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Categorical Proof of Holomorphic Atiyah-Bott Formula

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Categorical Proof of Holomorphic Atiyah-Bott Formula Categorical proof of Holomorphic Atiyah-Bott formula Grigory Kondyrev, Artem Prikhodko Abstract Given a 2-commutative diagram FX X / X ϕ ϕ Y / Y FY in a symmetric monoidal (∞, 2)-category E where X, Y ∈ E are dualizable objects and ϕ admits a right adjoint we construct a natural morphism TrE(FX ) / TrE(FY ) between the traces of FX and FY respectively. We then apply this formalism to the case when E is the (∞, 2)-category of k-linear presentable categories which in combination of various calculations in the setting of derived algebraic geometry gives a categorical proof of the classical Atiyah-Bott formula (also known as the Holomorphic Lefschetz fixed point formula). Contents 1 Dualizable objects and traces 4 1.1 Traces in symmetric monoidal (∞, 1)-categories .. .. .. .. .. .. .. 4 1.2 Traces in symmetric monoidal (∞, 2)-categories .. .. .. .. .. .. .. 5 2 Traces in algebraic geometry 11 2.1 Duality for Quasi-Coherent sheaves . .............. 11 2.2 Calculatingthetrace. .. .. .. .. .. .. .. .. .......... 13 3 Holomorphic Atiyah-Bott formula 16 3.1 Statement of Atiyah-Bott formula . ............ 17 3.2 ProofofAtiyah-Bottformula . ........... 19 arXiv:1607.06345v3 [math.AG] 12 Nov 2019 Introduction The well-known Lefschetz fixed point theorem [Lef26, Formula 71.1] states that for a compact manifold f M and an endomorphism M / M with isolated fixed points there is an equality 2dim M i ∗ L(f) := (−1) tr(f|Hi(X,Q))= degx(1 − f) (1) Xi=0 x=Xf(x) The formula (1) made huge impact on algebraic geometry in 20th century leading Grothendieck and co- authors to development of ´etale cohomology theory and their spectacular proof of Weil’s conjectures. Not only the fixed point theorem admits various generalizations, it can also be stated in different contexts than in the original work of Lefschetz. For example, if a fixed point x is simple, the index degx(1 − f) admits differential-geometric description, namely there is an equality degx(1 − f)= ± det(1 − dxf). This 1 observation has a vast generalization due to Atiyah and Bott [AB67], [AB68]: in the case when all the b fixed points of f are simple, given an elliptic complex E on M and a bundle map f −1E / E there is an equality i i L(E, b) := (−1) tr(H (b)|Hi(M,E))= µx Xi x=Xf(x) where µx are some explicit infinitesimal invariant of E and f at x (see below for a more concrete statement). • The original Lefschetz formula (1) can be recovered by taking E to be the de Rham complex ΩM,dR with its canonical equivariant structure. In this work we are concerned with the algebro-geometric version of the Atiyah-Bott formula. Namely let k be an algebraically closed field and X be a smooth proper variety over k together with an en- f Γf ∆ domorphism X / X such that its graph X / X × X intersects the diagonal X / X × X transversally so that the fixed point scheme Xf is a disjoint union of finitely many (simple) points. Let b us call a quasi-coherent sheaf E on X lax-equivariant if there is a fixed morphism f ∗E / E (prefix ”lax” corresponds to the fact that b is not required to be an equivalence). In this setting Atiyah-Bott for- mula (also known as holomorphic Lefschetz fixed point formula) says that for a dualizable lax-equivariant sheaf E there is an equality of two numbers bx Trk(Ex ≃ Ef(x) −→ Ex) L(E, b)= det(1 − dxf) x=Xf(x) dxf where TX,x / TX,x is the differential of f viewed as a map from the tangent space at a point x ∈ X to itself and L(E, b) ∈ k is the Lefschetz number ∗ ∗ Γ(b) L(E, b) := Trk Γ(X, E) / Γ(X,f∗f E) ≃ Γ(X,f E) / Γ(X, E) of b. Since both of the parts of the Atiyah-Bott formula are expressed in terms of traces, it is naturally to try to derive the Atiyah-Bott formula using some general mechanism which allows to compare traces of different objects. Recent development of higher category theory and derived algebraic geometry provide an appropriate context to formulate such ideas more concretely. In this paper, we provide a categorical proof of the Atiyah-Bott formula. Namely, we interpret both sides of the equality above as morphisms in the (∞, 1)-category of unbounded cochain complexes Vectk from k ∈ Vectk to itself. The desired equality then follows from the naturality of a certain construction in the world of (∞, 2)-categories. Plan of the paper. In the first section we introduce the main categorical tool. Namely, given a commutative up to a (not necessarily invertible) 2-morphism diagram of the form FX X / X ⑦⑦ ⑦⑦⑦⑦ ⑦⑦⑦⑦ T ⑦⑦⑦⑦ ϕ ⑦⑦⑦⑦ ϕ ⑦⑦⑦⑦ ⑦⑦⑦⑦ ⑦⑦⑦ zÒ⑦⑦⑦ Y / Y FY in a symmetric monoidal (∞, 2)-category E, where X,Y ∈ E are dualizable and ϕ admits a right adjoint, we construct a morphism of traces Tr(ϕ,T ) TrE(FX ) / TrE(FY ) 2 which is compatible with vertical compositions up to homotopy (see proposition 1.2.11 for a precise statement). In the second section of this paper we apply this formalism to the setting of derived algebraic geometry by considering the case E = 2 Catk, the (∞, 2)-category of k-linear stable presentable categories and continuous functors. It is well known (see e.g. [BZFN10]) that for a quasi-compact quasi-separated derived scheme X the (∞, 1)-category of unbounded cochain complexes of quasi-coherent sheaves QCoh(X) on X being compactly generated is a dualizable object in 2 Catk (see proposition 2.1.1 for more details), so f we can apply the machinery of traces. Namely, given an endomorphism X / X of a derived scheme f∗ X the functor f∗ induces an endomorphism QCoh(X) / QCoh(X) and we calculate (2.2.2) that the corresponding trace is simply f Tr2 Catk (f∗) ≃ Γ(X , OXf ) ∈ Hom2 Catk (Vectk, Vectk) ≃ Vectk where Xf is the derived fixed point scheme (see definition 2.2.1). Now a lax-equivariant sheaf E ∈ QCoh(X) as in the setting of the Atiyah-Bott formula allows us to construct a diagram Id Vectk Vectk / Vectk ❦❦❦ T ❦❦❦❦ E ❦❦❦❦ E ❦❦❦❦ qy ❦❦❦ QCoh(X) / QCoh(X) f∗ ∗ b where the 2-morphism T corresponds to the morphism f E / E . As Tr2 Catk (IdVectk ) ≃ k, the induced map of traces Tr (ϕ,T ) f k ≃ Tr2 Catk (IdVectk ) / Tr2 Catk (f∗) ≃ Γ(X , OXf ) f is just a choice of an element in Γ(X , OXf ). The main computation in the second section is another i characterization of this element: namely, if we denote by Xf / X the inclusion of the derived fixed points scheme, then proposition 2.2.3 establishes an equality ∗ ∗ ∗ ∗ i (b) ∗ Tr(ϕ, T )= Tr f i E ≃ i f E / i E QCoh(X ) which is extremely useful in further calculations. In the last section we apply the above categorical machinery to the particular case of the Atiyah-Bott formula. Considering the diagram Id Vectk Vectk / Vectk ❦❦❦ T ❦❦❦❦ E ❦❦❦❦ E ❦❦❦❦ qy ❦❦❦ QCoh(X) / QCoh(X) f∗ Γ Γ Vect / Vect k Id k Vectk we obtain a commutative triangle Tr(E,T ) Tr2 Catk (IdVectk ) / Tr2 Catk (f∗). ❯❯❯❯ ❯❯❯❯ ❯❯❯ Tr(Γ,IdΓ) Tr(Γ(X,E),Id ◦T ) ❯❯❯ Γ(X,E) ❯❯❯* Tr2 Catk (IdVectk ) 3 in Vectk. Since Tr2 Catk (IdVectk ) ≃ k this gives an equality of two numbers. It is then a combination of formal verifications and calculations from section 2 that the morphisms Tr(Γ(X, E), IdΓ(X,E)) and Tr(Γ, IdΓ) ◦ Tr(E, T ) are precisely the left-hand and the right-hand sides of the Atiyah-Bott formula respectively. Remark. After writing the paper we were pointed that results of a similar manner were obtained in the work [BZN13]. The main difference of the present work is a useful description of the chern character in geometric terms and a more concrete description of the functional on the derived stack of fixed points which allows to identify completely the categorical version of the statement with the classical Atiyah-Bott formula. Acknowledgments. We would like to thank Dennis Gaitsgory for suggesting the problem and numerous helpful discussions. We also want to thank anonymous referee for his thorough review and many useful suggestions. The first author was supported by the Russian Academic Excellence Project ’5-100’. The second author is partially supported by Laboratory of Mirror Symmetry NRUHSE, RF Government grant, ag. 14.641.31.0001. Conventions. 1) All the categories we work with are assumed to be (∞, 1)-categories. For an (∞, 1)-category C we will denote by (C)≃ the underlying ∞-groupoid of C obtained by discarding all the non-invertible morphisms from C. Analogously for an (∞, 2)-category E we will denote by E1-cat the maximal (∞, 1)-subcategory of E obtained by discarding all the non-invertible 2-morphisms. For a symmetric monoidal category C we will denote the full subcategory of dualizable objects by Cfd. 2) We will denote by S the symmetric monoidal (∞, 1)-category of spaces. For a field k we will de- note by Vectk the stable symmetric monoidal (∞, 1)-category of unbounded cochain complexes over k up ♥ to quasi-isomorphism with the canonical (∞, 1)-enhancement. We will also denote by Vectk the ordinary category of k-vector spaces considered as an (∞, 1)-category. L 3) We will denote by Pr∞ the (∞, 1)-category of presentable (∞, 1)-categories and continuous functors L,st with a symmetric monoidal structure from [Lur, Proposition 4.8.1.14.]. Similarly, we will denote by Pr∞ the (∞, 1)-category of stable presentable (∞, 1)-categories and continuous functors considered as a sym- L metric monoidal (∞, 1)-category with the monoidal structure inherited from Pr∞.
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