Virtual Residue and an Integral Formalism

Abstract

We generalize Grothendieck’s residues $Res\frac{\psi }{s}$ to virtual cases, namely cases when the zero loci of the section s has dimension larger than the expected dimension (zero). We also provide an exponential-type integral formalism for the virtual residue, which can be viewed as an analogue of the Mathai–Quillen formalism for localized Euler classes.

Appendix: Operators and Metrics on Exterior Algebra \({\mathbf B}\)

Let V be a rank n holomorphic bundle over a complex manifold M. Recall in section three ${\mathbf B}:=\oplus _{i,j,k,l}\Omega ^{(i,j)}(\wedge ^k V \otimes \wedge ^l V^*)$ is a graded commutative algebra extending the wedge products of $\Omega ^{*}, \wedge ^*V$, and $\wedge ^*V^*$. The degree of $\alpha \in \Omega ^{(i,j)}(\wedge ^k V \otimes \wedge ^l V^*)$ is $\sharp \alpha :=i+j+k-l$. We brief $A^0(\wedge ^k V \otimes \wedge ^l V^*)=\Omega ^{(0,0)}(\wedge ^k V \otimes \wedge ^l V^*)$.

Set $\kappa :{\mathbf B}\rightarrow \Omega ^*$ which sends $\omega (e\otimes e')$(for $\omega \in \Omega ^{(i,j)}, e\in \wedge ^k V, e'\in \wedge ^\ell V^*)$ to $\omega \langle e,e'\rangle$, where $\langle ,\rangle$ is the dual pairing between $\wedge ^k V,\wedge ^k V^*$ and $\langle e,e'\rangle =0$ when $k\ne \ell$. We further extend the pairing by setting $\langle \alpha ,\beta \rangle :=\kappa (\alpha \beta )$ for $\alpha ,\beta \in {\mathbf B}$. It is direct to verify

$$\begin{aligned} \overline{\partial }\langle \alpha ,\beta \rangle =\langle \overline{\partial }\alpha ,\beta \rangle +(-1)^{\sharp \alpha }\langle \alpha ,\overline{\partial }\beta \rangle .\end{aligned}$$

(5.1)

We now define three different types of contraction maps. Given $u\in \Omega ^{(i,j)}(\wedge ^k V)$ and $k\ge \ell$, we define

$$\begin{aligned} u \lrcorner : \Omega ^{(p,q)}(\wedge ^{l}V^*)\longrightarrow \Omega ^{(p+i,q+j)}(\wedge ^{k-l}V) \end{aligned}$$

(5.2)

where for $\theta \in \Omega ^{(p,q)}(\wedge ^{\ell }V^*)$, the $u\lrcorner \theta$ is determined by

$$\begin{aligned} \langle u\lrcorner \theta ,\nu ^*\rangle =(-1)^{(i+j)l+(p+q)\sharp u+\frac{l(l-1)}{2}}\langle u,\theta \wedge \nu ^*\rangle ,\qquad \forall \nu ^*\in A^0(\wedge ^{k-l}V^*). \end{aligned}$$

Given $\alpha \in A^0(V)$, we define

$$\begin{aligned} \iota _{\alpha } : \Omega ^{(i,j)}(\wedge ^{k}V^*) \longrightarrow \Omega ^{(i,j)}(\wedge ^{k-1}V^*) \end{aligned}$$

(5.3)

where for $w\in \Omega ^{(i,j)}(\wedge ^{k}V^*)$, the $\iota _{\alpha }(w)$ is determined by

$$\begin{aligned} \langle \nu ,\iota _{\alpha }(w)\rangle =\langle \alpha \wedge \nu ,w\rangle ,\qquad \forall \nu \in A^0(\wedge ^{k-1}V). \end{aligned}$$

For above $\alpha$, $\theta$, and w one has $\iota _\alpha (w\wedge \theta )=\iota _\alpha (w)\wedge \theta +(-1)^{\sharp w}w\wedge \iota _\alpha (\theta ).$

Given $\gamma \in A^0(V^*)$, also define

$$\begin{aligned} \iota _{\gamma } : \Omega ^{(i,j)}(\wedge ^{k}V) \longrightarrow \Omega ^{(i,j)}(\wedge ^{k-1}V), \end{aligned}$$

(5.4)

for $v\in \Omega ^{(i,j)}(\wedge ^{k}V)$, the $\iota _{\gamma }(v)$ is determined by

$$\begin{aligned} \langle \iota _{\gamma }(v),w\rangle =(-1)^{i+j}\langle v,\gamma \wedge w\rangle , \qquad \forall w\in A^0(\wedge ^{k-1}V^*). \end{aligned}$$

We have the following identities. Because the proofs of the identities are standard, we omit them here.

Lemma 5.1

Given $u\in \Omega ^{(i,j)}(\wedge ^n V)$, and $\theta ,\alpha , \gamma$ as above, one has

$$\begin{aligned} \alpha \wedge (u\lrcorner \theta )=u\lrcorner (\iota _\alpha (\theta ) ), \iota _{\gamma }(u\lrcorner \theta )=u\lrcorner (\gamma \wedge \theta ). \end{aligned}$$

Lemma 5.2

For $u\in \Omega ^{(i,j)} (\wedge ^k V), \theta \in \Omega ^{(p,q)} (\wedge ^\ell V^*)$, $k\ge l$ and smooth form $\alpha \in \Omega ^{(a,b)}(M)$, we have

$$\begin{aligned} \alpha \wedge (u\lrcorner \theta )= u\lrcorner (\alpha \theta ), \overline{\partial }(u\lrcorner \theta )=(-1)^{\sharp \theta }(\overline{\partial }u)\lrcorner \theta + u \lrcorner (\overline{\partial }\theta ). \end{aligned}$$

Now we study some simple metric inequalities on ${\mathbf B}$. Let h be a fixed hermitian metric over V. For arbitrary holomorphic local frame $\{e_i\}$ of V with $\{t^i\}$ its dual frame of $V^*$, one represents $h=\sum h_{i\bar{j}}t^i\otimes \bar{t}^j$. The induced metric $h^*$ on $V^*$ can be written as $h^*=\sum h^{i\bar{j}}e_i\otimes \bar{e}_{j}$, where $\sum h^{i\bar{k}}h_{j\bar{k}}=\delta _i^j$.

As in [22, p. 79 Ex 13], the induced metric $h_{\wedge ^k V}$ on $\wedge ^k V$ is

$$\begin{aligned} (\alpha _1\wedge \cdots \wedge \alpha _k,\beta _1\wedge \cdots \wedge \beta _k)_{h_{\wedge ^k V}}:=\det [h(\alpha _i,\beta _j)]. \end{aligned}$$

Similarly $h^*$ induced metrics $h^*_{\wedge ^l V^*}$ on $\wedge ^l V^*$ and $h_{\wedge ^k V}\otimes h_{\wedge ^l V}^*$ on $\wedge ^k V\otimes \wedge ^l V^*$. The induced metric on ${\mathbf B}=\oplus _{i,j,k,l}\Omega ^{(i,j)}(\wedge ^k V \otimes \wedge ^l V^*)$ would be denoted by $(\cdot ,\cdot )$ and $|\alpha |^2:=(\alpha ,\alpha )$ for $\alpha \in {\mathbf B}$. We have the following inequality.

Lemma 5.3

For $u\in \Omega ^{(n,0)}(\wedge ^k V)$ and $v^*\in \Omega ^{(0,q)}(\wedge ^l V^*)$ with $k\ge l$, one has

$$\begin{aligned} (u\lrcorner v^*,u\lrcorner v^*)\le & {} b(u,u) (v^*,v^*), \end{aligned}$$

where b depends on the ranks of the bundles correspond to $\Omega ^{(0,q)}\otimes \wedge ^l V^*, \wedge ^{k-l}V^*$.