A note on a question of Markman

1.1 Let $\displaystyle X$ be a smooth complex algebraic variety. Consider the first order deformation $\displaystyle\tilde{X}$ of $\displaystyle X$ associated to a class $\displaystyle\tilde{\alpha}\in H^{1}(X,T_{X})$.

In general, a vector bundle $\displaystyle\mathcal{F}$ may not deform to a bundle $\displaystyle\tilde{\mathcal{F}}$ on $\displaystyle\tilde{X}$. The obstruction $\displaystyle\alpha_{\mathcal{F}}\in\mathrm{Ext}^{2}(\mathcal{F},\mathcal{F})$ to the existence of a vector bundle $\displaystyle\tilde{\mathcal{F}}$ on $\displaystyle\tilde{X}$ such that $\displaystyle\tilde{\mathcal{F}}|_{X}\cong\mathcal{F}$ was described in [B94, T09] as the contraction

$$\displaystyle\alpha_{\mathcal{F}}=\tilde{\alpha}\lrcorner\,at_{\mathcal{F}}\in\mathrm{Ext}^{2}(\mathcal{F},\mathcal{F}).$$

Here $\displaystyle at_{\mathcal{F}}\in\mathrm{Ext}^{1}(\mathcal{F},\mathcal{F}\otimes\Omega_{X})$ is the Atiyah class of $\displaystyle\mathcal{F}$.

Moreover, if $\displaystyle\mathcal{F}$ does deform, then its Chern classes and hence its Mukai vector stay of Hodge type on the deformed space $\displaystyle\tilde{X}$. This implies that the class

$$\displaystyle\tilde{\alpha}\lrcorner\,v(\mathcal{F})\in\mathrm{H\Omega}_{*}(X)\overset{\mathrm{def}}{=}\bigoplus_{q-p=*}H^{p}(X,\wedge^{q}\Omega_{X})$$

vanishes, where $\displaystyle v(\mathcal{F})$ is the Mukai vector of $\displaystyle\mathcal{F}$.

Thus, in the simple case where $\displaystyle\tilde{\alpha}\in H^{1}(X,T_{X})$ we conclude that if $\displaystyle\tilde{\alpha}\lrcorner\,at_{\mathcal{F}}$ is zero, then $\displaystyle\tilde{\alpha}\lrcorner\,v(\mathcal{F})$ is zero.

n an email correspondence, Eyal Markman asked if the above statement can be generalized to the case where $\displaystyle\tilde{\alpha}$ is an arbitrary polyvector field in $\displaystyle\mathrm{HT}^{*}(X)=\bigoplus_{p+q=*}H^{p}(X,\wedge^{q}T_{X})$. According to Markman, this question is central to his study of the deformations of hyperkähler manifolds. We provide an answer to this question in this paper.

irst, there are two ways to generalize the class that appears in (LABEL:1.1) above. For a class $\displaystyle\tilde{\alpha}\in\mathrm{HT}^{*}(X)$ we can define classes in $\displaystyle\mathrm{Ext}^{*}(\mathcal{F},\mathcal{F})$ in the following two ways.

The first is defined by using the HKR isomorphism

$$\displaystyle I^{HKR}:\mathrm{HT}^{*}(X)\rightarrow\mathrm{HH}^{*}(X),$$

where the latter is the Hochschild cohomology of $\displaystyle X$. Since $\displaystyle\mathrm{HH}^{*}(X)$ can be interpreted as natural transformations of the identity functor at the dg level, this yields a natural map $\displaystyle\mathrm{HH}^{*}(X)\rightarrow\mathrm{Ext}^{*}(\mathcal{F},\mathcal{F})$.

The second construction was defined by Toda [T09]. Consider the exponential Atiyah class

$$\displaystyle\exp(at_{\mathcal{F}})=1+at_{\mathcal{F}}+\cdots+\frac{(at_{\mathcal{F}})^{k}}{k!}+\cdots,$$

where $\displaystyle(at_{\mathcal{F}})^{k}\in\mathrm{Ext}^{k}(\mathcal{F},\mathcal{F}\otimes\wedge^{k}\Omega_{X})$. Let $\displaystyle\tilde{\alpha}^{p,k}\in H^{p}(X,\wedge^{k}T_{X})$ be the homogenous degree $\displaystyle(p,k)$ part of $\displaystyle\tilde{\alpha}$. We can contract $\displaystyle\tilde{\alpha}^{p,k}$ with $\displaystyle\frac{(at_{\mathcal{F}})^{k}}{k!}$ to get an element in $\displaystyle\mathrm{Ext}^{p+k}(\mathcal{F},\mathcal{F})$. Taking the sum over all $\displaystyle(p,k)$, we get the desired class which will be denoted by $\displaystyle\tilde{\alpha\lrcorner\exp(at_{\mathcal{F}})}\in\mathrm{Ext}^{*}(\mathcal{F},\mathcal{F})$. When $\displaystyle\tilde{\alpha}$ is a class in $\displaystyle H^{1}(X,T_{X})$, we recover the previous contraction $\displaystyle\tilde{\alpha}\lrcorner\,at_{\mathcal{F}}$.

Our first result is below.

The two classes defined above are the same. In other words the diagram

is commutative.

There is an analogous result for Hopf algebras. See Theorem 2.7 in [CI17] and see [PW09] for more details.

he space $\displaystyle\mathrm{H\Omega}_{*}(X)$ is naturally a module over $\displaystyle\mathrm{HT}^{*}(X)$, mimicking the module structure of Hochschild homology over cohomology. For an object $\displaystyle\mathcal{F}$ in the derived category of $\displaystyle X$, its Mukai vector $\displaystyle v(\mathcal{F})$ lies in $\displaystyle\mathrm{H\Omega}_{*}(X)$. Thus we can act with the class $\displaystyle\tilde{\alpha}$ to obtain $\displaystyle\tilde{\alpha}\lrcorner\,v(\mathcal{F})\in\mathrm{H\Omega}_{*}(X)$.

Theorem B. If $\displaystyle\tilde{\alpha}\lrcorner\exp(at_{\mathcal{F}})=0$, then we have

$$\displaystyle D(\tilde{\alpha})\lrcorner\,v(\mathcal{F})=0.$$

Here $\displaystyle D$ is the Duflo operator,

$$\displaystyle D(\tilde{\alpha})=Td(X)^{\frac{1}{2}}\lrcorner\,\tilde{\alpha},$$

where $\displaystyle Td(X)$ is the Todd class of $\displaystyle X$.

he inspiration for Theorem A comes from a similar statement in Lie theory. Let $\displaystyle\mathfrak{g}$ be a finite dimensional Lie algebra and $\displaystyle V$ be a finite dimensional representation. One can draw the diagram

which is similar to the one in Theorem A. We will provide more details and prove that the diagram above is commutative in section 2.

ote that our statement in Theorem B appears to be different from the original one, which did not have the Duflo operator $\displaystyle D$. We will prove that the original statement follows easily from ours.

Section 2 contains the proof of Theorem A and of its Lie theoretic analogue.

Section 3 is devoted to the proof of Theorem B. It is a consequence of Theorem A. At the end we prove that we can recover the result in (LABEL:1.1) from Theorem B.

I would like to thank Andrei Căldăraru for discussing details with me during our weekly meeting. I have benefited from stimulating email correspondence with Dror Bar-Natan. I am also grateful to Eyal Markman for many valuable comments. The author is partially supported by the National Science Foundation under Grant No. DMS-1811925.

Let $\displaystyle\mathfrak{g}$ be a finite dimensional Lie algebra over a field of characteristic zero and let $\displaystyle V$ be a finite dimensional representation of $\displaystyle\mathfrak{g}$. There is a diagram

The PBW map from the symmetric algebra $\displaystyle S\mathfrak{g}$ to the universal enveloping algebra $\displaystyle U\mathfrak{g}$ is defined on the degree $\displaystyle n$-th component of $\displaystyle S\mathfrak{g}$ as follows

$$\displaystyle x_{1}\cdots x_{n}\rightarrow\frac{1}{n!}\sum_{\sigma\in S_{n}}x_{\sigma(1)}\cdots x_{\sigma(n)}.$$

Here $\displaystyle S_{n}$ is the symmetric group on a finite set of n symbols. The universal enveloping algebra $\displaystyle U\mathfrak{g}$ acts naturally on $\displaystyle V$. This natural action defines the map $\displaystyle(\mathrm{U}\mathfrak{g})^{\mathfrak{g}}\rightarrow\mathrm{Hom}(V,V)$ on the top of the diagram above. The map $\displaystyle(\mathrm{S}\mathfrak{g})^{\mathfrak{g}}\rightarrow\mathrm{Hom}(V,V)$ is defined as follows. We can rewrite the representation map $\displaystyle\mathfrak{g}\otimes V\rightarrow V$ as a map $\displaystyle\Lambda:V\rightarrow V\otimes\mathfrak{g}^{*}$. Take the exponent

$$\displaystyle\exp(\Lambda)=id_{V}+\Lambda+\cdots+\frac{\Lambda^{k}}{k!}+\cdots$$

of the map $\displaystyle\Lambda$. Then we can contract $\displaystyle\exp(\Lambda)$ with $\displaystyle S\mathfrak{g}$.

In algebraic geometry, Kapranov and Kontsevich [K99] observed that the shifted tangent bundle $\displaystyle T_{X}[-1]$ has a Lie algebra structure in the derived category of $\displaystyle X$. Roberts and Willerton [RW10] proved that the category of representations of $\displaystyle T_{X}[-1]$ is the derived category of $\displaystyle X$ and the universal enveloping algebra of $\displaystyle T_{X}[-1]$ is the Hochschild cochain complex $\displaystyle\mathcal{RH}om(\Delta_{*}\mathcal{O}_{X},\Delta_{*}\mathcal{O}_{X})$, where $\displaystyle\Delta$ is the diagonal embedding $\displaystyle\Delta:X\hookrightarrow X\times X$. The functor $\displaystyle(-)^{\mathfrak{g}}$ is the 0-th Lie algebra cohomology which is similar to $\displaystyle H^{*}(X,-)$. Setting $\displaystyle\mathfrak{g}$ to be equal to $\displaystyle T_{X}[-1]$ in the Lie algebra diagram, we get the diagram in Theorem A for a smooth complex variety $\displaystyle X$. The Hochschild cohomology $\displaystyle\mathrm{HH}^{*}(X)$ plays the role of $\displaystyle(U\mathfrak{g})^{\mathfrak{g}}$, $\displaystyle\mathrm{HT}^{*}(X)$ plays the role of $\displaystyle(S\mathfrak{g})^{\mathfrak{g}}$, and the HKR map is precisely the PBW map.

{Proof}

[Proof of the commutativity for the Lie algebra diagram.] We can prove that the diagram in (2) is commutative even before taking $\displaystyle\mathfrak{g}$-invariants, i.e., the diagram

is commutative. The map PBW factors through the tensor algebra $\displaystyle T\mathfrak{g}$

so we can replace $\displaystyle U\mathfrak{g}$ at the top left corner of the diagram by $\displaystyle T\mathfrak{g}$. It is easy to check that the map $\displaystyle S\mathfrak{g}\rightarrow\mathrm{Hom}(V,V)$ is equal to the following map

where the map $\displaystyle\varphi:T\mathfrak{g}\rightarrow\mathrm{Hom}(V,V)$ is defined as follows. Rewrite the representation map $\displaystyle\mathfrak{g}\otimes V\rightarrow V$ as a map $\displaystyle\Lambda:V\rightarrow V\otimes\mathfrak{g}^{*}$. Instead of taking the exponential of the map $\displaystyle\Lambda$, we compose the map $\displaystyle\Lambda$ with itself $\displaystyle k$ times. We get a map $\displaystyle\Lambda^{\otimes k}:V\rightarrow V\otimes(\mathfrak{g}^{*})^{\otimes k}$ in this way. Contract $\displaystyle\Lambda^{\otimes k}$ with $\displaystyle\mathfrak{g}^{\otimes k}$ and get a map $\displaystyle\mathfrak{g}^{\otimes k}\rightarrow\mathrm{Hom}(V,V)$. Adding the $\displaystyle k$-th components for all $\displaystyle k\in\mathbb{N}$, we obtain the desired map $\displaystyle\varphi:T\mathfrak{g}\rightarrow\mathrm{Hom}(V,V)$.

Now we have two maps $\displaystyle T\mathfrak{g}\rightarrow\mathrm{Hom}(V,V)$. One of them is the map $\displaystyle\varphi$, and the other one is $\displaystyle\Theta:T\mathfrak{g}\rightarrow U\mathfrak{g}\rightarrow\mathrm{Hom}(V,V)$. We want to show that they agree. This follows from Lemma 2 below by setting $\displaystyle W_{1}$ to be $\displaystyle V$ and $\displaystyle W_{2}$ to be $\displaystyle\mathfrak{g}^{\otimes k}$.

{Lemma}

Let $\displaystyle W_{1}$ and $\displaystyle W_{2}$ be finite dimensional vector spaces over a field $\displaystyle k$ and $\displaystyle f$ be a map $\displaystyle W_{2}\otimes W_{1}\rightarrow W_{1}$. Rewrite the map as $\displaystyle g:W_{1}\rightarrow W_{2}^{*}\otimes W_{1}$ by the adjunction formula $\displaystyle\mathrm{Hom}(W_{2}\otimes_{k}W_{1},W_{1})=\mathrm{Hom}(W_{1},W_{2}^{*}\otimes_{k}W_{1})$. Fix an element $\displaystyle x\in W_{2}$. Then $\displaystyle f(x\otimes-)$ is a map from $\displaystyle W_{1}$ to $\displaystyle W_{1}$. This map is precisely $\displaystyle g$ followed by the contraction with $\displaystyle x$.

{Proof}

[Proof of Theorem A.] The proof above reduces the commutativity of the Lie algebra diagram to a statement about tensor algebras. The statement about tensor algebras remains valid in the case of derived categories.

One can define a map $\displaystyle\mathrm{Sym}(T_{X}[-1])\rightarrow T(T_{X}[-1])$ given by the formula

$$\displaystyle x_{1}\wedge\cdots\wedge x_{n}\rightarrow\frac{1}{n!}\sum_{\sigma\in S_{n}}(-1)^{sgn(\sigma)}x_{\sigma(1)}\cdots x_{\sigma(n)},$$

where $\displaystyle T(T_{X}[-1])$ is the tensor algebra on $\displaystyle T_{X}[-1]$.

The map above is a differential graded version of the map $\displaystyle\psi$ in (2). Let $\displaystyle X^{(1)}$ be the first order neighborhood of $\displaystyle X$ in $\displaystyle X\times X$. There are embeddings $\displaystyle i:X\hookrightarrow X^{(1)}$ and $\displaystyle j:X^{(1)}\hookrightarrow X\times X$. Arinkin and Căldăraru [AC12] showed that $\displaystyle T(T_{X}[-1])$ is isomorphic to $\displaystyle(i^{*}i_{*}\mathcal{O}_{X})^{\vee}$, where $\displaystyle(-)^{\vee}$ is the dual. The map

$$\displaystyle(i^{*}i_{*}\mathcal{O}_{X})^{\vee}\rightarrow(i^{*}j^{*}j_{*}i_{*}\mathcal{O}_{X})^{\vee}=(\Delta^{*}\Delta_{*}\mathcal{O}_{X})^{\vee}=\mathcal{RH}om(\Delta_{*}\mathcal{O}_{X},\Delta_{*}\mathcal{O}_{X})$$

is defined by the adjunction $\displaystyle j^{*}\dashv j_{*}$. The composite map

$$\displaystyle\mathrm{Sym}(T_{X}[-1])\rightarrow T(T_{X}[-1])\cong(i^{*}i_{*}\mathcal{O}_{X})^{\vee}\rightarrow(i^{*}j^{*}j_{*}i_{*}\mathcal{O}_{X})^{\vee}$$

$$\displaystyle=(\Delta^{*}\Delta_{*}\mathcal{O}_{X})^{\vee}=\mathcal{RH}om(\Delta_{*}\mathcal{O}_{X},\Delta_{*}\mathcal{O}_{X})$$

is the sheaf version HKR isomorphism as showed in [AC12]. Taking cohomology on both sides of the equality above, we get the HKR isomorphism

$$\displaystyle I^{HKR}:\mathrm{HT}^{*}(X)=\bigoplus_{p+q=*}H^{p}(X,\wedge^{q}T_{X})\rightarrow\mathrm{HH}^{*}(X).$$

Now it is clear that we have a commutative diagram

which is similar to the Lie algebra diagram in (2). Taking cohomology on the diagram above, we get the diagram

that we start with in Theorem A.

enote $\displaystyle I^{hkr}(\tilde{\alpha})$ by $\displaystyle\alpha\in\mathrm{HH}^{*}(X)=\mathrm{Ext}_{X\times X}^{*}(\mathcal{O}_{\Delta},\mathcal{O}_{\Delta})$, where $\displaystyle\mathcal{O}_{\Delta}=\Delta_{*}\mathcal{O}_{X}$. Denote the image of $\displaystyle\alpha$ in $\displaystyle\mathrm{Ext}^{*}(\mathcal{F},\mathcal{F})$ by $\displaystyle\alpha_{\mathcal{F}}$. For any vector bundle $\displaystyle\mathcal{F}$ on $\displaystyle X$, Căldăraru and Willerton [CW10] defined an abstract Chern character $\displaystyle\mathrm{ch}(\mathcal{F})$ which lies in the degree zero part of the Hochschild homology $\displaystyle\mathrm{HH}_{*}(X)=\mathrm{Ext}_{X\times X}^{*}(S^{-1}_{\Delta},\mathcal{O}_{\Delta})$, where $\displaystyle S^{-1}_{\Delta}=\Delta_{*}(\omega_{X}^{\vee}[-\mathrm{dim}X])$. There is an HKR isomorphism for Hochschild homology

$$\displaystyle I_{HKR}:\mathrm{HH}_{*}(X)\rightarrow\mathrm{H\Omega}_{*}(X)=\bigoplus_{q-p=*}H^{p}(X,\wedge^{q}\Omega_{X}).$$

The image of the abstract Chern character under the map $\displaystyle I_{HKR}$ is the usual Chern character of $\displaystyle\mathcal{F}$ [C05]. We need the lemma below.

{Lemma}

If $\displaystyle\alpha_{\mathcal{F}}$ is zero, then $\displaystyle\alpha\circ\mathrm{ch}(\mathcal{F})$ is zero. Here $\displaystyle\circ$ is the composition of morphisms in $\displaystyle\mathbf{D}^{b}(X\times X)$ and $\displaystyle\mathrm{ch}(\mathcal{F})$ is the abstract Chern character.

The two HKR isomorphisms $\displaystyle I_{HKR}$ and $\displaystyle I^{HKR}$ can be twisted by the Todd class. We denote the resulting twisted isomorphisms by $\displaystyle I_{K}$ and $\displaystyle I^{K}$

$$\displaystyle I_{K}:\mathrm{HH}_{*}(X)\rightarrow\mathrm{H\Omega}_{*}(X)=\displaystyle{\bigoplus_{q-p=*}H^{p}(X,\wedge^{q}\Omega_{X})},$$

$$\displaystyle I^{K}:\mathrm{HT}^{*}(X)=\bigoplus_{p+q=*}H^{p}(X,\wedge^{q}T_{X})\rightarrow\mathrm{HH}^{*}(X).$$

They are given by the formula $\displaystyle I_{K}=(-\wedge Td(X)^{\frac{1}{2}})\circ I_{HKR}$ and $\displaystyle I^{K}=I^{HKR}\circ D^{-1}$, where $\displaystyle D^{-1}$ is the inverse of the Duflo operator.

The Mukai vector $\displaystyle v(\mathcal{F})$ of $\displaystyle\mathcal{F}$ is $\displaystyle I_{K}(\mathrm{ch}(\mathcal{F}))$ by definition. There are natural ring structures on $\displaystyle\mathrm{HH}^{*}(X)$ and $\displaystyle\mathrm{HT}^{*}(X)$: the product on $\displaystyle\mathrm{HH}^{*}(X)$ is the Yoneda product, and the product on $\displaystyle\mathrm{HT}^{*}(X)$ is the wedge product. Kontsevich [Kont03] claimed that the map $\displaystyle I^{K}$ is a ring isomorphism. This statement was proved by Calaque and Van den Bergh [CV10]. The Hochschild homology is a module over the Hochschild cohomology and similarly $\displaystyle\mathrm{H\Omega}_{*}(X)$ is a module over $\displaystyle\mathrm{HT}^{*}(X)$. Calaque, Rossi, and Van den Bergh [CRV12] proved that the maps $\displaystyle I_{K}$ and $\displaystyle I^{K}$ respect the module structures. {Proof}[Proof of Theorem B.] The commutative diagram in Theorem A shows that

$$\displaystyle\tilde{\alpha}\lrcorner\exp(at_{\mathcal{F}})=\alpha_{\mathcal{F}},$$

which is zero under the assumption of Theorem B. We conclude that $\displaystyle\alpha\circ\mathrm{ch}(\mathcal{F})$ is zero by Lemma 3. Since $\displaystyle I_{K}$ and $\displaystyle I^{K}$ respect the module structures, we have

$$\displaystyle 0=I_{K}(\alpha\circ\mathrm{ch}(\mathcal{F}))=(I^{K})^{-1}(\alpha)\lrcorner\,I_{K}(\mathrm{ch}(\mathcal{F}))=(I^{K})^{-1}(\alpha)\lrcorner\,v(\mathcal{F}).$$

The inverse map of $\displaystyle I^{K}$ is the composite map

As a consequence

$$\displaystyle 0=I_{K}(\alpha\circ\mathrm{ch}(\mathcal{F}))=(I^{K})^{-1}(\alpha)\lrcorner\,v(\mathcal{F})=D(\tilde{\alpha})\lrcorner\,v(\mathcal{F}).$$

The result in (LABEL:1.1) says that $\displaystyle\tilde{\alpha}\lrcorner\,v(\mathcal{F})$ is zero if $\displaystyle\tilde{\alpha}\lrcorner\exp(at_{\mathcal{F}})$ is zero for any $\displaystyle\tilde{\alpha}\in H^{1}(X,T_{X})$. We end this paper by proving that Theorem B implies the result in (LABEL:1.1).

From now on let $\displaystyle\tilde{\alpha}$ be an element in $\displaystyle H^{1}(X,T_{X})$. The only term in $\displaystyle\exp(at_{\mathcal{F}})=1+at_{\mathcal{F}}+\frac{(at_{\mathcal{F}})^{2}}{2!}+\cdots$ that can contract with $\displaystyle\tilde{\alpha}$ is $\displaystyle at_{\mathcal{F}}$, so $\displaystyle\tilde{\alpha}\lrcorner\exp(at_{\mathcal{F}})=\tilde{\alpha}\lrcorner\,at_{\mathcal{F}}$ in this case.

Choose $\displaystyle\mathcal{F}=O_{X}$. We have $\displaystyle\tilde{\alpha}\lrcorner\exp(at_{O_{X}})=0$. Therefore

$$\displaystyle D(\tilde{\alpha})\lrcorner\,v(O_{X})=(Td(X)^{\frac{1}{2}}\lrcorner\,\tilde{\alpha})\lrcorner\,Td(X)^{\frac{1}{2}}=0$$

according to Theorem B.

Expand the Todd class $\displaystyle Td(X)$ as $\displaystyle 1+\frac{c_{1}}{2}+\frac{c_{1}^{2}+c_{2}}{12}+\cdots$, and note that the only term of $\displaystyle(Td(X)^{\frac{1}{2}}\lrcorner\,\tilde{\alpha})\lrcorner\,Td(X)^{\frac{1}{2}}$ in $\displaystyle H^{2}(X,O_{X})$ is $\displaystyle\tilde{\alpha}\lrcorner\,\frac{c_{1}}{2}$. Since $\displaystyle(Td(X)^{\frac{1}{2}}\lrcorner\,\tilde{\alpha})\lrcorner\,Td(X)^{\frac{1}{2}}=0$, we can conclude that $\displaystyle\tilde{\alpha}\lrcorner\,c_{1}$ is zero for any $\displaystyle\tilde{\alpha}\in H^{1}(X,T_{X})$. The fact that $\displaystyle\tilde{\alpha}\lrcorner\,c_{1}=0$ for $\displaystyle\tilde{\alpha}\in H^{1}(X,T_{X})$ is also known due to Griffiths. Consider the first order deformation of $\displaystyle X$ corresponding to $\displaystyle\tilde{\alpha}$. The vanishing of $\displaystyle\tilde{\alpha}\lrcorner\,c_{1}$ is equivalent to the class $\displaystyle c_{1}$ remaining of type $\displaystyle(p,p)$.

The term $\displaystyle\tilde{\alpha}\lrcorner\,\frac{c_{1}}{4}$ is exactly the difference between $\displaystyle D(\tilde{\alpha})$ and $\displaystyle\tilde{\alpha}$ because

$$\displaystyle D(\tilde{\alpha})=Td(X)^{\frac{1}{2}}\lrcorner\,\tilde{\alpha}=(1+\frac{c_{1}}{4}+\cdots)\lrcorner\,\tilde{\alpha}=\tilde{\alpha}+\frac{c_{1}}{4}\lrcorner\,\tilde{\alpha}+0.$$

We conclude that $\displaystyle\tilde{\alpha}\lrcorner\,v(\mathcal{F})$ is zero if and only if $\displaystyle D(\tilde{\alpha})\lrcorner\,v(\mathcal{F})$ is zero for $\displaystyle\tilde{\alpha}\in H^{1}(X,T_{X})$.