Simplicial Chern-Weil theory for coherent analytic sheaves, part II

In the previous part of this diptych, we defined the notion of an admissible simplicial connection, as well as explaining how H.I. Green constructed a resolution of coherent analytic sheaves by locally free sheaves on the \v{C}ech nerve. This paper seeks to apply these abstract formalisms, by showing that Green's barycentric simplicial connection is indeed admissible, and that this condition is exactly what we need in order to be able to apply Chern-Weil theory and construct characteristic classes. We show that, in the case of (global) vector bundles, the simplicial construction agrees with what one might construct manually: the explicit \v{C}ech representatives of the exponential Atiyah classes of a vector bundle agree. Finally, we summarise how all the preceding theory fits together to allow us to define Chern classes of coherent analytic sheaves, as well as showing uniqueness in the compact case.


History and motivation
For an in-depth history and motivation, we refer the reader to the previous part of this diptych [Hos23,§1] (and we more generally assume that the reader is already somewhat familiar with its contents, although we always make explicit mention of any results that we use). We will however give a brief overview here.
In 1980, Green's thesis [Gre80] gave a construction of Chern classes for coherent analytic sheaves using twisting cochains, technology developed and put to great use by Toledo, Tong, and O'Brian in a series of work throughout the 70s and 80s, where they proved, amongst other things, a version of Grothendieck-Riemann-Roch for complex manifolds [OTT85]. Although Green's work was summarised from a more abstract point of view in [TT86], there was still no formal framework for the "simplicial connections" of which he made use, nor for his resolution, which turned a twisting cochain (an inherently homotopical object) into a complex of vector bundles on theČech nerve (something much more homotopically strict). The main purpose of [Hos23] was to rectify this situation, using the language of (∞, 1)-categories to turn Green's resolution into an equivalence between certain categories of complexes of coherent sheaves and that of vector bundles on theČech nerve endowed with simplicial connections, and also to formally define the ingredients necessary to develop a simplicial version of Chern-Weil theory. The purpose of this present work is to show that explicit calculations one can do "by hand", Green's construction, and this simplicial Chern-Weil theory, all give the sameČech representatives for the Chern classes of coherent analytic sheaves.

Purpose and overview
The calculations in this paper largely focus on global [1] vector bundles on the nerve; we delay the study of coherent sheaves and arbitrary vector bundles on the nerve until the last section. This is because Green vector bundles on the nerve (resp. simplicial connections generated in degree zero) are really a mild generalisation of pullbacks of global vector bundles (resp. Green's barycentric connections), and so, by studying the latter, we can mostly understand the former.
The main contributions of this paper are the following: 1. We manually compute explicit representatives in theČech-de Rham bicomplex of the first four Atiyah classes (from which we can recover the Chern classes) of an arbitrary holomorphic vector bundle on a complex manifold (Section 3).
2. We introduce and develop a simplicial version of Chern-Weil theory, and show that this can be applied in particular to Green's barycentric connection, placing it in a general formal framework. Explicit calculations show that this agrees with the manual computations above (Section 5).
3. This simplicial Chern-Weil theory allows us to constructČech simplicial Atiyah classes of complexes of coherent analytic sheaves. We then show that, in the case of a compact complex manifold, we recover exactlyČech representatives of the classical Chern classes (Corollary 6.3.2).
In Section 2 we recall the definition of Atiyah classes and study some of their properties, before recalling the definition of fibre integration of simplicial differential forms.
[1] That is, vector bundles on the nerve that are given by the pullback of vector bundles on X .
The purpose of Section 3 is simply to detail an inefficient algorithm for lifting exponential Atiyah classes of vector bundles to closed elements in theČech-de Rham bicomplex. This isn't particularly useful on its own, but will later act as an assurance that the simplicial construction does indeed given the results that we would hope for. Indeed, this section demonstrates the need for a more abstract approach.
As promised in the previous part of this diptych, we explain, in Section 4, how the admissibility condition of simplicial connections is exactly what we need in order to be able to apply Chern-Weil theory, and thus get well-defined characteristic classes from our simplicial connections.
Although the preceding section is general enough to apply to arbitrary vector bundles on the nerve, in Section 5, we apply the results to the specific case of Green's barycentric connection on global vector bundles. This allows us to compare the resultingČech representatives to those calculated in Section 3 to see that they do indeed agree.
Finally, Section 6 gives a summary of the whole story contained in this two-part work: we already know that we can resolve coherent analytic sheaves by vector bundles on the nerve, and that we even get sufficiently nice connections on the latter; now we can apply the results in this paper to obtain characteristic classes. There is an even stronger result, further reinforcing the fact that this simplicial construction is indeed the "good" one: whenever X is compact, we can appeal to an axiomatisation of Chern classes to show that the construction given here agrees with any other construction that one might construct in some other way.

Preliminaries
Throughout, let (X , O X ) be a paracompact complex-analytic manifold with its structure sheaf the sheaf of holomorphic functions; let U be a locally-finite Stein open cover such that all finite intersections are also Stein. Let E be a vector bundle of rank r on X , and assume that it is trivialised by U . We have trivialisation maps ϕ α : E U α ∼ − → (O X U α ) r , and transition maps given on overlaps by M αβ = ϕ α • ϕ −1 β . By picking some basis of sections {s α 1 , . . ., s α r } of E over U α we can realise the M αβ as (r×r)-matrices that describe the change of basis when we go from E U β to E U α , i.e.
We continue to use the conventions, definitions, and notation from the previous paper [Hos23], but recall particularly pertinent results when necessary. In addition, we write S k to mean the symmetric group on k elements, and |σ| to mean the sign of a permutation σ ∈ S k .

Standard and exponential Atiyah classes
Definition 2.1.1. The Atiyah exact sequence (or jet sequence) of a locally free sheaf E of O X -modules is the short exact sequence The Atiyah class at E of E is defined to be the extension class of this short exact sequence: Remark 2.1.2. We are interested in the Atiyah class, as well as the higher standard and exponential Atiyah classes, (which we define later), of a vector bundle because their trace is "equivalent", in some sense that we make precise below, to the Chern classes. • The traces of the Atiyah classes satisfy an axiomatisation of Chern classes that, in the case where X is compact, guarantees uniqueness. This is explained in Section 6.3.
Note that the treatment of the Atiyah class in [Huy05] is somehow reverse to that found here: it starts there with the Atiyah class as aČech cocycle and then shows that it is equivalent to some splitting of the Atiyah exact sequence.
Definition 2.1.3. A holomorphic (Koszul) connection ∇ on a vector bundle E on X is a (holomorphic) splitting of the Atiyah exact sequence of E. By enforcing the Leibniz rule ∇(s ⊗ ω) = ∇s ∧ ω + s ⊗ dω we can extend any connection ∇ : given by enforcing the above Leibniz rule.
Given a connection ∇ on E, we say that a section s ∈ Γ(U, E) is flat if ∇(s) = 0; we say that the connection ∇ is flat if κ(∇) = 0.
Remark 2.1.4. Given the definition of a connection as a splitting of the Atiyah exact sequence, and the Atiyah class as the extension class of the Atiyah exact sequence, we see that one way of understanding the Atiyah class is as the obstruction to admitting a (global) holomorphic connection.
Proof. First, recall that the difference of any two connections is exactly an O X -linear map. Secondly, note that the cocycle condition is indeed satisfied, since Combined, this tells us that {∇ β −∇ α } α,β ∈Č 1 U (Hom (E, E ⊗Ω 1 X )). Then we use the isomorphisms Finally, we have to prove that the class thus defined in homology agrees with that given in our definition of the Atiyah class. We show this by proving a more general fact. (i) consider A → B as a complex with B in degree 0, and take the quasi-isomorphism (A → B) ∼ − → C ; [2] Using the same symbol ∇ to denote the connection as well as any such extension is a common abuse of notation.
(ii) invert this quasi-isomorphism to get a map C ∼ − → (A → B) such that the composite When A is the category of locally free sheaves on X , we can realise the quasi-isomorphism by using theČech complex of a complex: , and so we can lift this difference to A , giving us the map This map we have constructed is exactly [B]. More precisely, Definition 2.1.7. We write ω αβ for the cocycle ( Remark 2.1.8. When E , F , and G are sheaves of O X -modules, with G locally free, we have the isomorphism In particular, Taking the trivialisation over U α , this lets us consider ω αβ as an endomorphismvalued 1-form: an (r × r)-matrix of (holomorphic) 1-forms on X (where r is the rank of E).
Remark 2.1.9. Recall that, for sheaves F and G of O X -modules, we have the cup product which is given inČech cohomology by the tensor product: Definition 2.1.10. We define the second exponential [3] Atiyah class at •2 E by the following construction: take the cup product and apply the composition map and then apply the wedge product (of differential forms) to get In general, the k-th exponential Atiyah class at •k E is the class where the product is given by applying composition and then the wedge product (of forms) as above.
Remark 2.1.11. As a general note on notation, we will omit the wedge symbol ∧ when talking about the wedge product of differential forms (or we will use · if we have to use any symbol at all), and reserve it solely for the wedge product of endomorphisms. In particular, for endomorphism-valued forms M and N, we write MN (or M · N) to mean the object given by composing the endomorphisms and wedging the forms, and M ∧ N to mean the object given by wedging the endomorphisms (following [Ble81, Definition 2.1.1]) and wedging the forms. In terms of (2 × 2)-matrices (i.e. taking E to be of rank 2), this means Note that, if we take the trace, then these two objects will both be 2-forms on U. Similarly, tr M k and tr ∧ k M are both just k-forms on U.
[3] In general, we will be interested in standard Atiyah classes, but we consider the manual construction of exponential ones since these can be written down very explicitly, as shown in Section 3.
Remark 2.1.12. The classical theory of Chern classes has two important "types" of Chern class: standard and exponential. For now, we are content with simply saying, as a definition, that the polynomial that gives the exponential classes is tr(M p ), and the polynomial that gives the standard classes is tr(∧ p M). Caution is needed when discussing the k-th Atiyah class though: there is no trace in the definition; to obtain characteristic classes we have to take the trace.
Definition 2.1.13. We define the second standard Atiyah class at ∧2 E by the following construction: take the cup product and then apply the wedge product of endomorphisms and the wedge product of forms to get at ∧2 In general, the k-th standard Atiyah class at ∧k E is the class Remark 2.1.14. We can find an explicit representative for at •2 E by using Remark 2.1.9: where Mat r×r (A) is the collection of A-valued (r×r)-matrices. But before composing these two matrices, as described in Definition 2.1.10, we first have to account for the change of trivialisation from U βγ to U αβ . That is, after applying composition and the wedge product, we have Remark 2.1.15. We know that at •3 E is represented locally by ω αβ ω βγ ω γδ , but where ω βγ and ω γδ undergo a base change to become Ω 1 (U αβ )-valued. But then there seem to be two choices of how we might calculate this: 2. base change both ω γδ and ω βγ to be Ω 1 (U αβ )-valued; and then perform the triple composition.
These two would give the same result if and, thankfully, this equality does indeed hold, thanks to the cocycle condition on the M αβ and some form of associativity [4] , and so we can use whichever one we so please.

Fibre integration
Here we recall some results summarised in [Hos23,§2.3]. Let Y • be a simplicial complex manifold. Following [Dup76], we define a simplicial differential r-form on Y • to be a family (ω p ) p∈N of forms, with ω p a global section of the sheaf where Ω Y p is the sheaf of holomorphic forms, and Ω ∆ p extd is the sheaf of smooth forms; and where π Y p and π ∆ p extd are the projection maps from Y p × ∆ p extd ) such that, for all coface , We write Ω r,∆ (Y • ) to mean the algebra of all simplicial differential r-forms on Y • . We can describe each ω p as a form of type (i, j), by writing ω p = ξ p ⊗ τ p , where ξ p is the Y p -part of ω p , and τ p is the ∆ p -part of ω p , and then setting i = |ξ p | and j = |τ p |. This lets us define a differential which is given by the Koszul convention with respect to the type of the form: Lemma 2.2.1 (Dupont's fibre integration). There is a quasi-isomorphism which, for each fixed degree r, consists of a map induced by fibre integration where the latter is given by integrating the type (r − p, p) part of a simplicial form over the geometric realisation of the p-simplex with its canonical orientation.
. Since the integral of a k-form over an ℓ-dimensional manifold is only non-zero when k = ℓ, we see that the fibre integral of some simplicial differential r-form ω • = {ω i, j p } p∈N,i+ j=r is determined entirely by the type-(r − p, p) parts on the p-simplices: where the signs come from the fact that we work with X U There are further sign confusions that can arise from the choice of orientation on the simplices, but this will not concern us here. This is discussed in [Hos20].

Manual construction for global vector bundles
Assume now that we have flat holomorphic (local) connections ∇ α on each E U α .

Truncated de Rham cohomology
Remark 3.1.1. Since U is Stein (and any finite intersection of these opens is also Stein) and the sheaves Ω r X are coherent, we can apply Cartan's Theorem B, which tells us . Since X is paracompact and U is Stein, we know thať Cech cohomology computes hypercohomology, and soȞ k (X , . In summary, we are in a nice enough setting thatČech-de Rham bicomplex lets us calculate singular cohomology: Definition 3.1.2. Given the de Rham complex Ω • X , we define the k-th Hodge complex Ω • k X as the truncation where Ω k X is in degree k. Definition 3.1.3. We define the k-th truncated de Rham cohomology (or tDR cohomology) to be H k , such that c i = 0 for i k + 1, then we can refine the corresponding class in singular cohomology [c] ∈ H 2k (X , C) to a class [c] ∈ H 2k tDR (X ) in tDR cohomology. Remark 3.1.5. Recalling Remark 3.1.1, we know that we can use theČech complex to calculate singular cohomology. Now, say we are given some -closed, and thus represents a cohomology class in H 2k Tot •Č⋆ (Ω ⋆ X ), and thus a cohomology class in H 2k (X , C).
In essence, given some "starting element" in theČech-de Rham bicomplex, we can try to manually lift it to some closed element of the same total degree.
Remark 3.1.6. A few important notes before we continue.
, we don't necessarily have an easy way of computing explicitly what a closed class in theČechde Rham complex maps to under this isomorphism, unless it has a non-zero degree-(0, r) component, in which case it maps to exactly that component.
• We will construct classes in singular cohomology, but note that they can actually all be considered as living in the corresponding tDR cohomology, thanks to Remark 3.1.4.
• The assumption that the local connections ∇ α are flat is not necessary for the abstract theory, but essential for these explicit calculations.
• We point out, once more, that the constructions given in this chapter are for the exponential Atiyah classes.

The first Atiyah class
We wish to calculate ω αβ (Definition 2.1.7), and we can use the fact that our local sections {s α 1 , . . ., s α r } are ∇ α -flat: The signs depend on the parity of k.
(where we omit the restriction notation on the local connections). But then, using (2.0.0.1) followed by the Leibniz rule, Since the s β ℓ are ∇ β -flat, the first part of each ℓ-term is zero, and then, using the inverse of (2.0.0.1), where we can move the M −1 αβ across the tensor product because the tensor is over O X , and the M m ℓ are exactly elements of this ring. Thus, in the U α trivialisation, This ω αβ is exactly the first Chern class d log M αβ of the bundle.
Proof. Using the fact that d A −1 = −A −1 · dA · A −1 , we see that Proof. Since d tr ω αβ = tr dω αβ = − tr ω 2 αβ , by Lemma 3.2.2, this lemma is a specific case of the fact [6] that tr A 2k = 0 for any k ∈ N.
[6] Using the skew-symmetry of forms, and writing matrix multiplication as a sum, we can show that we can cyclically permute matrices of forms inside a trace, up to a sign (tr A 1 A 2 · · · A k = (−1) k−1 tr A 2 · · · A k A 1 ); this fact is then a corollary, since tr A 2k = −tr A 2k . By Lemma 3.2.3, and the fact that ω αβ is aČech cocycle by definition (Lemma 2.1.6), we can draw the closed element at E in theČech-de Rham bicomplex as follows:

The second Atiyah class
By Definition 2.1.10, we know that αβ α,β and we introduce the notation so that at •2 E = A X , and everything can be thought of as living over U α . By Lemma 3.2.2, we know that dA = −A 2 , and similarly for B and X . Further, by differentiating the cocycle condition M αβ M βγ = M αγ of the transition maps, and then right-multiplying by M −1 αγ , we see that [7] tr(A + X ) = tr(B). Hence tr at •2 E = tr (A(B − A)). Using the fact that dA = A 2 we see that dA 2 = 0, whence such that δ f = −d tr at •2 E and d f = 0. It is necessary, but not sufficient, that f be the trace of a polynomial of homogeneous degree 3 in the one variable A = ω αβ . But then f (A) = tr A 3 is, up to a scalar multiple, our only option. We set f (A) = 1 3 tr A 3 and compute itsČech coboundary: [7] As previously mentioned, Mω βγ M −1 is the natural way of thinking of X as being a map into something lying over U α , so this equation should be read as a cocycle condition over U α by thinking of it as ω αβ + ω βγ = ω αγ , where the tilde corresponds to a base change. Note that this is also the result we expect, since ω αβ corresponds to ∇ β − ∇ α , and (as we have already noted) this clearly satisfies the additive cocycle condition.
Using the aforementioned (Lemma 3.2.3) fact that we can cyclically permute, up to a sign, inside the trace, we see that Now we just have to worry about whether or not d f is zero. But and we know, as in Lemma 3.2.3, that this is indeed zero. This calculation can be summarised by the following diagram: Remark 3.3.1. We know that tr at •2 E is a cocycle by definition, but we can still manually show that it isČech closed: [8] See Remark 3.3.1.

The third Atiyah class
We extend our previous notation, writing . It is then relatively simple to calculate that Trying to find some ϕ ∈Č 2 (Ω 4 X ) such thatδϕ = d tr at •3 E , however, is slightly harder. The most naive approach is to list all the monomials inČ 2 (Ω 4 X ), apply theČech differential to each one, and then equate coefficients. Using the fact that (up to a sign) we can cyclically permute under the trace, finding all the monomials is the same as finding all degree-2 monomials in non-commutative variables X and Y , modulo equivalence under cyclic permutation, and there are just four of these: is exactly such thatδϕ = d tr at •3 E . Factoring dϕ, we see that This calculation can be summarised in the following diagram: Taking the signs of the total differential into account, this gives us the closed element

The fourth Atiyah class and beyond
Looking at the first three Atiyah classes, there are some evident patterns: for example, theČech 1-cocycle always seems to be some multiple of tr A 2k−1 . Beyond some vague pattern recognition, however, it becomes increasingly hard to work with the k-th Atiyah class for k 4, mainly due to the cumbersome number of monomials whose coefficients we have to equate. It seems likely that there are two patterns -one for k odd, and one for k even -but even this is hard to verify, given that k = 0 and k = 1 are both rather trivial, and k 4 is so unwieldy that it becomes harder to recognise any patterns "by hand". [9] For the sake of completeness, we give below the lift of at •4 E , calculated by [9] Thanks to a comment by Mahmoud Zeinalian, we see that the coefficients 1, 1 3 , 1 10 , 1 35 , etc. of the degree (1, p − 1) terms (i.e. those that are simply the traces of odd powers of the Maurer-Cartan form A = g −1 dg) a basic Haskell implementation of the naive method used so far: calculating all noncommutative monomials of a certain degree, applying theČech differential, and then equating coefficients.
The lift of at •4 E in the total complex is 0 , −1 35 tr P (1,7) , 1 5 tr P (2,6) , 1 5 tr P (3,5) , tr P (4,4) , The moral of the above calculation is the following: we need to find a better way of doing this. are given by where B(x, y) is the beta function (or Euler integral of the first kind). It seems possible that the other coefficients could be expressed in terms of "iterated beta functions", using the formulas coming from the fibre integration approach we study in Section 5.

Characteristic classes via admissible simplicial connections
Remark 4.0.1. We recall, from [Hos23, §4.2], the motivating example for the definition of admissibility. Write C to mean the category whose objects are pairs (V , ϕ) of finite-dimensional vector spaces V along with an endomorphism ϕ, and whose morphisms f : Then we consider the Grothendieck group K (C ), and introduce the equivalence relation [(V , 0)] ∼ [(0, 0)] to obtain K (C )/∼. Letting E : C → C be the endofunctor that sends (V , ϕ) to (V / Ker ϕ, 0), we write L E C to mean the localisation of C at all morphisms that become isomorphisms after applying E (the wide subcategory of which we denote by W ).
Then a morphism f : 2. f restricts to a morphism V 1 → W 1 ; and

Generalised invariant polynomials
Definition 4.1.1. Using the notation of Remark 4.0.1, a sequence (P k ) k∈N of C-valued polynomials P k on C ⊗n is said to be a generalised invariant polynomial of degree n if the following conditions are satisfied: (ii) the P k each factor through K (C )/∼.
The second condition is equivalent to the following: the P k all satisfy the extension by zero property: This extension by zero property, when combined with additivity, tells us that P is fully additive: Mirroring classical notation, given some P • = (P k ) k∈N on C ⊗n , we write P • to mean the sequence of invariant and additive polynomials on C given by Example 4.1.2. The prototypical example of a generalised invariant polynomial is the degree-1 polynomial given by P k = tr for all r ∈ N, where the trace is of the endomorphism part of a pair; or, more generally, the degree n polynomial given by P n k = tr •µ n , where µ n is the multiplication map that sends an n-fold tensor product of endomorphisms to the endomorphism given by the composition of all the endomorphisms. [10] Lemma 4.1.3. Using the notation from Remark 4.0.1, let f : (V , ϕ) → (W, ψ) be an admissible morphism, and (P k ) k∈N a generalised invariant polynomial of degree n that is zero on an n-fold tensor product of flat objects. Then P k (V , ϕ) = P k (W, ψ) for all k ∈ N.
Proof. By definition, there exist decompositions V ∼ = V 1 ⊕ V 2 and W ∼ = W 1 ⊕ W 2 such that ϕ(V 1 ) = ψ(W 1 ) = 0 and f restricts to an isomorphism V /V 1 ∼ − → W/W 1 . Then, necessarily, dimV 2 = dimW 2 = s for some s ∈ N. Since P k is fully additive, But P k is assumed to be zero on an n-fold tensor product of flat objects, and so Then GL k -invariance tells us that P s (V 2 , ϕ V 2 ) = P s (W 2 , ϕ W 2 ) , and this is equal to P dimW (W, ψ) , again by full additivity and being zero on flat objects.
Corollary 4.1.4. Let ω • be an admissible endomorphism-valued simplicial r-form [11] on E • , and P ⋆ = (P k ) k∈N a generalised invariant polynomial of degree n that is zero on n-fold tensor products of flat objects. Then P ⋆ (ω • ) is a simplicial rn-form.
This means that we can define P r(p) (ω p ) by applying P r(p) to the endomorphism part of ω ⊗n p , and wedging the n-fold tensor product of the form part. Then we need to show that (X U can use the extension by zero property to replace P r(p−1) and P r(p) with P s , where s = [10] The fact that this is well defined relies on being able to view all the V i in the tensor product as being identical, but we can do this thanks to the GL-invariance of the trace, as well as the extension by zero property.
[11] See [Hos23, Definition 4.13]. max{r(p − 1), r(p)}. But then, by admissibility and Lemma 4.1.3, it suffices to show that P s commutes with pullbacks. Because P s is just given by the wedge product of forms on the form part of ω p , it commutes with pullbacks there. On the endomorphism part of ω p , since we are working locally, the pullback is simply a change of basis, and so commutes with the P s by GL-invariance.

Simplicial Chern-Weil
The simplicial version of Chern-Weil theory that we will now discuss is almost identical to the classical one, such as that found in [Huy05, §4.4]: we can apply the classical arguments "simplicial level by simplicial level", and then use the admissibility condition to ensure that these all glue. admissible endomorphism-valued simplicial d j -forms, and let P • be a generalised invariant polynomial of degree n. Then Proof. The first statement is found in the proof of [Huy05, Lemma 4.4.6], and so we simply apply it to each P r(p) (κ(∇ p )), where r(p) is the rank of E p ; the last four are [Huy05, Lemma 4.3.4, Lemma 4.3.5, Lemma 4.4.4, Corollary 4.4.5] (respectively) applied to each P r(p) (κ(∇ p )).

Lemma 4.2.2.
If ∇ • and ∇ ′ • are two admissible [12] simplicial connections on E • such that their difference is admissible, and P • is a generalised invariant polynomial of degree n, then is exact in the simplicial de Rham complex.
[12] That is, a connection ∇ p on each E p such that the curvatures define a simplicial differential form and By skew-symmetry, the last two terms will cancel when we apply P • , and so we want the difference B • = A ′ • − A • to be admissible, since then P ⋆ (B • ) will be an endomorphismvalued simplicial form (by Corollary 4.1.4), and so P ⋆ (dB • ) = dP ⋆ (B • ) will be closed in the simplicial de Rham complex. But this is the case, by hypothesis.

The Green example
Definition 4.3.1. Given a vector bundle on the nerve E • and an admissible simplicial connection ∇ • on E • , we define the k-th simplicial exponential Atiyah classât • on E •,1 is given by We see that the only non-trivial part of the simplicial connections is in (simplicial) degree 1, over U 12 , and so we calculate these curvatures: Recalling Remark 2.2.2, we know that the fibre integral is given by integrating the (k − p, p) term over the p-simplex (and here we are taking k = 1), and so we find that the [13] Atiyah classes are given by Note that the square of either curvature is zero, and so only the first Atiyah class is non-trivial -all higher ones are zero. Using the convention/definition that the zero-th Atiyah class is 1, we have the total Atiyah classes (that is, Chern characters), defined as the sums of all the Atiyah classes: Finally, if we say that, for a resolution the total Atiyah class (or Chern character) of R •,0 is given by the alternating sum of the total Atiyah classes (or Chern characters) of the R •,i , then [13] The first exponential and first standard Atiyah classes agree, by definition, so we do not need to say which we are using.
This agrees with what one might calculate using a short exact sequence (writing the skyscraper sheaf as a quotient) and the Whitney sum formula, but is stronger, since Green's method gives actual representatives of the cohomology class as well.

The barycentric connection
Throughout this section, • as always, let E be a vector bundle on X with local flat connections ∇ α ; • write π p : X U p × ∆ p → X U p to mean the projection map; • define E • , a vector bundle on the nerve, by setting E p = (X U p → X ) * E; • write E p to mean the pullback of E p along π p : is an isomorphism. [14] This means that, writing ζ i p : [0] → [p] to mean the map of simplices that sends 0 to i, we can identify (X U • ζ i p ) * E 0 with E p , by using the isomorphism E • ζ i p between them. This lets us, in particular, think of any connection ∇ α i on E 0 as a connection on E p , and we keep the same notation to denote both. Then the ∇ α -flat sections of E p are exactly those of the form π * p (s), where s is some ∇ α -flat section of E U α . Note also that the morphism O ∆ p → O X U p ×∆ p gives us an O ∆ p -action on E p .
It is actually an identity map.
which acts on a section s ⊗ f of E p = π −1 In essence, this tells us that, given a (flat) basis of sections of E p (U α 0 ...α p × ∆ p ) in the U α 0 trivialisation, the barycentric curvature acts simply as where dω p = (d X −d ∆ p )ω p (following the Koszul convention, as explained in Section 2.2).
Remark 5.1.8. Since E • is the pullback of a global vector bundle, it is in particular strongly cartesian, and in fact Green. Since the barycentric connection is defined exactly as a connection generated in degree zero, we can apply [Hos23,Theorem 4.29], which tells us that it is an admissible simplicial connection. Further, Remark 4.2.3 tells us that barycentric connections form a compatible family, and so Lemma 4.2.2 says that the resulting characteristic classes are well defined.
The barycentric curvature is, by definition (since the barycentric connection is admissible), admissible, but we can actually show that ω • itself is an admissible endomorphismvalued simplicial 2-form, by an argument very similar to that of the proof of [Hos23,Theorem 4.29].
Lemma 5.1.9. The trace of the k-th simplicial (standard or exponential) Atiyah class is closed in the simplicial de Rham complex.
Proof. We have already formally proven this in Section 4.2, but we give here an explicit proof for the case k = 1 as well. For any p ∈ N, But, recalling the proof of Lemma 3.2.3, we know that and Lemma 3.2.2 tells us that dω α 0 α i = −ω 2 α 0 α i , whence For fixed i, j, the first two terms both change sign under i ↔ j, whence they contribute zero to the trace, since they are equal when i = j. We also know [15] that tr ω 2 α 0 α j ω α 0 α i = tr ω α 0 α i ω 2 α 0 α j , whence the last two terms also both change sign under i ↔ j and are equal when i = j, as above. Thus the trace is zero.

The first simplicial Atiyah class
The first simplicial (exponential) Atiyah class is given bŷ by using the barycentric connection, and Remark 5.1.7. As explained in Remark 2.2.2, the fibre integral ofât •1 E depends only on the (2, 0), (1, 1), and (0, 2) parts, but here there is no (0, 2) part (i.e. there is no dt j dt i term), and so Continuing the calculation gives us the result that we expect: as in Section 3.2, we get that where we write that this term lives in degree p = 1 to remind us that the result is aČech 1-cocycle.
[15] The trace on endomorphism-valued simplicial forms is cyclic, up to a sign, as mentioned in the proof of Lemma 3.2.3.
Remark 5.2.1. In general, as in the manual construction, the fibre integral of the k-th simplicial (exponential) Atiyah class will have terms inČ k−i Ω k+i X for i = 0, . . ., k − 1, and so we make up for our notational laziness (namely, using + to mean ⊕) by labelling the terms with theirČech degree.
Note that, although the (2k, 0) part ofât •k E is, in general, non-zero, when we fibre integrate we look at it on the 0-simplex (again, by Remark 2.2.2), and there it is zero, since all the sums are trivially zero.

The second simplicial Atiyah class
Not forgetting the sign ǫ 2 = −1 from Definition 4.3.1, we have that but we also know that the only parts that will be non-zero after fibre integration are the (2, 2) parts on the 2-simplex, and the (3, 1) parts on the 1-simplex. The only (2, 2) part comes from the first half of the (dω 2 ) 2 term, which gives us This means that, so far, we have For the (3, 1) part, we work on the 1-simplex and get Finally then, we have which again agrees with the result of Section 3.3.

The third simplicial Atiyah class
Remark 5.4.1. There is a subtlety in the calculations when we reach the third simplicial Atiyah class due to our choice of conventions forČech cocycles: we don't assume skew-symmetry of cocycles (i.e. that exchanging two indices changes sign), but it is true that skew-symmetrisation of cocycles is a quasi-isomorphism, and so doesn't change the resulting cohomology class. If we had worked with skew-symmetricČech cocycles from the start then this calculation would seem somewhat simpler. In particular, the first two (exponential) Atiyah classes agree with those that we manually constructed in Sections 3.2 and 3.3 on the nose, whereas for k 3 we will have equality only in cohomology.
Definition 5.4.2. We write ς p to mean the skew-symmetrisation of aČech p-cochain, so that where |σ| denotes the sign of σ.
whence tr(−1) 2·2 Comparing this to Section 3.4, we see that we have the same, except for a missing + 1 2 tr ω 2 α 0 α 1 ω 2 α 1 α 2 term. But this missing term skew-symmetrises to zero, since it is invariant under the permutation that swaps 0 and 2. Thus the (4, 2) part is exactly what we wanted, and Finally, the (5, 1) part is which agrees exactly with the manual construction from Section 3.4. Thus (5.4.2.9)

Agreement with the manual construction
Theorem 5.5.1. The degree-(k, k) term in the trace of fibre integral of the k-th simplicial exponential Atiyah class agrees with the k-th exponential Atiyah class, up to skewsymmetrisation. That is, Proof. First we rewrite the left-hand side. Generalising the results of Section 5.4, we can write the term coming from fibre integration as where S k S k+1 acts on {0, 1, . . ., k} by fixing 0. But then, since σ(0) = 0, we can rewrite this as Now we can use the fact that multiplication by an element of S k S k+1 is an automorphism to perform a change of variables, giving us which is (trivially, since ω ii = 0) equal to Next we rewrite the right-hand side. The skew-symmetrisation is simply Now we prove equality. Since, for each fixed η, there are no relations satisfied between the ω η(0)η(i) , we have two homogeneous degree k polynomials in (k + 1) free noncommutative variables. To emphasise the fact the following argument is purely abstract, we write x i = ω 0i and define an action of S k+1 on the x i by x η(i) = ω η(0)η(i) . So showing that the left-and right-hand sides are equal amounts to showing that Write E to mean the Q-linear span of degree-k monomials in the (k + 1) free noncommutative variables x 0 , . . ., x k , and let σ p,q ∈ S k+1 be the transposition that swaps p with q. Then σ p,q gives an involution on E by swapping x p with x q , and thus E ∼ = E p,q (1) ⊕ E p,q (−1), where E p,q (λ) is the eigenspace corresponding to the eigenvalue λ.
Let H p,q be the Q-linear subspace of E spanned by monomials that contain at least one x p or x q . This subspace is stable [16] under σ p,q , and so this space also splits as H p,q ∼ = H p,q (1) ⊕ H p,q (−1). Further, we have the inclusion H p,q (−1) ⊆ E p,q (−1). But if X ∈ E p,q (−1), then, in particular, X must contain at least one [17] x p or x q , so X ∈ H p,q , whence X ∈ H p,q (−1). Thus E p,q (−1) = H p,q (−1).
The intersection H of the H p,q over all distinct pairs (p, q) ∈ {0, . . ., k} × {0, . . ., k} is the Q-linear span of all monomials containing all but one of the x i (and, in particular, containing k distinct x i ). But since H p,q (−1) = E p,q (−1), we see that the intersection E(−1) of all the E p,q (−1) is equal to H(−1). Now both A and B are in E p,q (−1) for all p, q (since the sign of ω p,q is −1), and so A, B ∈ E(−1) = H(−1). Since the coefficient of, for example, the x 1 · · · x k term is the same (and non-zero [18] ) in both A and B , it suffices to show that H(−1) is one-dimensional.
So let X , Y ∈ H(−1) be monomials. Then each one contains k distinct x i , and so there exists some (unique) σ ∈ S k+1 such that σX = ±Y . But, writing σ = σ p 1 ,q 1 · · · σ p r ,q r , we know that σX = (−1) r X , whence X = Y , up to some sign (and so up to some scalar in Z).

From vector bundles to coherent sheaves
applying the generalised invariant polynomial P • = {tr •µ n } n∈N (where µ n is the multiplication map that sends an n-fold tensor product of endomorphisms to the endomorphism given by composition (resp. wedge product)) to the curvatures of each of the simplicial connections, we obtain the simplicial exponential (resp. standard) Atiyah classes of each E •,i . Using the alternating-sum convention, this gives us the simplicial exponential (or standard) Atiyah class of F. Finally, by fibre integration of the trace of these classes, we obtain closed classes in theČech-de Rham bicomplex, and thus classes in de Rham (or even tDR) cohomology. Lemma 6.1.1. This construction is independent of the choice of twisting cochain (and thus cover) and of local connections.
Proof. The argument for this is of exactly the same form as the proof of [Gre80, Theorem 2.4], where we pass to the disjoint union of the covers and can inductively construct a twisting cochain which is a common extension of the two twisting cochains from which we start. Although Green only gives the proof for the (p, p)-term, his argument applies equally well to the fullČech-de Rham representative (C p,p , C p−1,p+1 , . . ., C 0,2p ) without modification.
Remark 6.1.2. Of particular importance is [Hos23, Lemma 4.39], which tells us that the homotopy colimit (over refinements of covers) of the localisation of the category Green ∇,0 (X U • ) of Green complexes is equivalent to the homotopy colimit of the localisation of the category of complexes Green ∇,0 (X U • ) which are locally quasi-isomorphic to Green complexes. In particular, we only know how to construct admissible simplicial connections for objects of the former, and so it is necessary that our construction of E •,⋆ in the above is an object of Green ∇,0 (X U • ). where c 1 (L ) denotes the first Chern class of the line bundle given by the connecting homomorphism from the Picard group in the long exact sequence associated to the exponential sheaf sequence Proof. This is basically the combination of the following facts: the derived pullback is exact; the simplicial Atiyah class is defined by taking a resolution; the derived pullback of a locally free sheaf agrees with the non-derived pullback; the non-simplicial Atiyah class of the non-derived pullback of a locally free sheaf is exactly the pullback of the non-simplicial Atiyah class of the locally free sheaf. We spell out how to join up these facts in slightly more detail below.

General properties
Since the derived pullback is exact, we know that, given Green's resolution E •,⋆ of F • = (X U • → X ) * F by vector bundles on the nerve, the derived pullback f * E •,⋆ is a resolution for f * F • . But the derived pullback on locally free objects agrees with the non-derived pullback (since we are tensoring with something locally free), and so f * E •,⋆ can be calculated by the non-derived pullback. Now we can use the fact [20] that, for a single vector bundle, the Atiyah class of the pullback is the pullback of the Atiyah class; for a single vector bundle on the nerve, the simplicial Atiyah class is determined entirely by the ω α 0 α i (which represent the non-simplicial Atiyah class). Combining all the above, we see that all the forms defining the simplicial Atiyah class of the derived pullback of F are exactly the pullbacks of the forms defining the simplicial Atiyah class of F.
In particular, then, taking the trace and then fibre integrating (both of which commute with the pullback of forms on X ) gives us the required result.
Lemma 6.2.3. If the total simplicial Atiyah classes are additive on every split exact sequence of coherent sheaves then they are additive on every short exact sequence of coherent sheaves.
[20] We are working with vector bundles, which are locally free, and so, as previously mentioned, the derived pullback is just exactly the non-derived pullback. There is some subtlety however, since we are using the word "pullback" to mean a few different things here. When we talk about pulling back the Atiyah class of E, we mean first applying the pullback of sheaves, to get some class in H 1 (X , f * Ω 1 X ⊗ End(f * E)) (using the fact that pullbacks commute with End for finite-dimensional locally free sheaves), and then applying the canonical map f * Ω 1 X → Ω 1 Y (which is what we really mean when we talk about pulling back forms). The fact that this construction sends at E to at f * E follows from the fact that short exact sequences are distinguished triangles, and so specifying a morphism between the first and last (non-zero) terms of two SESs extends uniquely to a morphism of the SESs (in that we get a unique morphism between the middle terms of the two SESs such that everything commutes).
Proof. Let 0 → F ι − → G π − → H → 0 be a short exact sequence of coherent sheaves on X , and t ∈ C (which can be thought of as t ∈ Γ(C, P 1 )). Write p : X ×P 1 → X to mean the projection map. Define We claim that this gives a short exact sequence of sheaves over X × P 1 , where the maps are the "obvious" ones: To prove surjectivity, let h ∈ Γ(U, p * H). Then h ⊗ t ∈ Γ(U, p * H(1)). But π : G → H is surjective, and thus so too is the induced map π(1) : p * G(1) → p * H(1). Hence there exists g ⊗ y ∈ Γ(U, p * G(1)) such that π(g ⊗ y) = h ⊗ t. So (g ⊗ y, h) ∈ N maps to h.
To prove injectivity (and that this map is indeed well defined), we use the fact that tensoring with O(1) is exact, and so, in particular, ι(1) : p * F(1) → p * G(1) is injective. The inclusion into the direct sum p * G(1) ⊕ p * H is injective by the definition of a direct sum, so all that remains to show is that the image of this composite map is contained inside N . Let f ⊗ x ∈ Γ(U, p * F(1)). Then this maps to (ι( f ) ⊗ x, 0) ∈ p * G(1) ⊕ p * H, but this is clearly in the kernel of π(1) − t · id since πι( f ) = 0.
To prove exactness, it suffices to show that Ker(N → p * F) ∼ = p * F(1). But Ker(N → p * F) = {(g ⊗ y, h) ∈ N | h = 0} = {(g ⊗ y, h) ∈ p * G(1) ⊕ p * H | h = 0 and π(g) ⊗ y − h ⊗ t = 0} = {(g ⊗ y, h) ∈ p * G(1) ⊕ p * H | π(g) ⊗ y = 0} = {(g ⊗ y, h) ∈ p * G(1) ⊕ p * H | (g ⊗ y) ∈ Im ι(1)} ∼ = p * F(1). Now we claim that the short exact sequence is split for t = 0, and has N ∼ = p * G for t = 0. Formally, we do this by looking at the pullback of the map X × {t} → X × P 1 , but we can think of this as just "picking a value for t". For t = 0, define the injective morphism ϕ : p * G → N of coherent sheaves by ϕ(g) = (g ⊗ t, π(g)). To see that this is also surjective, let (g ⊗ y, h) ∈ N . If y = 0 then we must have h = 0, and so (g ⊗ y, h) = (0, 0) = ϕ(0⊗0). If y = 0 then π(g)⊗ y− h⊗ t = 0, with y, t = 0, whence π(g) = y t h. Then (g ⊗ y, h) = ( t y g ⊗ t, π( t y g)) = ϕ( t y g). As one final ingredient, note that any coherent sheaf on X pulled back to a sheaf on X × P 1 is flat over P 1 , and so N is flat over P 1 , since both F(1) and H are. Thus, for τ t : X ×{t} → X ×P 1 given by a choice of t ∈ C, the derived pullback Lτ * t N agrees with the usual pullback τ * t N . Now we use the P 1 -homotopy invariance of de Rham cohomology, which is the following statement: the induced map does not depend on the choice of t. Since X × {t} is (canonically) homotopic to X , we can identify (pτ t ) * with the identity on H • (X , Ω • X ). Since the SES splits for t = 0, by our hypothesis, flatness, and the fact that Green's construction is functorial under derived pullback, we know that But we also know that N ∼ = p * G for t = 0, and so c(G) = (pτ t ) * c(G) = τ * t c(N ).
So, finally, the t-invariance of τ * tells us that c(G) = c(F) ∧ c(H).
Lemma 6.2.4. The total simplicial Atiyah classes are additive on every split exact sequence of coherent sheaves.
Proof. This is exactly [Gre80, Lemma 2.6], but the essence of the proof is simple: twisting cochains behave nicely with respect to direct sums, as do all of the constructions giving the simplicial connections generated in degree zero.
Corollary 6.2.5. The total simplicial Atiyah classes are additive on every short exact sequence of coherent sheaves.

The compact case
In the case where X is compact, we can appeal to [Gri09, Theorem 6.5], which gives a list of conditions that, if satisfied, ensure uniqueness of Chern classes: if we can show that the fibre integrals of traces of the simplicial Atiyah classes satisfy certain axioms, then we know that they are exactly the same classes as given by any other construction of Chern classes. Since tDR cohomology satisfies conditions (α) to (δ) of [Gri09, §6.2], it suffices to check that the following things hold true: (i) The construction agrees with 'the' classical one for line bundles. Although [Gre80, Lemma 2.5] tells us that the construction actually agrees with the classical one for arbitrary vector bundles, we explain the specific case of line bundles in Lemma 6.2.1.
(ii) The construction is functorial under pullbacks. This is Lemma 6.2.2.
(iii) The construction gives us the Whitney sum formula for short exact sequences. We have split the proof of this into two steps: firstly, showing in Lemma 6.2.3 that it suffices to prove that we have the Whitney sum formula for split exact sequences; then showing in Lemma 6.2.4 that we do indeed have the Whitney sum formula for split exact sequences.
(iv) The construction gives us the Grothendieck-Riemann-Roch formula for closed immersions. Although this is given as a requirement in [Gri09, Theorem 6.5], it is actually not necessary, thanks to e.g. [Gri12, Proposition 3.1], whose proof relies only on the three properties above, following some classical algebraic geometry methods (such as deformation to the normal cone).
Remark 6.3.1. Again, when X is compact, [Gre80, Lemma 2.7] gives a direct proof of the fact that the (p, p)-term of the fibre integral of the trace of these simplicial Atiyah classes agrees (up to a constant factor) with the Chern classes of Atiyah-Hirzenbruch.
Corollary 6.3.2. If X is a compact complex-analytic manifold, then, for any coherent analytic sheaf, the Chern classes given by the fibre integrals of the traces of the simplicial Atiyah classes (following the construction described in Section 6.1) agree with all other constructions of Chern classes of coherent analytic sheaves.