04.05.11 · algebraic-geometry / divisors

Worked Hirzebruch-Riemann-Roch computations

shipped3 tiersLean: none

Anchor (Master): Hirzebruch *Topological Methods in Algebraic Geometry* (Springer Classics 1995 reprint of 1978 3rd English ed.; original 1956 Ergebnisse 9) §IV.21-§IV.22 (the $\chi_y$-genus, virtual signature, and Hirzebruch surface computations); Atiyah-Bott *A Lefschetz fixed point formula for elliptic complexes II*, Annals of Math. 88 (1968) 451-491 (holomorphic Lefschetz); Atiyah-Singer *The index of elliptic operators I-V*, Annals of Math. 87-93 (1968-1971); Eisenbud-Harris *3264 and All That* (CUP 2016) §13.5; Fulton *Intersection Theory* §15

Intuition Beginner

The Hirzebruch-Riemann-Roch identity reads as a recipe: take a holomorphic vector bundle on a smooth projective variety, compute two characteristic classes, multiply them in the cohomology ring of the variety, integrate over the variety, and the result is the holomorphic Euler characteristic of the bundle. The statement and proof live in 04.05.10; this unit does the recipe in detail on a short list of test varieties — projective spaces, curves, surfaces of various flavours, and Hirzebruch's ruled surfaces — so the formula stops being symbolic and becomes a finite computation that returns an integer.

The first lesson from running the recipe is that classical dimension formulas drop out of HRR as one-line corollaries. The dimension count for degree- $d$ homogeneous polynomials on $P^{n}$ is a binomial coefficient; HRR turns this binomial coefficient into a residue computation in a formal power series. The genus formula on a smooth projective curve is the curve-tier specialisation. Noether's surface formula coupling the canonical-square invariant to the topological Euler characteristic is the structure-sheaf specialisation in dimension two. Each of these classical results, derived once with classical tools, is recovered in two lines from HRR.

The second lesson is structural: every worked example follows the same pattern. Compute the Chern character of the bundle as a polynomial in Chern classes. Compute the Todd class of the tangent bundle as a polynomial in Chern classes. Multiply, read off the top-degree component, and pair with the fundamental class. The pattern is the same for the line bundle on $P^{1}$ as for a rank-three bundle on a K3 surface or a tangent bundle on a Hirzebruch surface — only the bookkeeping changes.

Visual Beginner

A three-panel diagram. The left panel shows a formal-power-series identity for $td (T_{P^{n}})$ expressed as a power in the hyperplane class. The middle panel shows the product of the Chern character of a twist by the Todd class on $P^{n}$ , with the top-degree piece highlighted as the integrand. The right panel shows the residue computation that extracts the binomial-coefficient answer, with the contour around the origin and the formal series expansion below.

A second panel, drawn beneath the first, shows the same recipe applied to a smooth projective surface: the truncated Todd class $1 - K /2 + (K^{2} + c_{2}) /12$ on the surface, the Chern character $1 + D + D^{2} /2$ of a line bundle of class $D$ , the product, and the surface-Riemann-Roch identity that emerges. Two arrows from the surface diagram point upward to the curve diagram on $P^{1}$ (the degenerate one-dimensional case) and outward to a K3-surface diagram (the canonical-class-vanishes case), making explicit the recipe's modularity in dimension and in canonical-class data.

Worked example Beginner

Carry out the HRR recipe on the smallest non-curve example: the projective plane, the line bundle of degree $d$ , with the answer to be read off as a binomial coefficient.

Step 1. Set-up. The projective plane $P^{2}$ has hyperplane class $H$ with $H^{2} = 1$ . The canonical class is $K = - 3 H$ from the Euler sequence. The Chern character of $O (d)$ is $1 + d H + d^{2} H^{2} /2$ , truncated in dimension two. The Todd class of the tangent bundle of $P^{2}$ is $1 + 3 H /2 + H^{2}$ (the constant 1, the linear term $- K /2 = 3 H /2$ , and the quadratic term $(K^{2} + c_{2}) /12 = (9 + 3) /12 \cdot H^{2} = H^{2}$ ).

Step 2. Multiply. The product of the Chern character and the Todd class is $(1 + d H + d^{2} H^{2} /2) (1 + 3 H /2 + H^{2})$ . Carrying out the multiplication and collecting the degree- $2$ piece gives $d^{2} /2 + 3 d /2 + 1$ (the $1 \times H^{2}$ , $d H \times 3 H /2$ , and $d^{2} H^{2} /2 \times 1$ contributions, summing to $1 + 3 d /2 + d^{2} /2$ ).

Step 3. Integrate. The integral of $H^{2}$ over $P^{2}$ is $1$ (the fundamental-class normalisation). So the answer is $d^{2} /2 + 3 d /2 + 1 = (d + 1) (d + 2) /2 = (2 d + 2)$ .

Step 4. Check. The number of monomials of degree $d$ in three variables is $(2 d + 2)$ , which is the dimension of the space of global sections of $O_{P^{2}} (d)$ on $P^{2}$ when $d \geq 0$ . Higher cohomology vanishes in this range by 04.03.04. So the holomorphic Euler characteristic agrees with the dimension count, and HRR returns the right answer with the right algebraic structure.

What this tells us. The recipe is mechanical: write the Chern character of the bundle, write the Todd class of the tangent bundle, multiply, pick off the top-degree piece, integrate. The answer is a closed expression in the data of the bundle and the variety. The same pattern produces the binomial coefficient on $P^{n}$ , the surface Riemann-Roch identity on a surface, and the arithmetic-genus expression on every smooth projective variety.

Check your understanding Beginner

Formal definition Intermediate+

Throughout this unit, let $k$ be an algebraically closed field of characteristic zero, let $X$ be a smooth projective variety over $k$ of complex dimension $n$ , and let $E$ be a coherent sheaf on $X$ . The HRR identity (see 04.05.10) reads $$ \chi(X, \mathcal{E}) = \int_X \mathrm{ch}(\mathcal{E}) \cdot \mathrm{td}(T_X), $$ with $ch (E) = \sum_{i} e^{x_{i}}$ in formal Chern roots and $td (F) = \prod_{i} x_{i} / (1 - e^{- x_{i}})$ for a holomorphic vector bundle $F$ on $X$ .

Definition (truncated Todd-class expansion). On a smooth projective variety of complex dimension $n$ , the Todd class $td (T_{X})$ truncates in cohomological degree to a polynomial of weighted degree $n$ in the Chern classes $c_{i} (T_{X})$ : $$ \mathrm{td}(T_X) = 1 + \tfrac{c_1}{2} + \tfrac{c_1^2 + c_2}{12} + \tfrac{c_1 c_2}{24} + \tfrac{-c_1^4 + 4 c_1^2 c_2 + c_1 c_3 + 3 c_2^2 - c_4}{720} + \cdots, $$ with the displayed terms describing the cases $n = 0, 1, 2, 3, 4$ .

Definition (Todd class on projective space). On $P^{n}$ with hyperplane class $H$ ( $H^{n + 1} = 0$ , $\int_{P^{n}} H^{n} = 1$ ), the Todd class of the tangent bundle is $$ \mathrm{td}(T_{\mathbb{P}^n}) = \left(\frac{H}{1 - e^{-H}}\right)^{n+1}, $$ extracted from the Euler exact sequence $0 \to O \to O (1)^{\oplus (n + 1)} \to T_{P^{n}} \to 0$ and the multiplicativity of the Todd class on short exact sequences. Equivalently, in low dimensions: $td (T_{P^{1}}) = 1 + H$ ; $td (T_{P^{2}}) = 1 + \frac{3 H}{2} + H^{2}$ ; $td (T_{P^{3}}) = 1 + 2 H + \frac{11 H ^{2}}{6} + H^{3}$ .

Definition ( $χ_{y}$ -genus). The $χ_{y}$ -genus of a smooth projective variety $X$ of complex dimension $n$ is the polynomial $$ \chi_y(X) = \sum_{p = 0}^{n} y^p , \chi(X, \Omega_X^p) = \sum_{p, q} (-1)^q y^p h^{p,q}(X) \in \mathbb{Z}[y], $$ where $Ω_{X}^{p} = \land^{p} Ω_{X}^{1}$ is the sheaf of holomorphic $p$ -forms and $h^{p, q} (X) = dim H^{q} (X, Ω_{X}^{p})$ is the $(p, q)$ -Hodge number. The polynomial $χ_{y} (X)$ specialises to the arithmetic genus $χ (O_{X}) = χ_{0} (X) = χ_{y} (X) ∣_{y = 0}$ , to the signature $σ (X) = χ_{1} (X)$ (for $n$ even), and to the topological Euler characteristic $χ_{- 1} (X) = \int_{X} c_{n} (T_{X}) = e (X)$ .

Definition ( $td_{y}$ -class). The $td_{y}$ -class of a holomorphic vector bundle $F$ of rank $r$ is the multiplicative class $$ \mathrm{td}y(\mathcal{F}) = \prod{i = 1}^{r} \frac{x_i (1 + y e^{-x_i})}{1 - e^{-x_i}}, $$ attached to the power series $Q_{y} (x) = x (1 + y e^{- x}) / (1 - e^{- x})$ . The Hirzebruch identity $$ \chi_y(X) = \int_X \mathrm{td}_y(T_X) $$ specialises to HRR at $y = 0$ (since $td_{0} = td$ and $χ_{0} (X) = χ (O_{X})$ ), to the Hirzebruch signature theorem at $y = 1$ (the $L$ -genus computes the signature), and to the Gauss-Bonnet-Chern identity $e (X) = \int_{X} c_{n} (T_{X})$ at $y = - 1$ .

Counterexamples to common slips

The Chern character of $O (d)$ on $P^{n}$ is $e^{d H} = 1 + d H + d^{2} H^{2} /2! + \dots + d^{n} H^{n} / n!$ , truncated at degree $n$ because $H^{n + 1} = 0$ on $P^{n}$ . Forgetting the truncation gives a power series rather than an element of the cohomology ring, and the integral has no meaning.
The Todd class of $T_{P^{n}}$ is not simply $1 + (n + 1) H /2 + \dots$ — the linear coefficient is correctly $(n + 1) /2 \cdot H$ from the multiplicative-sequence expansion, but the higher coefficients require the Euler-sequence calculation. Computing only the leading two terms misses the integer answer on $P^{n}$ for $n \geq 2$ .
Noether's formula $12 χ (O_{S}) = K_{S}^{2} + c_{2} (S)$ uses $c_{1} (T_{S}) = - K_{S}$ as the first Chern class of the tangent bundle, not the canonical class itself. Conflating the two introduces a sign error that propagates to the surface table on K3, $P^{2}$ , and Hirzebruch surfaces.
The $χ_{y}$ -genus identity $χ_{y} (P^{n}) = 1 - y + y^{2} - \dots + (- y)^{n}$ uses the Hodge numbers $h^{p, q} (P^{n}) = 1$ if $p = q \leq n$ and $0$ otherwise. The signature contribution at $y = 1$ gives $χ_{1} (P^{n}) = 1$ for $n$ even and $0$ for $n$ odd, matching the topological signature.

Key theorem with proof Intermediate+

Theorem (worked HRR specialisations). Each of the following identities is the value of the right-hand side of HRR for the indicated pair of variety and sheaf.

(1) (Line bundles on projective space.) On $P^{n}$ with hyperplane class $H$ , $$ \chi(\mathbb{P}^n, \mathcal{O}(d)) = \binom{n + d}{n}. $$

(2) (Tangent bundle on $P^{2}$ .) On $P^{2}$ , $$ \chi(\mathbb{P}^2, T_{\mathbb{P}^2}) = 8. $$

(3) (Riemann-Roch on a smooth projective curve.) On a smooth projective curve $C$ of genus $g$ and a line bundle $L$ of degree $d$ , $$ \chi(C, L) = d - g + 1. $$

(4) (Noether's formula.) On a smooth projective surface $S$ , $$ \chi(\mathcal{O}_S) = \tfrac{1}{12}(K_S^2 + c_2(S)). $$

(5) ( $χ_{y}$ -genus of projective space.) For every $n$ , $$ \chi_y(\mathbb{P}^n) = 1 - y + y^2 - y^3 + \cdots + (-y)^n. $$

Proof.

(1) The Chern character is $ch (O (d)) = e^{d H} = \sum_{k = 0}^{n} d^{k} H^{k} / k!$ (truncated at $k = n$ since $H^{n + 1} = 0$ ). The Todd class is $td (T_{P^{n}}) = (H / (1 - e^{- H}))^{n + 1}$ by the Euler-sequence calculation in the Formal-definition section. The HRR product is $$ \mathrm{ch}(\mathcal{O}(d)) \cdot \mathrm{td}(T_{\mathbb{P}^n}) = e^{d H} \cdot \left(\frac{H}{1 - e^{-H}}\right)^{n+1}, $$ and the integral picks off the coefficient of $H^{n}$ . By the residue formulation, this coefficient equals $Res_{H = 0} (e^{d H} / (1 - e^{- H})^{n + 1}) \cdot d H / H \cdot H^{n + 1}$ , which after substituting $u = 1 - e^{- H}$ (so $H = - lo g (1 - u)$ and $d H / d u = 1/ (1 - u)$ ) becomes the coefficient of $u^{n}$ in $(1 - u)^{- d - 1}$ , equal to $(n - d - 1) (- 1)^{n} = (n n + d)$ by the standard binomial identity. So $χ (P^{n}, O (d)) = (n n + d)$ , the classical projective-space dimension formula recovered as the polynomial value on the positive-twist range and the analytic-continuation value elsewhere.

(2) From the Euler sequence $0 \to O \to O (1)^{\oplus 3} \to T_{P^{2}} \to 0$ on $P^{2}$ , the Chern character is additive: $ch (T_{P^{2}}) = ch (O (1)^{\oplus 3}) - ch (O) = 3 ch (O (1)) - 1 = 3 (1 + H + H^{2} /2) - 1 = 2 + 3 H + 3 H^{2} /2$ . The Todd class is $td (T_{P^{2}}) = 1 + 3 H /2 + H^{2}$ . The HRR product is $$ \mathrm{ch}(T_{\mathbb{P}^2}) \cdot \mathrm{td}(T_{\mathbb{P}^2}) = (2 + 3 H + 3 H^2 / 2)(1 + 3 H / 2 + H^2). $$ Expanding and collecting the degree- $2$ part: $2 \cdot H^{2} + 3 H \cdot 3 H /2 + 3 H^{2} /2 \cdot 1 = 2 + 9/2 + 3/2 = 8$ . So $χ (P^{2}, T_{P^{2}}) = 8$ . The direct cohomology check: $H^{0} (P^{2}, T_{P^{2}})$ is the space of holomorphic vector fields on $P^{2}$ , which has dimension $dim PGL (3, k) = 8$ ; higher cohomology of $T_{P^{2}}$ vanishes by the Bott vanishing theorem for the tangent bundle on $P^{n}$ . So the holomorphic Euler characteristic equals the dimension of $H^{0}$ , which is $8$ , matching the HRR computation.

(3) On a smooth projective curve $C$ of genus $g$ , the Todd class of the tangent bundle truncates in dimension one to $td (T_{C}) = 1 + c_{1} (T_{C}) /2 = 1 - K_{C} /2$ , and the integral of the degree-one piece is $\int_{C} (- K_{C} /2) = - (2 g - 2) /2 = 1 - g$ (using the curve-genus formula $de g K_{C} = 2 g - 2$ ). The Chern character of a line bundle $L$ of degree $d$ truncates to $ch (L) = 1 + c_{1} (L) = 1 + L$ (where $L$ denotes also its first Chern class on the curve). The HRR product is $ch (L) \cdot td (T_{C}) = (1 + L) (1 + (1 - g) [pt]) = 1 + L + (1 - g) [pt]$ (after identifying the degree-one piece $c_{1} (T_{C}) /2$ with $(1 - g) [pt]$ via integration). The integral picks off the degree-one part: $χ (C, L) = \int_{C} (L + (1 - g) [pt]) = d + (1 - g) = d - g + 1$ . This is the classical Riemann-Roch identity for line bundles on a smooth projective curve; see 04.04.01.

(4) Apply HRR to $E = O_{S}$ with $ch (O_{S}) = 1$ . The integrand reduces to $td (T_{S})$ alone, and the integral picks off the top-degree part. On a smooth projective surface, $$ \mathrm{td}(T_S) = 1 + \tfrac{c_1(T_S)}{2} + \tfrac{c_1(T_S)^2 + c_2(T_S)}{12}, $$ with $c_{1} (T_{S}) = - K_{S}$ and $c_{2} (T_{S}) = c_{2} (S)$ (the second Chern class of the tangent bundle, which on a smooth projective surface equals the topological Euler-class evaluation $\int_{S} c_{2} (T_{S}) = e (S)$ ). The top-degree part is $(K_{S}^{2} + c_{2} (S)) /12$ , and integrating gives $χ (O_{S}) = (K_{S}^{2} + c_{2} (S)) /12$ . Equivalently $12 χ (O_{S}) = K_{S}^{2} + c_{2} (S)$ , which is Noether's formula (Noether 1875).

Apply to three surfaces. On $S = P^{2}$ : $K_{S} = - 3 H$ , $K_{S}^{2} = 9$ , $c_{2} (P^{2}) = 3$ (the topological Euler characteristic of $P^{2}$ is $3$ ), so $χ (O_{P^{2}}) = (9 + 3) /12 = 1$ , matching $H^{0} = k$ and higher cohomology vanishing. On $S = P^{1} \times P^{1}$ with rulings $h_{1}, h_{2}$ ( $h_{1}^{2} = h_{2}^{2} = 0$ , $h_{1} \cdot h_{2} = 1$ ): $K_{S} = - 2 h_{1} - 2 h_{2}$ , $K_{S}^{2} = (- 2) (- 2) (2) = 8$ , $e (S) = 4$ (Euler characteristic of a product is the product of the Eulers, $2 \cdot 2 = 4$ ), so $χ (O_{S}) = (8 + 4) /12 = 1$ , again matching. On a smooth quartic $S \subset P^{3}$ (the K3 case): $K_{S} = 0$ by adjunction (since $- 4 + 4 = 0$ , 04.05.07), and $c_{2} (S) = 24$ by the conormal-sequence calculation (the smooth-quartic $S$ has $e (S) = 24$ , a classical computation of Lefschetz), so $χ (O_{S}) = (0 + 24) /12 = 2$ , the K3 arithmetic genus.

(5) The $χ_{y}$ -genus identity is $χ_{y} (X) = \int_{X} td_{y} (T_{X})$ , with $td_{y} (T_{X})$ the multiplicative class attached to the power series $Q_{y} (x) = x (1 + y e^{- x}) / (1 - e^{- x})$ . On $P^{n}$ with hyperplane class $H$ , the Euler sequence gives $td_{y} (T_{P^{n}}) = (H (1 + y e^{- H}) / (1 - e^{- H}))^{n + 1}$ . The integral picks off the coefficient of $H^{n}$ . By a residue computation parallel to (1) — substitute $u = 1 - e^{- H}$ , so $1 - u = e^{- H}$ , $e^{- H} = 1 - u$ , $1 + y e^{- H} = 1 + y (1 - u) = (1 + y) - y u$ — the integrand becomes $(((1 + y) - y u) / u)^{n + 1}$ in the formal variable $u$ , multiplied by the Jacobian $1/ (1 - u)$ . Reading off the coefficient of $u^{n}$ in $((1 + y) - y u)^{n + 1} / (u^{n + 1} (1 - u))$ and using $((1 + y) - y u)^{n + 1} = \sum_{k} (k n + 1) (1 + y)^{n + 1 - k} (- y u)^{k}$ , after the residue extraction the answer collapses to $$ \chi_y(\mathbb{P}^n) = \sum_{k = 0}^{n} (-y)^k = 1 - y + y^2 - y^3 + \cdots + (-y)^n. $$ The cross-check is the Hodge-diamond computation: $h^{p, q} (P^{n}) = 1$ if $p = q \leq n$ and $0$ otherwise (Dolbeault cohomology of projective space is concentrated on the diagonal). So $χ_{y} (P^{n}) = \sum_{p} (- 1)^{p} y^{p} h^{p, p} (P^{n}) = \sum_{p = 0}^{n} (- 1)^{p} y^{p} = 1 - y + y^{2} - \dots + (- y)^{n}$ , matching the HRR computation.

$□$

Bridge. These five worked computations build toward the body of classical algebraic geometry: the projective-space binomial-coefficient identity in (1) appears again in the Hilbert-polynomial theory of projective embeddings, the foundational reason every projectively embedded variety has a polynomial dimension-table for high-degree twists; the tangent-bundle computation in (2) is exactly the dimension-of-automorphisms calculation that identifies $PGL (n + 1, k) ≅ Aut (P^{n})$ via $H^{0} (P^{n}, T_{P^{n}})$ , and this is dual to the deformation-theoretic content of $H^{1} (P^{n}, T_{P^{n}}) = 0$ (projective space is rigid); the curve identity in (3) is the foundational reason the Riemann surface theory of 04.04.01 sits inside the HRR framework, and generalises to coherent sheaves on smooth projective curves of any genus; Noether's surface formula in (4) is dual to the Hodge index theorem on a surface (04.05.09) via the coupling of the canonical-square invariant to the topological Euler characteristic; and the $χ_{y}$ -genus identity in (5) is the bridge between HRR and the Hirzebruch signature theorem (03.06.11), the central insight being that one parameter-family of multiplicative genera interpolates between the holomorphic Euler characteristic at $y = 0$ , the signature at $y = 1$ , and the topological Euler characteristic at $y = - 1$ . Putting these together, every classical genus-and-dimension table in algebraic geometry — the Plücker formula on plane curves, the Castelnuovo and Beauville tables on surfaces, the arithmetic-genus tables on complete intersections — is one residue computation away from a worked HRR application, and the bridge is the recipe encoded by the Chern-character / Todd-class product.

Exercises Intermediate+

Exercise 5 (medium, numeric).

Verify Noether's formula on the smooth quartic K3 surface $S \subset P^{3}$ by computing $K_{S}^{2} + c_{2} (S)$ and checking the answer agrees with $12 χ (O_{S}) = 24$ .

Hint

On the K3 surface, $K_{S} = 0$ by adjunction (since $- 4 + 4 = 0$ for the smooth quartic in $P^{3}$ ). The Euler characteristic of the underlying smooth manifold of a smooth quartic in $P^{3}$ is $24$ , computable from the Chern data of the quartic via the conormal sequence.

Answer

On a smooth quartic $S \subset P^{3}$ : $K_{S} = 0$ , hence $K_{S}^{2} = 0$ . The topological Euler characteristic $e (S) = c_{2} (S)$ is computed by the conormal sequence $0 \to T_{S} \to T_{P^{3}} ∣_{S} \to N_{S / P^{3}} \to 0$ with $N_{S / P^{3}} = O_{S} (4)$ .

Using $c (T_{P^{3}}) = (1 + H)^{4} = 1 + 4 H + 6 H^{2} + 4 H^{3} + H^{4}$ and $c (N) = 1 + 4 H$ , the Whitney sum formula gives $c (T_{S}) = c (T_{P^{3}} ∣_{S}) / c (N) = (1 + 4 H + 6 H^{2} + 4 H^{3}) / (1 + 4 H) = (1 + 4 H + 6 H^{2} + \dots) (1 - 4 H + 16 H^{2} - \dots)$ modulo terms of $H$ -degree $\geq 3$ on $S$ (a surface), giving $c (T_{S}) = 1 + 0 + 6 H^{2} \cdot (coefficient corrections) = 1 + 0 + (6 - 16 + \dots) H^{2}$ .

A careful track: $c_{1} (T_{S}) = c_{1} (T_{P^{3}} ∣_{S}) - c_{1} (N) = 4 H - 4 H = 0$ , matching $K_{S} = 0$ ; and $c_{2} (T_{S}) = c_{2} (T_{P^{3}} ∣_{S}) - c_{1} (T_{S}) \cdot c_{1} (N) - c_{2} (N) = 6 H^{2} - 0 - 0 = 6 H^{2}$ on $S$ , integrated against the surface class $[S] = 4 H \cdot [P^{3}]$ giving $\int_{S} c_{2} (T_{S}) = 6 \cdot 4 = 24$ . So $K_{S}^{2} + c_{2} (S) = 0 + 24 = 24$ , and Noether's formula reads $12 χ (O_{S}) = 24$ , hence $χ (O_{S}) = 2$ , the K3 arithmetic genus. Rubric: full credit for $24$ with the conormal-sequence Chern computation; partial credit for $K_{S}^{2} = 0$ and the K3 condition alone.

Exercise 6 (medium, symbolic).

Verify the curve specialisation of HRR by direct cohomology on $P^{1}$ : compute $χ (P^{1}, O (d))$ for all $d \in Z$ from the Čech complex, and check it matches $d - g + 1 = d + 1$ .

Hint

On $P^{1}$ with the standard two-piece affine cover, the Čech complex computes $H^{0} = k {x_{0}^{a} x_{1}^{b} : a + b = d, a, b \geq 0}$ and $H^{1} = k {x_{0}^{a} x_{1}^{b} : a + b = d, a, b < 0}$ . Count monomials.

Answer

For $d \geq 0$ : $H^{0} (P^{1}, O (d))$ has basis ${x_{0}^{d}, x_{0}^{d - 1} x_{1}, \dots, x_{1}^{d}}$ , dimension $d + 1$ . $H^{1} = 0$ (no Laurent monomials of total degree $d$ with both exponents strictly negative when $d \geq - 1$ ). So $χ = d + 1 - 0 = d + 1$ . For $d \leq - 2$ : $H^{0} = 0$ (no monomials of negative total degree with both exponents non-negative). $H^{1}$ has basis ${x_{0}^{a} x_{1}^{b} : a + b = d, a, b \leq - 1}$ , which under the substitution $a = - i, b = - j$ with $i, j \geq 1$ and $- i - j = d$ , gives dimension $∣ d ∣ - 1 = - d - 1$ . So $χ = 0 - (- d - 1) = d + 1$ . For $d = - 1$ : both $H^{0}$ and $H^{1}$ vanish, so $χ = 0 = - 1 + 1$ . In every case, $χ (O (d)) = d + 1 = d - g + 1$ at $g = 0$ . Rubric: full credit for the full Čech-complex computation and the match with the curve specialisation.

Exercise 7 (hard, short-answer).

Verify the $χ_{y}$ -genus identity $χ_{y} (P^{2}) = 1 - y + y^{2}$ by direct computation of the Hodge numbers of $P^{2}$ and reading the alternating polynomial.

Hint

The Hodge diamond of $P^{2}$ has $h^{p, q} (P^{2}) = 1$ for $(p, q) \in {(0, 0), (1, 1), (2, 2)}$ and $h^{p, q} = 0$ otherwise. Read $χ_{y} (P^{2}) = \sum_{p} (- 1)^{p} y^{p} h^{p, p}$ (since $h^{p, q} = 0$ for $p \neq = q$ , only the diagonal contributes; and $χ (Ω^{p}) = (- 1)^{p} h^{p, p}$ via Hodge symmetry and the diagonal concentration).

Answer

The Hodge diamond of $P^{2}$ : $$ \begin{array}{ccccc} & & h^{0,0} & & \ & h^{1,0} & & h^{0,1} & \ h^{2,0} & & h^{1,1} & & h^{0,2} \ & h^{2,1} & & h^{1,2} & \ & & h^{2,2} & & \end{array}

\begin{array}{ccccc} & & 1 & & \ & 0 & & 0 & \ 0 & & 1 & & 0 \ & 0 & & 0 & \ & & 1 & & \end{array}. $$ So $χ (O_{P^{2}}) = χ (Ω_{P^{2}}^{0}) = h^{0, 0} - h^{0, 1} + h^{0, 2} = 1 - 0 + 0 = 1$ ; $χ (Ω_{P^{2}}^{1}) = - h^{1, 0} + h^{1, 1} - h^{1, 2} = - 0 + 1 - 0 = 1$ , but the $χ_{y}$ -genus formula picks the sign $(- 1)^{q}$ on $h^{p, q}$ and the variable $y$ to the power $p$ , so $χ_{1} (P^{2}) = h^{1, 1} - 0 - 0 = 1$ contributing $- y$ (the $y^{1}$ piece, with the alternating sign convention $\sum (- 1)^{q} y^{p} h^{p, q} = - y \cdot h^{1, 1} = - y$ ); $χ_{2} (P^{2}) = h^{2, 2} = 1$ contributing $y^{2}$ . Summing: $χ_{y} (P^{2}) = 1 - y + y^{2}$ , matching the HRR formula at $n = 2$ .

The cross-check at $y = 0$ gives $χ_{0} (P^{2}) = 1 = χ (O_{P^{2}})$ , at $y = 1$ gives the signature $χ_{1} (P^{2}) = 1$ (the signature of the intersection form $H^{2} (P^{2}, R) = R$ is $1$ ), and at $y = - 1$ gives the topological Euler characteristic $χ_{- 1} (P^{2}) = 1 + 1 + 1 = 3 = e (P^{2})$ , all three correct. Rubric: full credit for the polynomial identity with the three specialisations verified.

Exercise 8 (hard, short-answer).

On the Hirzebruch surface $F_{n}$ with section $C_{0}$ ( $C_{0}^{2} = - n$ ), ruling $F$ ( $F^{2} = 0$ , $C_{0} \cdot F = 1$ ), and canonical class $K = - 2 C_{0} - (n + 2) F$ , verify $χ (O_{F_{n}}) = 1$ using Noether's formula, and compute $c_{2} (F_{n})$ .

Hint

Noether's formula reads $12 χ (O_{S}) = K_{S}^{2} + c_{2} (S)$ . Compute $K_{S}^{2}$ on $F_{n}$ first, then use the structure-sheaf calculation $χ (O_{F_{n}}) = 1$ (the irregularity $q = 0$ and geometric genus $p_{g} = 0$ on a rational surface) to deduce $c_{2}$ .

Answer

Compute $K_{F_{n}}^{2} = (- 2 C_{0} - (n + 2) F)^{2} = 4 C_{0}^{2} + 4 (n + 2) C_{0} \cdot F + (n + 2)^{2} F^{2} = 4 (- n) + 4 (n + 2) + 0 = - 4 n + 4 n + 8 = 8$ . So $K_{F_{n}}^{2} = 8$ for every $n \geq 0$ . The Hirzebruch surface is rational ( $F_{n}$ is birational to $P^{2}$ via a sequence of elementary transformations), hence has irregularity $q = 0$ and geometric genus $p_{g} = 0$ , so $χ (O_{F_{n}}) = 1 - q + p_{g} = 1$ . Noether's formula gives $12 = K^{2} + c_{2} = 8 + c_{2}$ , hence $c_{2} (F_{n}) = 4$ . The cross-check: the topological Euler characteristic of $F_{n}$ is $4$ for every $n$ (the surface is topologically $P^{1} \times P^{1}$ if $n$ is even and $P^{2} # \overline{P^{2}}$ if $n$ is odd, both with $e = 4$ ), and $\int_{F_{n}} c_{2} (T_{F_{n}}) = e (F_{n}) = 4$ , matching. Rubric: full credit for $K^{2} = 8$ , $χ (O) = 1$ , $c_{2} = 4$ , with the topological cross-check.

Lean formalization Intermediate+

lean_status: none — Mathlib has the Euler-characteristic and Chern-class scaffolding of the general HRR unit 04.05.10 but does not ship any of the worked computations as named lemmas. The intended formalisation reads schematically:

import Mathlib.AlgebraicGeometry.EulerCharacteristic
import Mathlib.AlgebraicGeometry.RiemannRoch
import Mathlib.AlgebraicGeometry.Projective

namespace Codex.AlgebraicGeometry.Divisors.HRRWorked

variable {k : Type*} [Field k] [IsAlgClosed k] [CharZero k]

/-- Hirzebruch-Riemann-Roch on projective space: the closed binomial
formula for the holomorphic Euler characteristic of O(d). -/
theorem chi_O_d_projective (n d : ℕ) :
    eulerCharacteristic (ProjectiveSpace k n) (lineBundle d) =
      Nat.choose (n + d) n := by
  -- expand td(T_P^n) via the Euler sequence
  -- compute the residue of e^{dH} / (1 - e^{-H})^{n+1}
  sorry

/-- Tangent-bundle Euler characteristic on the projective plane. -/
theorem chi_T_P2 :
    eulerCharacteristic (ProjectiveSpace k 2) (tangentBundle _) = 8 := by
  sorry  -- via the Euler sequence ch(T) = 3 ch(O(1)) - 1

/-- Curve specialisation: Riemann-Roch on a smooth projective curve. -/
theorem chi_curve_line_bundle
    (C : SmoothProjectiveCurve k) (L : LineBundle C) :
    eulerCharacteristic C L = L.degree - C.genus + 1 := by
  sorry  -- td(T_C) = 1 - K_C/2, ch(L) = 1 + L

/-- Noether's formula on a smooth projective surface. -/
theorem noether_surface (S : SmoothProjectiveSurface k) :
    12 * eulerCharacteristic S (structureSheaf S) =
      canonicalClass S |>.selfIntersection +
      secondChernNumber S := by
  sorry  -- HRR at E = O_S, td_2(T_S) = (c1^2 + c2)/12

/-- The chi_y-genus of projective space. -/
theorem chi_y_projective (n : ℕ) :
    chiY (ProjectiveSpace k n) = ∑ k in Finset.range (n + 1), (-1)^k * y^k := by
  sorry  -- via the residue computation on (H(1 + y e^{-H}) / (1 - e^{-H}))^{n+1}

The proof gap is substantive but each step compiles cleanly once the general HRR theorem is packaged: the binomial-coefficient identity reduces to a residue computation in formal power series, the tangent-bundle case uses the Euler exact sequence and Chern-character additivity, the curve case uses the truncation of the Todd class in dimension one, Noether's formula is the structure-sheaf specialisation of HRR with the surface-tier Todd-class truncation, and the $χ_{y}$ -genus identity is a residue computation parallel to the binomial-coefficient one. Each formalisation target unlocks a concrete corollary of HRR and is a natural follow-on to the general theorem.

Advanced results Master

Theorem (Hirzebruch-Riemann-Roch on $P^{n}$ via residues; Hirzebruch §IV.22). On $P^{n}$ with hyperplane class $H$ , the holomorphic Euler characteristic of $O (d)$ is $$ \chi(\mathbb{P}^n, \mathcal{O}(d)) = \mathrm{Res}_{H = 0} \frac{e^{d H}}{(1 - e^{-H})^{n+1}} \cdot \frac{dH}{H^{n+1}} \cdot H^{n+1} = \binom{n + d}{n}. $$

The residue evaluation goes through the substitution $u = 1 - e^{- H}$ , which is a biholomorphism between a small disk around $H = 0$ and a small disk around $u = 0$ (the Jacobian $d u / d H = e^{- H} = 1 - u$ is non-vanishing near the origin). The integrand becomes $((1 - u)^{- d}) \cdot u^{- n - 1} \cdot (1 - u)^{- 1} \cdot d u$ (a function of $u$ alone), and the residue at $u = 0$ is the coefficient of $u^{n}$ in $(1 - u)^{- d - 1}$ , equal to $(n - d - 1) (- 1)^{n} = (n n + d)$ by the binomial-coefficient identity for upper-negative arguments. The residue computation thus replaces the cohomological computation by a one-variable contour-integral evaluation, and the binomial-coefficient answer emerges as a coefficient extraction in a formal power series. The same residue method applies to vector bundles on $P^{n}$ that decompose as direct sums of line bundles after a finite cover (or whose Chern character can be expanded in formal Chern roots), and produces closed Euler-characteristic formulas for symmetric powers, exterior powers, and tensor products of $O (d_{i})$ .

Theorem (HRR on Hirzebruch surfaces $F_{n}$ ; Eisenbud-Harris §13.5). Let $F_{n} = P (O_{P^{1}} \oplus O_{P^{1}} (- n))$ be the Hirzebruch surface, with the standard ruling $F$ (fibre class) and section $C_{0}$ ( $C_{0}^{2} = - n$ , $C_{0} \cdot F = 1$ , $F^{2} = 0$ ). For a line bundle $L = O_{F_{n}} (a C_{0} + b F)$ , $$ \chi(\mathbb{F}n, L) = (a + 1) \left(b - \tfrac{n}{2} a + 1\right) + \tfrac{n}{2} a \left(\tfrac{n}{2} a + 1\right) \cdot \delta{n \mathrm{\ even}} + \cdots $$ more compactly, $$ \chi(\mathbb{F}n, \mathcal{O}(a C_0 + b F)) = 1 + \tfrac{1}{2}((a C_0 + b F)^2 - (a C_0 + b F) \cdot K), $$ *by surface Riemann-Roch with $K = - 2 C_{0} - (n + 2) F$ , $K^{2} = 8$ , $\chi(\mathcal{O}{\mathbb{F}_n}) = 1 $,$ c_2 = 4$.*

The intersection-number computation: $(a C_{0} + b F)^{2} = a^{2} C_{0}^{2} + 2 ab C_{0} \cdot F + b^{2} F^{2} = - n a^{2} + 2 ab + 0 = 2 ab - n a^{2}$ ; $(a C_{0} + b F) \cdot K = (a C_{0} + b F) \cdot (- 2 C_{0} - (n + 2) F) = - 2 a C_{0}^{2} - a (n + 2) C_{0} \cdot F - 2 b C_{0} \cdot F - b (n + 2) F^{2} = 2 an - a (n + 2) - 2 b + 0 = - 2 b - 2 a + an - an = - 2 a - 2 b$ after simplification. So $χ (F_{n}, O (a C_{0} + b F)) = 1 + \frac{1}{2} (2 ab - n a^{2} + 2 a + 2 b) = 1 + ab - n a^{2} /2 + a + b$ . The classical Hirzebruch-surface dimension table follows by specialisation; the well-known case $n = 0$ ( $F_{0} = P^{1} \times P^{1}$ ) reproduces $χ (O (a, b)) = (a + 1) (b + 1)$ after the cancellation of the $- n a^{2} /2$ term.

Theorem ( $\hat{A}$ -genus and the integrality of the index; Hirzebruch §I). On a smooth $Spin$ manifold $M$ of dimension $4 k$ , the $\hat{A}$ -genus $\hat{A} (M) = \int_{M} \hat{A} (T M) \in Z$ is the index of the Dirac operator on $M$ , and is an integer.

The connection to HRR: on a smooth projective variety of complex dimension $n$ , the Todd genus $\int_{X} td (T_{X}) = χ (O_{X})$ is the holomorphic Euler characteristic of the structure sheaf, which is an integer. The $\hat{A}$ -genus is the parallel real-manifold invariant attached to the power series $Q (x) = (x /2) / sinh (x /2)$ , the unique multiplicative sequence among those starting $1 + a_{1} x + \dots$ that computes the index of the Dirac operator on a $Spin$ manifold. The two genera are related on a smooth complex projective variety with a $Spin$ structure: $td (T_{X}) = \hat{A} (T_{X}) \cdot e^{c_{1} (T_{X}) /2}$ . The $\hat{A}$ -genus computation, like the Todd-genus computation, is a discrete integer-valued polynomial in Chern classes, and Atiyah-Singer (1963) provides the index-theoretic identification that makes the integrality manifest as a Fredholm index of an elliptic operator; see 03.09.10.

Theorem (holomorphic Lefschetz fixed-point formula; Atiyah-Bott 1968). Let $G$ act on a smooth projective variety $X$ over $C$ with isolated fixed points ${p_{1}, \dots, p_{m}}$ , all of which are non-degenerate (the action linearises around each $p_{i}$ with eigenvalues ${λ_{i, 1}, \dots, λ_{i, n}}$ on the tangent space $T_{p_{i}} X$ ). Let $E$ be a $G$ -equivariant holomorphic vector bundle on $X$ . The $G$ -equivariant holomorphic Euler characteristic of $E$ is $$ \chi^G(X, \mathcal{E}) = \sum_{i = 1}^{m} \frac{\mathrm{tr}(g | \mathcal{E}{p_i})}{\prod{j = 1}^{n} (1 - \lambda_{i, j}^{-1})}, $$ for a generic group element $g \in G$ .

The constant-class summand is the HRR identity itself: setting $g = e$ the identity and integrating, the right-hand side reduces (after careful limit analysis) to $\int_{X} ch (E) \cdot td (T_{X})$ , and the equation $χ^{G} (X, E) ∣_{g = e} = χ (X, E)$ is HRR. The non-equivariant statement is thus a degeneration of the equivariant one. The fixed-point formula computes the equivariant Euler characteristic from purely local data — the trace of $g$ on the fibre of $E$ over each fixed point, and the linearisation eigenvalues of $g$ on the tangent space — without computing the cohomology globally. The classical applications include the Weyl character formula (apply to $G = T$ a maximal torus acting on $G / B$ a flag variety, with $E$ the line bundle of a dominant weight, recovering the trace of the Weyl character as a localised sum over $∣ W ∣$ fixed points indexed by the Weyl group), and the Atiyah-Singer index formula in its $G$ -equivariant form, which packages the fixed-point sum as an elliptic-operator index.

Theorem (HRR for symmetric products of curves; Macdonald 1962). Let $C$ be a smooth projective curve of genus $g$ and let $Sym^{n} C = C^{n} / S_{n}$ be the $n$ -th symmetric product, a smooth projective variety of complex dimension $n$ . The Hodge numbers and the $χ_{y}$ -genus of $Sym^{n} C$ are computable via HRR applied to the universal line bundle on $Sym^{n} C \times C$ .

The Macdonald formula reads $χ_{y} (Sym^{n} C) =$ a polynomial in $y$ with coefficients in the Hodge numbers of $C$ , and specialises to $χ (O_{Sym^{n} C}) = (n g)$ (with the convention $(n g) = 0$ if $n > g$ ). The HRR derivation uses the natural Abel-Jacobi map $Sym^{n} C \to Pic^{n} (C)$ and the fact that the symmetric product is a fibre bundle in projective spaces over the Jacobian for $n \geq 2 g - 1$ . The cohomology of $Sym^{n} C$ is then computed inductively in $n$ via HRR applied to the line bundle of degree- $n$ divisors on $C$ .

Theorem (HRR for the Hilbert scheme of points; Göttsche 1990). Let $S$ be a smooth projective surface and let $S^{[n]}$ be the Hilbert scheme of $n$ points on $S$ , a smooth projective variety of complex dimension $2 n$ . The generating function for the Euler characteristics of structure sheaves on $S^{[n]}$ is $$ \sum_{n \geq 0} \chi(\mathcal{O}{S^{[n]}}) \cdot t^n = \prod{m \geq 1} \frac{1}{(1 - t^m)^{\chi(\mathcal{O}_S)}}. $$

Göttsche's formula is one of the most striking applications of HRR in higher-dimensional algebraic geometry: the generating series is a closed expression in $χ (O_{S})$ , the arithmetic genus of the underlying surface, and the formula generalises to a $y$ -variable refinement that computes the full $χ_{y}$ -genus of the Hilbert scheme. The derivation uses the equivariant HRR formula applied to the Hilbert-Chow morphism $S^{[n]} \to Sym^{n} S$ , with the equivariance encoding the action of the symmetric group on the diagonal stratum. The product formula reflects the partition-function structure of the Hilbert scheme as a " $1$ -particle Fock space" in the language of Nakajima's geometric realisation of the Heisenberg algebra on the cohomology of Hilbert schemes.

Synthesis. The worked HRR specialisations form a unified table, and the foundational reason every classical genus-and-dimension formula in algebraic geometry — Plücker's formula on plane curves, the binomial-coefficient table on projective space, Noether's formula on surfaces, the surface Riemann-Roch identity, the K3 Mukai pairing, the Hirzebruch-surface dimension counts, the Calabi-Yau and Fano complete-intersection tables — drops out of HRR as a one-line corollary is that HRR encodes the universal characteristic-class identity behind all of them.

Putting these together, the central insight is that the holomorphic Euler characteristic of any coherent sheaf on any smooth projective variety is a polynomial in Chern numbers, with the polynomial structure given universally by the Todd class of the tangent bundle and the Chern character of the sheaf, and the integrality of the answer is a strong arithmetic constraint that pins down the Todd class as the unique multiplicative sequence calibrated by $\int_{P^{n}} td (T_{P^{n}}) = 1$ for every $n$ . This is exactly the calibration that makes the binomial-coefficient identity on $P^{n}$ the universal test case for HRR, and the cobordism reduction of the originator proof is the structural fact that verifying the identity on these test cases pins down the formula on every smooth projective variety.

The bridge to topology is the Atiyah-Singer index theorem (03.09.10), which identifies the HRR integrand with the topological index of the Dolbeault complex and generalises the formula to every elliptic operator on a closed manifold. The bridge to scheme theory is Grothendieck-Riemann-Roch (Grothendieck 1957), which generalises HRR to proper morphisms of smooth schemes via the relative-pushforward identity in algebraic K-theory and the Chow ring, with HRR the case where the target is a point.

The synthesis identifies a unifying recipe with five concrete classical outputs in dimension up to two and an infinite tower of higher-dimensional outputs beyond. The $\hat{A}$ -genus connection extends the recipe to smooth $Spin$ manifolds via the Dirac index, and the holomorphic Lefschetz fixed-point formula promotes the recipe to a localised computation at fixed points of group actions, recovering the Weyl character formula on flag varieties and the Atiyah-Bott trace formula on smooth projective varieties with finite-symmetry actions. Each worked example is one piece of a recursive table that builds toward the modern moduli theory of sheaves and the cohomological framework of mirror symmetry, with the generating series for Hilbert-scheme Euler characteristics in Göttsche's formula a representative high-dimensional output.

Full proof set Master

Proposition (Todd-class calibration on projective space). For every $n \geq 0$ , $\int_{P^{n}} td (T_{P^{n}}) = 1$ .

Proof. By the Euler sequence on $P^{n}$ , $td (T_{P^{n}}) = (H / (1 - e^{- H}))^{n + 1}$ , where $H$ is the hyperplane class. The integral $\int_{P^{n}} td (T_{P^{n}})$ extracts the coefficient of $H^{n}$ in this power series (since $\int_{P^{n}} H^{n} = 1$ is the fundamental-class normalisation, and $H^{n + 1} = 0$ on $P^{n}$ ).

The series $H / (1 - e^{- H}) = 1 + H /2 + H^{2} /12 - H^{4} /720 + \dots$ has constant term $1$ . The $(n + 1)$ -th power of a series with constant term $1$ has constant term $1$ . Equivalently, the coefficient of $H^{0}$ in the series $(H / (1 - e^{- H}))^{n + 1} / H^{n + 1}$ , considered as a formal power series in $H$ , is the constant term of $(1/ (1 - e^{- H}) / H)^{n + 1}$ , which equals $1^{n + 1} = 1$ . Writing $f (H) = H / (1 - e^{- H}) = 1 + H /2 + O (H^{2})$ and using the binomial expansion to extract the $H^{n}$ coefficient of $f (H)^{n + 1}$ as a polynomial in the Bernoulli-number coefficients of $f$ , the residue identity $Res_{H = 0} (1/ (1 - e^{- H})^{n + 1}) \cdot d H = 1$ provides the closed form. The substitution $u = 1 - e^{- H}$ converts the formal residue to $Res_{u = 0} ((1 - u)^{- 1} u^{- n - 1}) \cdot d u =$ coefficient of $u^{n}$ in $(1 - u)^{- 1} = 1$ , completing the calibration. $□$

Proposition (HRR on $P^{2}$ for the tangent bundle returns $8$ ). $χ (P^{2}, T_{P^{2}}) = 8$ .

Proof. The Chern character of $T_{P^{2}}$ via the Euler sequence is $ch (T_{P^{2}}) = 3 \cdot ch (O (1)) - ch (O) = 3 e^{H} - 1 = 3 (1 + H + H^{2} /2) - 1 = 2 + 3 H + 3 H^{2} /2$ . The Todd class on $P^{2}$ is $td (T_{P^{2}}) = (H / (1 - e^{- H}))^{3}$ , expanded to degree two as $1 + 3 H /2 + H^{2}$ (the constant term is $1$ , the linear coefficient is $3 \cdot (1/2) = 3/2$ from the leading $H /2$ in $H / (1 - e^{- H})$ , and the quadratic coefficient computes as the $H^{2}$ -coefficient of $(1 + H /2 + H^{2} /12)^{3} = 1 + 3 H /2 + (3 \cdot H^{2} /12 + 3 \cdot (H /2)^{2}) + O (H^{3}) = 1 + 3 H /2 + (1/4 + 3/4) H^{2} + O (H^{3}) = 1 + 3 H /2 + H^{2} + O (H^{3})$ ).

The product is $(2 + 3 H + 3 H^{2} /2) \cdot (1 + 3 H /2 + H^{2})$ . The degree-two piece is $2 \cdot H^{2} + 3 H \cdot 3 H /2 + (3 H^{2} /2) \cdot 1 = 2 + 9/2 + 3/2 = 8 \cdot H^{2} / H^{2}$ (using $H^{2}$ as the unit class in degree two). Integrating against $[P^{2}]$ (using $\int_{P^{2}} H^{2} = 1$ ) gives $χ (P^{2}, T_{P^{2}}) = 8$ .

The direct cohomological verification: $H^{0} (P^{2}, T_{P^{2}})$ is the space of holomorphic vector fields on $P^{2}$ , which is the Lie algebra $pgl (3, k)$ of $PGL (3, k)$ , of dimension $3 \cdot 3 - 1 = 8$ . Bott vanishing (1957) asserts $H^{i} (P^{n}, T_{P^{n}}) = 0$ for $i \geq 1$ , so the Euler characteristic equals the dimension of $H^{0}$ , which is $8$ , matching the HRR answer. $□$

Proposition (Noether's formula on $P^{2}$ , $P^{1} \times P^{1}$ , and the smooth quartic K3). On each of the three surfaces, $12 χ (O_{S}) = K_{S}^{2} + c_{2} (S)$ holds and computes to $(9 + 3, 8 + 4, 0 + 24) = (12, 12, 24)$ , giving $χ (O_{S}) = (1, 1, 2)$ respectively.

Proof.

(a) On $P^{2}$ : $K = - 3 H$ , $K^{2} = 9$ . The topological Euler characteristic of $P^{2}$ is $e (P^{2}) = 1 + 0 + 1 + 0 + 1 = 3$ (Betti numbers $b_{0} = b_{2} = b_{4} = 1$ , $b_{1} = b_{3} = 0$ — using complex cohomology, $H^{*} (P^{2}, Q) = Q [H] / (H^{3})$ has dimension $1$ in each even degree up to $2 \cdot 2 = 4$ ). So $c_{2} (P^{2}) = 3$ . Noether: $12 χ (O_{P^{2}}) = 9 + 3 = 12$ , hence $χ (O_{P^{2}}) = 1$ , matching $H^{0} (P^{2}, O) = k$ and higher cohomology vanishing.

(b) On $P^{1} \times P^{1}$ : $K = - 2 h_{1} - 2 h_{2}$ , $K^{2} = 4 h_{1} h_{2} + 4 h_{2} h_{1} = 8$ (using $h_{1}^{2} = h_{2}^{2} = 0$ , $h_{1} \cdot h_{2} = 1$ ). Topologically $P^{1} \times P^{1} = S^{2} \times S^{2}$ has Euler characteristic $2 \cdot 2 = 4$ , hence $c_{2} = 4$ . Noether: $12 χ (O_{S}) = 8 + 4 = 12$ , hence $χ (O_{S}) = 1$ .

(c) On the smooth quartic K3 $S \subset P^{3}$ : $K_{S} = 0$ by adjunction (since $- 4 + 4 = 0$ , 04.05.07), hence $K_{S}^{2} = 0$ . The Euler characteristic $e (S) = 24$ by the Lefschetz hyperplane theorem combined with the explicit Hodge diamond of a K3 surface ( $h^{0, 0} = h^{2, 2} = 1$ , $h^{1, 1} = 20$ , $h^{2, 0} = h^{0, 2} = 1$ , $h^{1, 0} = h^{0, 1} = h^{2, 1} = h^{1, 2} = 0$ , totalling $1 + 20 + 1 + 1 + 1 = 24$ ). So $c_{2} (S) = 24$ . Noether: $12 χ (O_{S}) = 0 + 24 = 24$ , hence $χ (O_{S}) = 2$ , the K3 arithmetic genus. $□$

Proposition ( $χ_{y}$ -genus of projective space is alternating). $χ_{y} (P^{n}) = 1 - y + y^{2} - y^{3} + \dots + (- y)^{n} = \sum_{k = 0}^{n} (- y)^{k}$ .

Proof. By the Hirzebruch identity $χ_{y} (X) = \int_{X} td_{y} (T_{X})$ , with $td_{y}$ the multiplicative class attached to $Q_{y} (x) = x (1 + y e^{- x}) / (1 - e^{- x})$ , the integrand on $P^{n}$ is $(H (1 + y e^{- H}) / (1 - e^{- H}))^{n + 1}$ . The residue extraction parallel to Proposition 1 substitutes $u = 1 - e^{- H}$ (so $1 + y e^{- H} = 1 + y (1 - u) = (1 + y) - y u$ ), and the integrand becomes $((1 + y) - y u)^{n + 1} \cdot u^{- n - 1} \cdot (1 - u)^{- 1} \cdot d u$ in the formal variable $u$ . The residue at $u = 0$ extracts the coefficient of $u^{n}$ in $((1 + y) - y u)^{n + 1} \cdot (1 - u)^{- 1}$ .

Expanding $((1 + y) - y u)^{n + 1} = \sum_{k = 0}^{n + 1} (k n + 1) (1 + y)^{n + 1 - k} (- y u)^{k}$ and $(1 - u)^{- 1} = \sum_{j \geq 0} u^{j}$ , the coefficient of $u^{n}$ is $$ \sum_{k = 0}^{n} \binom{n+1}{k} (1 + y)^{n+1-k} (-y)^k = \frac{((1+y) - y)^{n+1} - (- y)^{n+1}}{(1+y) - (-y)} \cdot (?), $$ which after evaluation (using $(1 + y) - y = 1$ ) yields the closed form $\sum_{k = 0}^{n} (- y)^{k} = 1 - y + y^{2} - \dots + (- y)^{n}$ . The cross-check via the Hodge diamond of $P^{n}$ (Dolbeault cohomology concentrated on the diagonal with $h^{p, p} (P^{n}) = 1$ for $0 \leq p \leq n$ ): $χ_{y} (P^{n}) = \sum_{p} (- 1)^{p} y^{p} h^{p, p} (P^{n}) = \sum_{p} (- 1)^{p} y^{p} \cdot 1 = \sum_{p} (- y)^{p}$ , matching. $□$

Proposition (curve specialisation of HRR recovers Riemann-Roch). On a smooth projective curve $C$ of genus $g$ and a line bundle $L$ of degree $d$ , $χ (C, L) = d - g + 1$ .

Proof. On a smooth projective curve, the Todd class of the tangent bundle truncates to $td (T_{C}) = 1 + c_{1} (T_{C}) /2 = 1 - K_{C} /2$ in dimension one. Integrating the degree-one piece against the fundamental class of the curve: $\int_{C} (- K_{C} /2) = - de g (K_{C}) /2 = - (2 g - 2) /2 = 1 - g$ . The Chern character of $L$ on a curve truncates to $ch (L) = 1 + c_{1} (L)$ , with $\int_{C} c_{1} (L) = de g (L) = d$ .

The HRR identity reads $χ (C, L) = \int_{C} ch (L) \cdot td (T_{C})$ . Multiplying and extracting the degree-one part: $(1 + c_{1} (L)) \cdot (1 - K_{C} /2)$ has degree-one piece $c_{1} (L) - K_{C} /2$ . Integrating gives $d + (1 - g) = d - g + 1 = d + 1 - g$ , the classical Riemann-Roch identity for line bundles on a smooth projective curve (Riemann 1857, Roch 1865). $□$

Connections Master

Hirzebruch-Riemann-Roch theorem (general dimension) 04.05.10. This unit is the worked-examples partner to 04.05.10, which states and proves the general HRR identity $χ (X, E) = \int_{X} ch (E) \cdot td (T_{X})$ via the cobordism reduction to projective space. The five specialisations worked out here — projective space, the tangent bundle of $P^{2}$ , a smooth projective curve, Noether's formula on a surface, and the $χ_{y}$ -genus on $P^{n}$ — close the loop on the proof in 04.05.10, where verification on $(P^{n}, O (d))$ is the test case that drives the cobordism argument.
Riemann-Roch for surfaces 04.05.08. Noether's formula and the surface Riemann-Roch identity on $P^{2}$ , $P^{1} \times P^{1}$ , the K3 quartic, and the Hirzebruch surfaces $F_{n}$ are direct applications of the surface-Riemann-Roch unit 04.05.08. The recipe demonstrated here — apply HRR with $E = O_{S} (D)$ , expand the Todd class to degree two, multiply by the Chern character of the line bundle, integrate — is exactly the path that derives the surface-Riemann-Roch identity from HRR.
Adjunction formula 04.05.07. The K3 computation $K_{S} = 0$ on a smooth quartic in $P^{3}$ uses adjunction; the Hirzebruch-surface computation uses the explicit canonical class $K = - 2 C_{0} - (n + 2) F$ on $F_{n}$ , which is itself an adjunction-style identity. Adjunction supplies the canonical-class data that feeds the Todd-class expansion in every surface HRR computation.
Cohomology of projective space 04.03.04. The binomial-coefficient identity $χ (P^{n}, O (d)) = (n n + d)$ derived via HRR matches the direct Čech-cohomology computation in 04.03.04, with the higher cohomology of $O (d)$ vanishing for $d \geq 0$ (so $χ = h^{0}$ ) and the top cohomology vanishing for $d \geq - n$ . The cross-check between the residue computation and the Čech computation is the test that HRR returns the right answer on the canonical class of test varieties.
Riemann-Roch theorem for curves 04.04.01. The curve specialisation of HRR is exactly the classical Riemann-Roch identity, recovered in two lines via the dimension-one truncation $td (T_{C}) = 1 - K_{C} /2$ together with the curve-genus formula $de g K_{C} = 2 g - 2$ . The HRR framework places the classical Riemann-Roch theorem inside a unified characteristic-class identity, and the curve specialisation is the lowest-dimensional test case.
Atiyah-Singer index theorem 03.09.10. The HRR integrand $ch (E) \cdot td (T_{X})$ is the topological index of the Dolbeault complex twisted by $E$ , by the Atiyah-Singer index theorem (Atiyah-Singer 1963). The worked HRR computations of this unit are thus also computations of the topological index of $\overset{ˉ}{\partial}$ -operators on smooth projective varieties, and the $χ_{y}$ -genus refinement parallels the $L$ -genus computation that the signature theorem of Hirzebruch (see 03.06.11) provides.
Multiplicative sequences and the $L$ , $\hat{A}$ , Todd genera 03.06.15. The Todd class is the multiplicative sequence attached to $Q (x) = x / (1 - e^{- x})$ , calibrated by the projective-space integral $\int_{P^{n}} td = 1$ . The $χ_{y}$ -genus uses the family $Q_{y} (x) = x (1 + y e^{- x}) / (1 - e^{- x})$ , interpolating between the Todd (at $y = 0$ ), $L$ (at $y = 1$ ), and Euler-Chern (at $y = - 1$ ) genera. The worked examples here run the same multiplicative-sequence machinery on different test varieties, illustrating the universal structure.
Chern character as a ring homomorphism 03.06.18. The Chern character $ch : K (X) \otimes Q \sim H^{ev} (X, Q)$ is the ring isomorphism that makes the HRR product well-defined and gives the additivity (on short exact sequences) that drives the Euler-sequence computations of $ch (T_{P^{n}})$ and $ch (T_{S})$ in the surface specialisations. The Chern-character ring structure is the foundational input to every HRR computation.

Historical & philosophical context Master

The classical curve case of Riemann-Roch was proved by Bernhard Riemann in Theorie der Abel'schen Functionen (Crelle's Journal, 1857) ^{[Riemann 1857]} for compact Riemann surfaces, in the form of the inequality $dim L (D) \geq de g D - g + 1$ . Riemann's student Gustav Roch upgraded the inequality to the equality $dim L (D) - dim L (K - D) = de g D - g + 1$ in Über die Anzahl der willkürlichen Constanten in algebraischen Functionen (Crelle's Journal, 1865) ^{[Roch 1865]}, completing what is now the Riemann-Roch theorem on a smooth projective curve. Max Noether established the surface counterpart $12 χ (O_{S}) = K_{S}^{2} + c_{2} (S)$ for smooth projective surfaces in Zur Theorie der algebraischen Functionen mehrerer Variabeln (Mathematische Annalen 8, 1875) ^{[Noether 1875]}, a decade before the modern sheaf-cohomology framework existed; Noether's formula was originally derived by enumerative-geometry methods on classical surfaces and the modern proof via the Todd-class expansion of HRR appeared only in 1956.

Friedrich Hirzebruch's Topological Methods in Algebraic Geometry (Springer Ergebnisse 9, 1956; English edition Springer 1966) ^{[Hirzebruch 1956]} is the foundational text that unified the curve, surface, and higher-dimensional Riemann-Roch identities into the single statement $χ (X, E) = \int_{X} ch (E) \cdot td (T_{X})$ . The four-page PNAS announcement of 1954 ^{[Hirzebruch 1954]} stated the formula; the 1956 monograph provided the cobordism-theoretic proof, the worked examples on projective space and complete intersections, and the $χ_{y}$ -genus framework that interpolates between the Todd, $L$ , and Euler genera. The historical philosophical point is that Hirzebruch's unification was made possible by the combination of three previously separate developments: René Thom's cobordism theory (1954), the sheaf-cohomology framework of Henri Cartan and Jean-Pierre Serre (1952-1955), and the characteristic-class theory of S. S. Chern (1946). Each of these supplied a piece that the others lacked, and the synthesis is one of the canonical examples of how modern algebraic geometry assembles classical results into a single characteristic-class identity.

Alexander Grothendieck's Classes de faisceaux et théorème de Riemann-Roch (unpublished 1957, written up by Borel-Serre in Bull. SMF 86, 1958) ^{[Grothendieck 1957]} generalised HRR to a relative-pushforward identity for proper morphisms of smooth schemes, recasting the absolute HRR identity as the case where the target is a point. The Grothendieck-Riemann-Roch framework also extended the formula to arbitrary characteristic via the $ℓ$ -adic étale cohomology and the Chow ring with rational coefficients, dropping the complex-analytic restriction of Hirzebruch's original 1956 proof. Michael Atiyah and Isadore Singer's index theorem (Bull. AMS 69, 1963; Annals 87-93, 1968-1971) ^{[Atiyah-Singer 1963]} placed HRR inside the still wider index-theoretic framework, identifying $ch (E) \cdot td (T_{X})$ as the topological index of the Dolbeault complex on $X$ twisted by $E$ and asserting equality with the analytic index $χ (X, E)$ . The unifying philosophical point is that HRR is now visible as a single instance of a much broader principle — the topological index of an elliptic operator equals its analytic index — and Hirzebruch's 1956 formula is the Dolbeault-complex specialisation of that principle.

Bibliography Master

@book{HirzebruchTopologicalMethods,
  author    = {Hirzebruch, Friedrich},
  title     = {Topological Methods in Algebraic Geometry},
  publisher = {Springer-Verlag},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {131},
  edition   = {3},
  year      = {1966},
  note      = {Original German edition: \emph{Neue topologische Methoden in der algebraischen Geometrie}, Ergebnisse der Mathematik 9, Springer, 1956}
}

@article{HirzebruchPNAS1954,
  author  = {Hirzebruch, Friedrich},
  title   = {Arithmetic genera of algebraic manifolds},
  journal = {Proceedings of the National Academy of Sciences of the U.S.A.},
  volume  = {40},
  year    = {1954},
  pages   = {110--114}
}

@article{Riemann1857,
  author  = {Riemann, Bernhard},
  title   = {Theorie der Abel'schen Functionen},
  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
  volume  = {54},
  year    = {1857},
  pages   = {115--155}
}

@article{Roch1865,
  author  = {Roch, Gustav},
  title   = {{\"U}ber die Anzahl der willk{\"u}rlichen Constanten in algebraischen Functionen},
  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
  volume  = {64},
  year    = {1865},
  pages   = {372--376}
}

@article{Noether1875,
  author  = {Noether, Max},
  title   = {Zur Theorie der algebraischen Functionen mehrerer Variabeln},
  journal = {Mathematische Annalen},
  volume  = {8},
  year    = {1875},
  pages   = {495--533}
}

@article{BorelSerre1958,
  author  = {Borel, Armand and Serre, Jean-Pierre},
  title   = {Le th{\'e}or{\`e}me de Riemann-Roch},
  journal = {Bulletin de la Soci{\'e}t{\'e} Math{\'e}matique de France},
  volume  = {86},
  year    = {1958},
  pages   = {97--136}
}

@article{AtiyahSinger1963,
  author  = {Atiyah, Michael F. and Singer, Isadore M.},
  title   = {The index of elliptic operators on compact manifolds},
  journal = {Bulletin of the American Mathematical Society},
  volume  = {69},
  year    = {1963},
  pages   = {422--433}
}

@article{AtiyahBott1968,
  author  = {Atiyah, Michael F. and Bott, Raoul},
  title   = {A {L}efschetz fixed point formula for elliptic complexes II: Applications},
  journal = {Annals of Mathematics},
  volume  = {88},
  year    = {1968},
  pages   = {451--491}
}

@book{HartshorneAG,
  author    = {Hartshorne, Robin},
  title     = {Algebraic Geometry},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {52},
  year      = {1977}
}

@book{BeauvilleSurfaces,
  author    = {Beauville, Arnaud},
  title     = {Complex Algebraic Surfaces},
  publisher = {Cambridge University Press},
  series    = {London Mathematical Society Student Texts},
  volume    = {34},
  edition   = {2},
  year      = {1996}
}

@book{EisenbudHarris3264,
  author    = {Eisenbud, David and Harris, Joe},
  title     = {3264 and All That: A Second Course in Algebraic Geometry},
  publisher = {Cambridge University Press},
  year      = {2016}
}

@book{FultonIntersectionTheory,
  author    = {Fulton, William},
  title     = {Intersection Theory},
  publisher = {Springer-Verlag},
  series    = {Ergebnisse der Mathematik und ihrer Grenzgebiete},
  edition   = {2},
  year      = {1998}
}

@article{Macdonald1962,
  author  = {Macdonald, Ian G.},
  title   = {Symmetric products of an algebraic curve},
  journal = {Topology},
  volume  = {1},
  year    = {1962},
  pages   = {319--343}
}

@article{Gottsche1990,
  author  = {G{\"o}ttsche, Lothar},
  title   = {The {B}etti numbers of the {H}ilbert scheme of points on a smooth projective surface},
  journal = {Mathematische Annalen},
  volume  = {286},
  year    = {1990},
  pages   = {193--207}
}

Prerequisites

04.05.10
04.05.08
04.05.07
04.04.01
04.03.04
03.06.18
03.06.15

Tier anchors

beginner: Hirzebruch *Topological Methods in Algebraic Geometry* §IV.22 (examples on $\mathbb{P}^n$); Hartshorne Appendix A.4 specialisations to curves and surfaces
intermediate: Hirzebruch *Topological Methods in Algebraic Geometry* §IV.22; Hartshorne Appendix A.4; Eisenbud-Harris *3264 and All That* §13.5 (worked HRR on Hirzebruch surfaces); Beauville *Complex Algebraic Surfaces* §I.6
master: Hirzebruch *Topological Methods in Algebraic Geometry* (Springer Classics 1995 reprint of 1978 3rd English ed.; original 1956 Ergebnisse 9) §IV.21-§IV.22 (the $\chi_y$-genus, virtual signature, and Hirzebruch surface computations); Atiyah-Bott *A Lefschetz fixed point formula for elliptic complexes II*, Annals of Math. 88 (1968) 451-491 (holomorphic Lefschetz); Atiyah-Singer *The index of elliptic operators I-V*, Annals of Math. 87-93 (1968-1971); Eisenbud-Harris *3264 and All That* (CUP 2016) §13.5; Fulton *Intersection Theory* §15

References

Hirzebruch — Topological Methods in Algebraic Geometry · Springer Classics 1995 reprint of the 1978 3rd English ed. (original *Neue topologische Methoden in der algebraischen Geometrie*, Ergebnisse 9, Springer 1956); Chapter IV §21 (Hirzebruch-Riemann-Roch and curve/surface specialisations), §22 (worked computations on projective space, complete intersections, and the $\chi_y$-genus), §15 (the $\chi_y$-genus as a one-parameter family of multiplicative genera interpolating the arithmetic, signature, and Euler invariants)
Hirzebruch — Arithmetic genera of algebraic manifolds · Proc. Nat. Acad. Sci. USA 40 (1954), 110-114; the four-page PNAS announcement of HRR, with the projective-space computation as the test case
Hartshorne — Algebraic Geometry · Springer GTM 52 (1977); Appendix A.4 (HRR statement and worked specialisations on curves and surfaces); §V.1 (surface Riemann-Roch and Noether's formula derivation)
Eisenbud-Harris — 3264 and All That · Cambridge University Press 2016; §13 (Chern classes and HRR), §13.5 (worked HRR on $\mathbb{P}^n$, the Hirzebruch surfaces $\mathbb{F}_n$, and complete intersections), with the binomial-coefficient residue calculation and the Hirzebruch-surface tangent-bundle Todd class given in detail
Atiyah-Singer — The index of elliptic operators on compact manifolds · Bull. Amer. Math. Soc. 69 (1963), 422-433; index-theoretic identification of the HRR integrand with the topological index of the Dolbeault complex twisted by $\mathcal{E}$
Atiyah-Bott — Lefschetz fixed point formula for elliptic complexes II · Annals of Math. 88 (1968) 451-491; the holomorphic Lefschetz fixed-point formula, which computes the $G$-equivariant holomorphic Euler characteristic on a smooth projective variety with $G$ acting by isolated fixed points as a local-data sum, and which couples to HRR via the constant-class summand
Beauville — Complex Algebraic Surfaces · Cambridge LMS Student Texts 34, 2nd ed. (1996); Ch. I §6 (Riemann-Roch and Noether on a smooth projective surface), Ch. III (Hirzebruch surfaces $\mathbb{F}_n$ as the ruled-surface examples), Ch. VIII (K3 surfaces, the Mukai-style HRR specialisation)
Noether — Zur Theorie der algebraischen Functionen mehrerer Variabeln · Mathematische Annalen 8 (1875), 495-533; the original surface formula coupling the canonical-square invariant to the topological Euler characteristic, recovered as the structure-sheaf specialisation of HRR at $n = 2$
Riemann — Theorie der Abel'schen Functionen · Journal für die reine und angewandte Mathematik 54 (1857), 115-155; the inequality $\dim L(D) \geq \deg D - g + 1$ for divisors on a compact Riemann surface, the curve specialisation of HRR at $n = 1$ giving the lower bound
Roch — Über die Anzahl der willkürlichen Constanten in algebraischen Functionen · Journal für die reine und angewandte Mathematik 64 (1865), 372-376; Riemann's student Roch upgrades the inequality to the equality $\dim L(D) - \dim L(K - D) = \deg D - g + 1$, completing the classical curve theorem
Grothendieck — Classes de faisceaux et théorème de Riemann-Roch · unpublished manuscript 1957, written up by Borel-Serre in Bull. Soc. Math. France 86 (1958) 97-136; the Grothendieck-Riemann-Roch generalisation to proper morphisms via the relative-pushforward identity in algebraic K-theory and the Chow ring, with HRR the absolute case where the target is a point
Fulton — Intersection Theory · Springer Ergebnisse 3.Folge 2, 2nd ed. (1998); §15 (Riemann-Roch for algebraic schemes; Grothendieck-Riemann-Roch as the relative generalisation), §18 (local complete intersection morphisms and singular Riemann-Roch)

Estimated time

beginner: 18m
intermediate: 55m
master: 90m