite when restricted to the kernel of the wedge product with $\omega^{n-2q+1}$.

Analogous statements hold for $H^{q_1,q_2}(M)$ with $q_1 \neq q_2$, and these provide a way to show that certain compact complex manifolds cannot admit any Kähler form. For deeper applications, see [Voi10].

3. Projective varieties

3.1

Let $k$ be an algebraically closed field, and let $\mathbb{P}^m$ be the $m$-dimensional projective space over $k$. A projective variety over $k$ is a subset of the form
\[
X=\{h_1=h_2=\ldots=h_k=0\} \subseteq \mathbb{P}^m,
\]
where $h_i$ are homogeneous polynomials in $m+1$ variables. One can define the dimension, connectedness, and smoothness of projective varieties in a way that is compatible with our intuition when $k=\mathbb{C}$. One can also define what it means for a map between two projective varieties, each living in two possibly different ambient projective spaces, to be algebraic.

Let $K$ be another field, not necessarily algebraically closed but of characteristic zero. A Weil cohomology theory with coefficients in $K$ is an assignment
\[
X \longmapsto H^*(X)=\bigoplus_k H^k(X),
\]
where $X$ is a smooth and connected projective variety over $k$ and $H^*(X)$ is a graded-commutative algebra over $K$. This assignment is required to satisfy certain rules similar to those satisfied by the singular cohomology of compact complex manifolds, such as functoriality, finite dimensionality, Poincar&eacute duality, K&uumlnneth formula, etc. For this reason the product of two elements in $H^*(X)$ will be written
\[
\xi_1 \cup \xi_2 \in H^*(X).
\]
For algebraic geometers, the most important of the rules is that to every codimension $q$ subvariety $Y \subseteq X$ there be a corresponding cohomology class
\[
\text{cl}(Y) \in H^{2q}(X).
\]
These classes should have the property that, for example,
\[
\text{cl}(Y_1 \cap Y_2)=\text{cl}(Y_1) \cup \text{cl}(Y_2)
\]
whenever $Y_1$ and $Y_2$ are subvarieties intersecting transversely, and that
\[
\text{cl}(H_1)=\text{cl}(H_2)
\]
whenever $H_1$ and $H_2$ are two hyperplane sections of $X \subseteq \mathbb{P}^m$. Though not easy, it is possible to construct a Weil cohomology theory for any $k$ for some $K$. For example, when both $k$ and $K$ are the field of complex numbers, one can take the de Rham cohomology of smooth differential forms.

Definition
For each nonnegative integer $q$, let $A^q(X) \subseteq H^{2q}(X)$ be the set of rational linear combinations of cohomology classes of codimension $q$ subvarieties of $X$.

One of the rules for $H^*(X)$ implies that, if $n$ is the dimension of $X$, there is an isomorphism
\[
\text{deg}: A^n(X) \longrightarrow \mathbb{Q}
\]
determined by the property that
\[
\text{deg}(\text{cl}(\text{p}))=1 \ \ \text{for every} \ \ \text{p} \in X.
\]
Writing $h$ for the class in $A^1(X)$ of any hyperplane section of $X \subseteq \mathbb{P}^m$, the degree of $X \subseteq \mathbb{P}^m$ satisfies the formula
\[
\text{deg}(X \subseteq \mathbb{P}^m)=\text{deg}(h^n),
\]
the number of points in the intersection of $X$ with a sufficiently general subspace $\mathbb{P}^{m-n} \subseteq \mathbb{P}^m$. Compare the corresponding statements of the previous sections
\[
\text{Vol}(P)=\frac{1}{n!}\text{Vol} (\texttt{p}^n) \quad \text{and} \quad \text{Vol}(M)=\frac{1}{n!} \int_M w^n.
\]

3.2

Let $\mathscr{K}(X)$ be the set of cohomology classes of hyperplane sections of $X$ under all possible embeddings of $X$ into projective spaces. Classical projective geometers knew that $\mathscr{K}(X)$ is a convex cone in a certain sense; if you are curious, read about the Segre embedding and the Veronese embedding.

Conjecture (Grothendieck)
Let $h$ be an element in $\mathscr{K}(X)$, and let $q$ be a nonnegative integer $\le n/2$.

  1. Lefschetz standard: The multiplication by $h^{n-2q}$ defines an isomorphism
    \[
    A^q(X) \longrightarrow A^{n-q}(X), \quad \xi \longmapsto h^{n-2q} \cup \xi.
    \]
  2. Hodge standard: The multiplication by $h^{n-2q}$ defines a symmetric bilinear form
    \[
    A^q(X) \times A^q(X) \longrightarrow \mathbb{Q}, \quad (\xi_1,\xi_2) \longmapsto (-1)^q \text{deg}\big(h^{n-2q} \cup \xi_1 \cup \xi_2\big),
    \]
    that is positive definite when restricted to the kernel of the cup product with $h^{n-2q+1}$.

The above statements are at the core of Grothendieck’s approach to Weil’s conjecture on zeta functions and other important problems in algebraic geometry [Gro69].

4. Matroids

4.1

As we know, a matroid $\mathrm{M}$ is given by a closure operator defined on all subsets of a finite set $E$ satisfying the Steinitz-MacLane exchange property:


For every subset $I$ of $E$ and every element $a$ not in the closure of $I$, if $a$ is in the closure of ${I \cup\{ b\}}$, then $b$ is in the closure of $I \cup \{a\}$.

It is remarkable that this single sentence leads to an intricate algebraic structure of the kind we have seen above. This structure reveals certain properties of matroids that are not easy to see by other means.

Let’s write $S_\mathrm{M}$ for the polynomial ring with real coefficients and variables $x_F$, one for each nonempty proper flat $F$ of $\mathrm{M}$.

Definition
The Chow ring of a loopless matroid $\mathrm{M}$ is defined to be the quotient
\[
A^*(\mathrm{M}):=S_\mathrm{M}/(I_\mathrm{M}+J_\mathrm{M}),
\]
where $I_\mathrm{M}$ is the ideal generated by the quadratic monomials
\[
x_{F_1}x_{F_2}, \ \ \text{$F_1$ and $F_2$ are two incomparable nonempty proper flats of $\mathrm{M}$,}
\]
and $J_\mathrm{M}$ is the ideal generated by the linear forms
\[
\sum_{i_1 \in F} x_F – \sum_{i_2 \in F} x_F, \ \ \text{$i_1$ and $i_2$ are distinct elements of the ground set $E$.}
\]
We write $A^q(\mathrm{M}) \subseteq A^*(\mathrm{M})$ for the subspace spanned by all degree $q$ monomials.

Let $n+1$ be the rank of $\mathrm{M}$. An important step is to identify the map analogous to the volume in section $1$, the integral in section $2$, and the degree in section $3$.

Theorem
There is an isomorphism $\text{deg}: A^n(\mathrm{M}) \longrightarrow \mathbb{R}$ uniquely determined by the property
\[
\text{deg}(x_{F_1}x_{F_2}\cdots x_{F_n})=1 \ \ \text{for every flag of nonempty proper flats} \ \ F_1 \subsetneq F_2 \subsetneq \cdots \subsetneq F_n.
\]

In particular, any two monomials corresponding to a complete flag of nonempty proper flats are equal in the Chow ring of a loopless matroid.

4.2

What should be the convex cone $\mathscr{K}(\mathrm{M})$? In fact, there is a certain piecewise linear space associated to $\mathrm{M}$, the tropical linear space of $\mathrm{M}$, and one takes $\mathscr{K}(\mathrm{M})$ to be the set of all strictly convex piecewise linear functions on the tropical linear space. For known applications, the following more restrictive definition is sufficient.

Definition
A function $c$ on the set of nonempty proper subsets of $E$ is said to be strictly submodular if
\[
c_{I_1}+c_{I_2} > c_{I_1 \cap I_2} +c_{I_1 \cup I_2} \ \ \text{for any two incomparable subsets $I_1,I_2 \subseteq E$,}
\]
where we replace $c_\varnothing$ and $c_E$ by zero whenever they appear in the above inequality.

A strictly submodular function $c$ defines an element
\[
\ell(c):= \sum_F c_F x_F\in A^1(\mathrm{M}),
\]
where the sum is over all nonempty proper flats of $\mathrm{M}$; the set of all such is a convex cone in the obvious sense. Note that the rank function of any matroid on $E$ can be obtained as a limit of strictly submodular functions.

Theorem [AHK]
Let $\ell$ be an element of $A^1(\mathrm{M})$ associated to a strictly submodular function, and let $q$ be a nonnegative integer $\le \frac{n}{2}$.

  1. Hard Lefschetz theorem: The multiplication by $\ell^{n-2q}$ defines an isomorphism
    \[
    A^q(\mathrm{M}) \longrightarrow A^{n-q}(\mathrm{M}), \qquad a \longmapsto \ell^{n-2q} \ a.
    \]
  2. Hodge-Riemann relations: The multiplication by $\ell^{n-2q}$ defines a symmetric bilinear form
    \[
    A^q(\mathrm{M}) \times A^q(\mathrm{M}) \longrightarrow \mathbb{R}, \qquad (a_1,a_2) \longmapsto (-1)^q \ \text{deg}(\ell^{n-2q}\ a_1 a_2)
    \]
    that is positive definite when restricted to the kernel of the multiplication by
    $\ell^{n-2q+1}$.

In fact, the theorem applies more generally to elements $\ell$ in the cone $\mathscr{K}(\mathrm{M})$ mentioned above. Below are two applications presented in [AHK], which use the Hodge-Riemann relations in the special case when $q=1$.

  1. Let $w_k$ be the absolute value of the coefficient of $\lambda^{n-k+1}$ in the characteristic polynomial of $\mathrm{M}$. Then the sequence $w_k$ is log-concave:
    \[
    w_{k-1} w_{k+1} \le w_k^2 \ \ \text{for all $1 \le k\le n$.}
    \]
    In particular, the sequence $w_k$ is unimodal:
    \[
    w_0 \le w_1 \le \cdots \le w_l \ge \cdots \ge w_n \ge w_{n+1} \ \ \text{for some index $l$.}
    \]
    This verifies a conjecture of Heron, Rota, and Welsh.
  2. Let $f_k$ be the number of independent subsets of $E$ with cardinality $k$. Then the sequence $f_k$ is log-concave:
    \[
    f_{k-1} f_{k+1} \le f_k^2 \ \ \text{for all $1 \le k \le n$.}
    \]
    In particular, the sequence $f_k$ is unimodal:
    \[
    f_0 \le f_1 \le \cdots \le f_l \ge \cdots \ge f_n \ge f_{n+1} \ \ \text{for some index $l$.}
    \]
    This verifies a conjecture of Mason and Welsh.

These applications only use the Hodge-Riemann relations for $q=1$ and for one carefully chosen $\ell$. The general Hodge-Riemann relations for all $\ell$ in $\mathscr{K}(\mathrm{M})$ may contain more interesting information on $\mathrm{M}$.

References

[AHK] Karim Adiprasito, June Huh, and Eric Katz, Hodge theory for combinatorial geometries, arXiv:1511.02888.

[Gro69] Alexander Grothendieck, Standard conjectures on algebraic cycles, 1969 Algebraic Geometry, 193-199, Oxford University Press.

[Karu04] Kalle Karu, Hard Lefschetz theorem for nonrational polytopes, Inventiones Mathematicae 157 (2004), 419-447.

[McM89] Peter McMullen, The polytope algebra, Advances in Mathematics 78 (1989), 76-130.

[McM93] Peter McMullen, On simple polytopes, Inventiones Mathematicae 113 (1993), 419-444.

[Stan80] Richard Stanley, The number of faces of a simplicial convex polytope, Advances in Mathematics 35 (1980), 236-238.

[vLin82] Jack van Lint, The van der Waerden conjecture: two proofs in one year, The Mathematical Intelligencer 4 (1982), 72-77.

[Voi10] Claire Voisin, On the cohomology of algebraic varieties, Proceedings of the International Congress of Mathematicians I, 476-503, New Delhi, 2010.

2 thoughts on “Hard Lefschetz theorems and Hodge-Riemann relations

  1. This is remarkable stuff.

    I have 3 nits to pick. 1. Please don’t perpetuate Rota’s unfortunate use of the general term “combinatorial geometry” for simple matroid. 2. Please use the italic el, $l$, instead of the non-italic, handwriting-style \ell, which originated in the days when people had to type math on typewriters and use the el for 1 because there was no 1. You can fix this with the command \renewcommand\ell{l}. 3. It is unwise to use the letter w, customarily used for the actual coefficient with sign, to mean the absolute value. Would you change the meaning of s(n,k), the Stirling number of the first kind, to the absolute value? I hope not; it would spoil many properties.. That is exactly analogous; Rota introduced the letters w_k and W_k in analogy with Stirling numbers. One popular solution is to write $w_k^+$ for $|w_k|$.

  2. Nit #4. A “polytope” does not have to be convex. If you want to use the term “polytope” to mean “convex polytope”, you should state that at the beginning. That’s all that’s needed. For a partial proof that a polytope need not be convex, see the book “Polyhedron Models” of Magnus Wenninger.

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