Toric Hirzebruch-Riemann-Roch via Ishida's theorem on the Todd genus
aa r X i v : . [ m a t h . AG ] A ug TORIC HIRZEBRUCH-RIEMANN-ROCH VIAISHIDA’S THEOREM ON THE TODD GENUS
HAL SCHENCK
Abstract.
We give a simple proof of the Hirzebruch-Riemann-Roch theoremfor smooth complete toric varieties, based on Ishida’s result in [5] that theTodd genus of a smooth complete toric variety is one. Introduction
The Hirzebruch-Riemann-Roch theorem relates the Euler characteristic of a co-herent sheaf F on a smooth complete n − dimensional variety X to intersectiontheory, via the formula(1) χ ( F ) = Z ch ( F ) T d ( T X ) . In [2], Brion-Vergne prove an equivariant Hirzebruch-Riemann-Roch theorem forcomplete simplicial toric varieties. If the toric variety is actually smooth, it ispossible to derive (1) from their result. In this note, we give a simple direct proofof (1) when X is a smooth complete toric variety. Such a variety is determined bya smooth complete rational polyhedral fan Σ ⊆ N R , where N ≃ Z n is a lattice;we write X for the associated toric variety X Σ . We will make use of the followingstandard facts about toric varieties. First,(2) T d ( X Σ ) = Y ρ ∈ Σ(1) D ρ − e − D ρ , where Σ( k ) denotes the set of k -dimensional faces of Σ. For τ ∈ Σ( k ) there is anassociated torus invariant orbit O ( τ ), and we use V ( τ ) to denote the orbit closure O ( τ ), which has dimension n − k . A key fact is that (see [4], Proposition 3.2.7)(3) V ( τ ) = O ( τ ) ≃ X Star( τ ) . Since Σ is smooth, all orbits are also smooth, and if ρ i , ρ j are distinct elements ofΣ(1), then (see [4], Lemma 12.5.7)[ D ρ i | V ( ρ j ) ] = ( V ( τ ) τ = ρ i + ρ j ∈ Σ0 ρ i , ρ j are not both in any cone in Σ . The final ingredient we need is a result of Ishida: building on work of Brion [1], in[5] Ishida shows that (1) holds for the structure sheaf of a smooth complete toricvariety X :(4) 1 = Z T d ( T X ) = h Y ρ ∈ Σ(1) D ρ − e − D ρ i n . Mathematics Subject Classification.
Key words and phrases.
Toric variety, Chow ring, cohomology.Schenck supported by NSF 1068754, NSA H98230-11-1-0170. The proof
For a smooth complete toric variety, any coherent sheaf has a resolution by linebundles [3], so it suffices to consider the case F = O X ( D ). Let X = X Σ , and recallthat Pic( X ) is generated by the classes of the divisors D ρ , ρ ∈ Σ(1). We will showthat if (1) holds for a divisor D , then it also holds for D + D ρ and D − D ρ , for any ρ ∈ Σ(1). We begin with the case D − D ρ , and induct on the dimension of X .A smooth complete toric variety of dimension one is simply P , so the basecase holds by Riemann-Roch for curves. Suppose the theorem holds for all smoothcomplete fans of dimension < n , and let Σ be a smooth complete fan of dimension n . When D = 0 the result holds by Ishida’s theorem. Let ρ ∈ Σ(1), and partitionthe rays of Σ as Σ(1) = ρ ∪ Σ ′ (1) ∪ Σ ′′ (1) , where the rays in Σ ′ (1) are in one to one correspondence with the rays of the fanStar( ρ ). Let X ′ = X Star( ρ ) ≃ V ( ρ ). Tensoring the standard exact sequence0 −→ O X ( − D ρ ) −→ O X −→ O X ′ −→ O X ( D ) yields the sequence0 −→ O X ( D − D ρ ) −→ O X ( D ) −→ O X ′ ( D ) −→ . From the additivity of the Euler characteristic, we have χ ( O X ( D )) − χ ( O X ( D − D ρ )) = χ ( O X ′ ( D )) . Our hypotheses imply that Z X ′ e D T d ( T X ′ ) = χ ( O X ′ ( D )) Z X e D T d ( T X ) = χ ( O X ( D )) , so it suffices to show that(5) Z X ′ ch ( D ) T d ( T X ′ ) = Z X ( e D − e D − D ρ ) T d ( T X )= Z X e D (cid:16) − e − D ρ D ρ (cid:17) D ρ T d ( T X )Break the Todd class of X into two parts: T d ( T X ) = Y γ ∈ Σ ′ (1) ∪ ρ D γ − e − D γ · Y γ ∈ Σ ′′ (1) D γ − e − D γ In (5), the term − e − Dρ D ρ cancels with the corresponding term in T d ( T X ), so that(6) Z X e D (cid:16) − e − D ρ D ρ (cid:17) D ρ T d ( T X ) = Z X e D D ρ Y γ ∈ Σ ′ (1) ∪ Σ ′′ (1) D γ − e − D γ = Z X e D D ρ Y γ ∈ Σ ′ (1) D γ − e − D γ . ORIC HIRZEBRUCH-RIEMANN-ROCH VIA ISHIDA’S THEOREM ON THE TODD GENUS 3
The second equality follows since D ρ · D γ = 0 if γ ∈ Σ ′′ (1). By smoothness, allintersections are either zero or one, and thus Z X e D D ρ Y γ ∈ Σ ′ (1) D γ − e − D γ = h e D D ρ Y γ ∈ Σ ′ (1) D γ − e − D γ i n = h e D | V ( ρ ) Y γ ∈ Σ ′ (1) D γ − e − D γ i n − = Z X ′ e D · T d ( T X ′ ) . This proves the result for D − D ρ . For D + D ρ , the result follows using the substi-tution e D ρ − e D ρ (1 − e − D ρ ) . Question
Ishida’s proof (4) is not easy; does there exist a simple proof of (4)?
Acknowledgements
I thank David Cox for pointing out Ishida’s result to me,and the referee for useful comments.
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Points entiers dans les poly´edres convexes , Ann. Sci. ´Ecole Norm. Sup. (1988),653–663.[2] M. Brion, M. Vergne, An equivariant Riemann-Roch theorem for complete, simplicial toricvarieties , J. Reine Angew. Math. (1997), 67–92.[3] D. Cox,
The homogeneous coordinate ring of a toric variety , J. Algebraic Geom. (1995),15–50.[4] D. Cox, J. Little and H. Schenck, Toric Varieties , Graduate Studies in Mathematics, Amer-ican Mathematical Society (2011).[5] M. Ishida,
Polyhedral Laurent series and Brion’s equalities , Intl. J. Math. (1990), 251–265. Schenck: Mathematics Department, University of Illinois, Urbana, IL 61801, USA
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