Cohomology of a smooth complete toric variety

Intuition [Beginner]

A smooth complete toric variety is built from a fan, and the fan already tells you how the space is glued together. The cohomology ring asks a different question: what are the basic shape-measuring classes, and how do they multiply? In the toric case the answer is almost shockingly finite. Put one variable on each ray of the fan. A product of variables survives exactly when those rays sit together in a cone. Products from rays that never meet in a cone vanish.

There is one more rule. Some divisor classes are the same because they differ by a character of the dense torus. Those character relations are linear. After imposing the missing-cone products and the character relations, the resulting polynomial quotient is the rational cohomology ring of the variety.

For projective space this recipe recovers the familiar answer: one generator $H$ in degree two, with $H^{n+1}=0$. For the product of two projective lines it recovers two generators $A$ and $B$, with $A^2=0$ and $B^2=0$. The fan is doing the bookkeeping that a cell decomposition or a spectral sequence would otherwise do.

Visual [Beginner]

Imagine the fan of ${\mathbb{P}}^2$ as three rays in the plane. Label the rays $x_0$, $x_1$, and $x_2$. Any two rays span a cone, so products like $x_0{x}_1$ are allowed. All three rays do not lie in one cone, so $x_0{x}_1{x}_2$ is forbidden. The linear relations then identify the three labels with one class $H$, leaving the rule $H^3=0$.

The picture is the whole recipe in miniature. Rays give degree-two cohomology classes. Cones tell you which products can be nonzero. Characters collapse linearly equivalent ray labels. The quotient ring remembers both the additive classes and their multiplication.

Worked example [Beginner]

Compute the cohomology ring of ${\mathbb{P}}^2$ from its fan. The fan has three rays, so start with three labels $x_0$, $x_1$, and $x_2$.

Step 1. Record which sets of rays form cones. In the fan of ${\mathbb{P}}^2$, every pair of rays forms a two-dimensional cone. The set of all three rays does not form a cone. So the only missing-cone product is $x_0{x}_1{x}_2$.

Step 2. Record the character relations. The three toric divisors of ${\mathbb{P}}^2$ are linearly equivalent. In cohomology that means $x_0=x_1=x_2$. Call the common class $H$.

Step 3. Rewrite the missing-cone rule. Since $x_0=x_1=x_2=H$, the product $x_0{x}_1{x}_2$ becomes $H^3$. So the rule says $H^3=0$.

What this tells us. The cohomology ring is $\mathbb{Q}[H]/(H^3)$. The classes are $1$, $H$, and $H^2$. Geometrically, $1$ is the whole space, $H$ is a line, and $H^2$ is a point. Multiplying by one more $H$ would ask for the intersection of three general lines in the plane, which has no room to exist.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $\Sigma$ be a smooth complete fan in $N_{\mathbb{R}}$, with $N$ a rank-$n$ lattice, $M={\mathrm{Hom}}_{\mathbb{Z}}(N,\mathbb{Z})$, and primitive ray generator $v_{\rho }\in N$ for each $\rho \in \Sigma (1)$. Let $X_{\Sigma }$ be the associated smooth complete complex toric variety.

Assign one degree-two variable $x_{\rho }$ to each ray $\rho \in \Sigma (1)$. The Stanley-Reisner ideal of the fan is $$ I_\Sigma=\bigl(x_{\rho_1}\cdots x_{\rho_k}:\rho_1,\ldots,\rho_k\text{ do not span a cone of }\Sigma\bigr) \subseteq \mathbb{Q}[x_\rho:\rho\in\Sigma(1)]. $$ The linear ideal is $$ J_\Sigma=\left(\sum_{\rho\in\Sigma(1)}\langle m,v_\rho\rangle x_\rho\in M\right). $$ It is generated by $n$ independent linear forms whenever the rays span $N_{\mathbb{R}}$, in particular for complete fans.

The Danilov-Jurkiewicz ring of the smooth complete fan is $$ R_\Sigma=\mathbb{Q}[x_\rho:\rho\in\Sigma(1)]/(I_\Sigma+J_\Sigma), \qquad \deg x_\rho=2. $$ The class $x_{\rho }$ is sent to the cohomology class $[D_{\rho }]\in H^2(X_{\Sigma };\mathbb{Q})$ of the torus-invariant divisor from 04.11.09.

For ${\mathbb{P}}^n$, the fan has rays $e_1,\dots ,e_n,-e_1-\cdots -e_n$. The linear relations identify all ray variables with one class $H$, and the unique minimal missing cone is the product of all $n+1$ rays. Thus $$ H^*(\mathbb{P}^n;\mathbb{Q})\cong \mathbb{Q}[H]/(H^{n+1}),\qquad \deg H=2. $$

For ${\mathbb{P}}^1\times {\mathbb{P}}^1$, the fan has rays $\pm e_1,\pm e_2$. Opposite rays do not span cones, so $x_{+}{x}_{-}=0$ in each factor. The linear relations identify the two horizontal divisors with a class $A$ and the two vertical divisors with a class $B$. Hence $$ H^*(\mathbb{P}^1\times\mathbb{P}^1;\mathbb{Q})\cong \mathbb{Q}[A,B]/(A^2,B^2),\qquad \deg A=\deg B=2. $$

Counterexamples to common slips

• The Stanley-Reisner ideal alone is not the cohomology ring. It gives the face ring of the fan. The character relations $J_{\Sigma }$ must also be imposed. Without $J_{\Sigma }$, the ring still has one independent degree-two generator per ray, whereas cohomology has degree-two rank $\rho (X_{\Sigma })=|\Sigma (1)|-n$ in the smooth complete case.

• Completeness matters. For a smooth affine toric variety $U_{\sigma }$, the same face-ring quotient does not compute the ordinary cohomology ring of a compact manifold. Completeness is the compactness input behind Poincare duality and finite-dimensional cohomology.

• The theorem is rational in this unit. Smooth complete toric varieties have integral presentations under additional hypotheses and with Chow-ring language, but rational coefficients avoid torsion and keep the cycle-class comparison clean.

Key theorem with proof [Intermediate+]

Theorem (Danilov-Jurkiewicz presentation). Let $X_{\Sigma }$ be a smooth complete toric variety over $\mathbb{C}$. The assignment $x_{\rho }\mapsto [D_{\rho }]$ induces an isomorphism of graded $\mathbb{Q}$-algebras $$ \Phi\Sigma\longrightarrow H^*(X_\Sigma;\mathbb{Q}). $$

Proof. First, the assignment is well-defined. If ${\rho }_1,\dots ,{\rho }_k$ do not span a cone of $\Sigma$, then the corresponding torus-invariant divisors have empty common intersection: orbit-cone correspondence 04.11.06 identifies an intersection of orbit closures with the orbit closure of the cone spanned by the rays, and no such cone exists. Therefore $$ [D_{\rho_1}]\cdots [D_{\rho_k}]=0, $$ so $I_{\Sigma }$ lies in the kernel.

For every $m\in M$, the character ${\chi }^m$ has principal divisor $$ \mathrm{div}(\chi^m)=\sum_{\rho\in\Sigma(1)}\langle m,v_\rho\rangle D_\rho $$ as in 04.11.09. Principal divisors define linearly equivalent line bundles, and their first Chern classes vanish. Hence the image of every generator of $J_{\Sigma }$ is zero. Thus $\Phi$ is a well-defined graded ring homomorphism.

Surjectivity follows from the orbit filtration. The toric stratification decomposes $X_{\Sigma }$ into torus orbits, and the closures of codimension-$k$ orbits are intersections of $k$ torus-invariant divisors in the smooth case. The cycle classes of orbit closures span the Chow group. Since a smooth complete toric variety has a cellular decomposition by even-dimensional torus-orbit cells after choosing a generic one-parameter subgroup, the cycle-class map from the rational Chow ring to singular cohomology is surjective, and every cohomology class is represented by a polynomial in the divisor classes $[D_{\rho }]$.

It remains to identify the kernel. The same orbit-closure calculus gives the Chow ring presentation $$ A^*(X_\Sigma)\mathbb{Q}\cong \mathbb{Q}[x\rho:\rho\in\Sigma(1)]/(I_\Sigma+J_\Sigma): $$ the face ideal records empty intersections of invariant divisors, and the linear ideal records rational equivalence by principal divisors of characters. The cycle-class map $$ A^k(X_\Sigma)\mathbb{Q}\to H^{2k}(X\Sigma;\mathbb{Q}) $$ is an isomorphism for smooth complete toric varieties because the orbit-cell decomposition has only even real cells. Therefore the kernel of $\Phi$ is exactly $I_{\Sigma }+J_{\Sigma }$, proving the stated isomorphism.

Bridge. This presentation builds toward toric intersection theory 04.11.13, where products of divisor classes become mixed volumes, and appears again in Bernstein-Kushnirenko theory 04.11.14. The foundational reason is that the orbit-cone correspondence identifies intersections with cones, while the Picard exact sequence identifies characters with linear relations. Putting these together, the bridge is the quotient ring that generalises the cohomology computations for projective space 04.03.04 and is dual to the face combinatorics of the fan.

Exercises [Intermediate+]

Advanced results [Master]

The quotient $R_{\Sigma }$ is not merely a convenient presentation. It is the Chow ring, the rational cohomology ring, and the combinatorial face ring modulo the lattice of characters: $$ A^(X_\Sigma)_\mathbb{Q}\cong H^{2}(X_\Sigma;\mathbb{Q})\cong \mathbb{Q}[\Sigma]/(\Theta), $$ where $\mathbb{Q}[\Sigma ]=\mathbb{Q}[x_{\rho }]/I_{\Sigma }$ is the Stanley-Reisner face ring and $\Theta$ is the linear system of parameters coming from $M$. Smoothness says the fan is a simplicial sphere with unimodular maximal cones, so the quotient is an Artinian Poincare duality algebra.

When $X_{\Sigma }$ is projective, an ample divisor class $\omega \in H^2(X_{\Sigma };\mathbb{Q})$ obtained from a strictly convex support function satisfies hard Lefschetz: $$ \omega^{n-2k}^{2k}(X_\Sigma;\mathbb{Q})\xrightarrow{\sim}H^{2n-2k}(X_\Sigma;\mathbb{Q}),\qquad 0\leq k\leq n/2, $$ and the Hodge-Riemann bilinear relations on primitive classes. Translated through $R_{\Sigma }$, these become inequalities on the face numbers of the underlying simplicial polytope. Stanley's 1980 proof of the necessity direction of McMullen's $g$-theorem uses exactly this translation: projective toric varieties turn hard Lefschetz into unimodality and nonnegativity statements for the $h$-vector.

The class of an orbit closure has a monomial representative. If $\sigma =⟨{\rho }_1,\dots ,{\rho }_k⟩$ is a cone of a smooth fan, then the orbit closure $V(\sigma )$ has codimension $k$, and $$ [V(\sigma)]=x_{\rho_1}\cdots x_{\rho_k}\in H^{2k}(X_\Sigma;\mathbb{Q}). $$ This formula is the computational reason the presentation is powerful: intersection products are computed by multiplying monomials and reducing by $I_{\Sigma }+J_{\Sigma }$.

Projective toric geometry also identifies the top-degree pairing with polytope volume. If $D_P$ is the ample divisor attached to a lattice polytope $P$ as in 04.11.10, then $$ \int_{X_\Sigma} c_1(\mathcal{O}(D_P))^n=n!\operatorname{Vol}(P), $$ with volume normalised so a fundamental lattice simplex has volume $1/n!$. The mixed version for several polytopes is the content of 04.11.13, and the generic zero-counting version is Bernstein-Kushnirenko 04.11.14.

Synthesis. The Danilov-Jurkiewicz presentation builds toward the intersection-theoretic volume formula, appears again in the Cox quotient construction 04.11.15 pending, and generalises the projective-space cohomology computation 04.03.04. The central insight is that torus-invariant divisors identify cohomology with fan combinatorics, while Poincare duality 03.12.16 is dual to the spherical nature of a complete smooth fan. Putting these together identifies toric topology with a finite quotient of a polynomial ring.

Full proof set [Master]

Proposition (orbit-closure monomials). Let $\Sigma$ be smooth and complete. If $\sigma =⟨{\rho }_1,\dots ,{\rho }_k⟩$ is a cone of $\Sigma$, then $[V(\sigma )]=x_{{\rho }_1}\cdots x_{{\rho }_k}$ under the Danilov-Jurkiewicz isomorphism.

Proof. For a smooth fan, the divisors $D_{{\rho }_i}$ meet with normal crossings whenever the rays ${\rho }_i$ span a cone. Their common intersection is the orbit closure $V(\sigma )$ by the orbit-cone correspondence: $D_{{\rho }_i}=V({\rho }_i)$, and intersections of orbit closures correspond to joins of cones when that join exists. Because the cone is smooth, the local affine chart $U_{\sigma }$ is isomorphic to ${\mathbb{A}}^k\times ({\mathbb{C}}^{\ast }{)}^{n-k}$ along the orbit, and the divisors $D_{{\rho }_i}$ are the coordinate hyperplanes in the affine factor. Their intersection multiplicity is therefore one. The cup product of their divisor classes is the fundamental cohomology class of the transverse intersection, namely $[V(\sigma )]$.

Proposition (Betti numbers from the $h$-vector). If $\Sigma$ is the normal fan of a simple $n$-polytope $P$, then $b_{2k}(X_{\Sigma })=h_k(P)$ and $b_{2k+1}(X_{\Sigma })=0$.

Proof. The quotient $R_{\Sigma }$ has one variable of degree two for each ray and is the face ring of the dual simplicial polytope modulo a linear system of parameters. The Hilbert series of the Stanley-Reisner ring of a simplicial $(n-1)$-sphere is $$ \operatorname{Hilb}{\mathbb{Q}[\Sigma]}(t)=\frac{h_0+h_1t+\cdots+h_nt^n}{(1-t)^n}. $$ Quotienting by the $n$ independent linear forms from $J\Sigma$multipliestheHilbertseriesby$(1-t)^n$,leaving$h_0+h_1t+\cdots+h_nt^n$.Since$x_\rho$hascohomologicaldegreetwo,thecoefficient$h_k$is$\dim_\mathbb{Q}H^{2k}(X_\Sigma;\mathbb{Q})$. No odd degrees occur.

Proposition (projective-space presentation). For the standard fan of ${\mathbb{P}}^n$, the Danilov-Jurkiewicz presentation is $\mathbb{Q}[H]/(H^{n+1})$.

Proof. Let the rays be $v_0=-e_1-\cdots -e_n$ and $v_i=e_i$ for $1\le i\le n$. The only subset of rays not contained in a cone and minimal for inclusion is the full set $\{v_0,\dots ,v_n\}$, so $I_{\Sigma }=(x_0{x}_1\cdots x_n)$. The character $e_i^{\ast }\in M$ gives the linear relation $-x_0+x_i=0$. Thus all $x_i$ equal a common class $H$. Substituting into the monomial relation gives $H^{n+1}=0$, and no smaller power is killed by the presentation because the quotient has dimension $n+1$ as a vector space, with basis $1,H,\dots ,H^n$.

Connections [Master]

• 04.11.09 supplies the divisor-class exact sequence whose character relations become the ideal $J_{\Sigma }$ in the cohomology presentation.

• 04.11.10 supplies ample divisors from lattice polytopes; their top self-intersections become normalised lattice volumes in 04.11.13.

• 03.12.16 explains the Poincare duality algebra structure that the smooth complete fan quotient exhibits combinatorially.

• 04.03.04 is the projective-space special case: $\mathbb{Q}[H]/(H^{n+1})$ reappears here as the fan presentation for the simplex fan.

• 04.12.01 and tropical intersection theory use the same face-ring instinct: combinatorial incidence data governs intersection products after the correct balancing or linear relations are imposed.

Historical & philosophical context [Master]

Danilov's 1978 survey ^{[Danilov 1978]} gave one of the first systematic accounts of toric varieties as a meeting point of convex geometry, algebraic geometry, and topology, including the orbit-closure and cohomology calculations that became standard. Jurkiewicz's 1980 paper ^{[Jurkiewicz 1980]} isolated the Chow-ring presentation for nonsingular torus embeddings, and Fulton's 1993 text put the result into the concise Stanley-Reisner form now used in the subject.

Stanley's 1980 proof ^{[Stanley 1980]} made the cohomology presentation famous outside toric geometry. The hard Lefschetz theorem for projective toric varieties turned the cohomology ring into a proof of inequalities for face numbers of simplicial polytopes, settling the necessity half of McMullen's $g$-theorem. This is the point where a quotient of a polynomial ring became a bridge between algebraic topology and enumerative combinatorics.

Bibliography [Master]

@article{Danilov1978Toric,
  author = {Danilov, V. I.},
  title = {The Geometry of Toric Varieties},
  journal = {Russian Mathematical Surveys},
  volume = {33},
  number = {2},
  pages = {97--154},
  year = {1978}
}

@article{Jurkiewicz1980Chow,
  author = {Jurkiewicz, Jerzy},
  title = {Chow Ring of Projective Nonsingular Torus Embeddings},
  journal = {Colloquium Mathematicum},
  volume = {43},
  pages = {261--270},
  year = {1980}
}

@book{Fulton1993Toric,
  author = {Fulton, William},
  title = {Introduction to Toric Varieties},
  publisher = {Princeton University Press},
  year = {1993}
}

@book{CoxLittleSchenck2011Toric,
  author = {Cox, David A. and Little, John B. and Schenck, Henry K.},
  title = {Toric Varieties},
  publisher = {American Mathematical Society},
  year = {2011}
}

@article{Stanley1980Faces,
  author = {Stanley, Richard P.},
  title = {The Number of Faces of a Simplicial Convex Polytope},
  journal = {Advances in Mathematics},
  volume = {35},
  pages = {236--238},
  year = {1980}
}