Serre Duality for the Cohomology of Landau-Ginzburg models

TL;DR

This paper extends Serre duality to the cohomology of Landau-Ginzburg models, specifically focusing on the Koszul complex associated with holomorphic bundles and sections over complex manifolds.

Contribution

It proves a generalized Serre duality theorem for the cohomology of Koszul complexes in Landau-Ginzburg models, advancing the understanding of their duality properties.

Findings

Established a duality theorem for Koszul complex cohomology

Extended Serre duality to Landau-Ginzburg model context

Provided mathematical framework for future research in complex geometry

Abstract

Let V and F be holomorphic bundles over a complex manifold M, and s be a holomorphic section of V. We study the cohomology associated to the Koszul complex induced by s, and prove a generalized Serre duality theorem for them.

Equations

$$0\to {\wedge }^n{V}^{\vee }\stackrel{{\iota }_s}{⟶}\cdots \stackrel{{\iota }_s}{⟶}{\wedge }^2{V}^{\vee }\stackrel{{\iota }_s}{⟶}{V}^{\vee }\stackrel{{\iota }_s}{⟶}\mathbb{C}\to 0,$$

$$0\to {\wedge }^n{V}^{\vee }\otimes F\stackrel{{\iota }_s\otimes 1}{⟶}\cdots \stackrel{{\iota }_s\otimes 1}{⟶}{\wedge }^2{V}^{\vee }\otimes F\stackrel{{\iota }_s\otimes 1}{⟶}{V}^{\vee }\otimes F\stackrel{{\iota }_s\otimes 1}{⟶}F\to 0.$$

$$(-,-{)}_{\psi }:{\mathbb{H}}^{\bullet }(M;V,F)\times {\mathbb{H}}^{\bullet }(M;V,F^{\vee })\to \mathbb{C},$$

$${\mathbb{H}}^k(M;V,F)\cong {\mathbb{H}}^{-k}(M;V,F^{\vee }{)}^{\vee }.$$

$$H^{p,q}(M,F)\cong H^{n-p,n-q}(M,F^{\vee }{)}^{\vee }.$$

$${\mathbb{H}}^0(M;V,F)\cong \Gamma (M,{\mathcal{O}}_M)/(f_1,\cdots ,f_n);{\mathbb{H}}^k(M;V,F)=0,k\ne 0.$$

$$(g,h{)}_{\psi }$$

$${\mathrm{res}}_s:\Gamma (M,{\mathcal{O}}_M)/(f_1,\cdots ,f_n)\times \Gamma (M,{\mathcal{O}}_M)/(f_1,\cdots ,f_n)\to \mathbb{C}$$

$${\Omega }_c^{(i,j)}({\wedge }^l{V}^{\vee }\otimes F):=\{\alpha |\alpha \in \Gamma (M,{\mathcal{A}}^{i,j}({\wedge }^l{V}^{\vee }\otimes F))\text{with compact support}\}.$$

$$\mathbf{B}:={\oplus }_{i,j,l}{\Omega }^{(i,j)}({\wedge }^l{V}^{\vee }\otimes F)$$

$${\mathcal{C}}_M^k={\oplus }_{i-j=k}{C}_M^{i,j}\subset \mathbf{B}\text{mbox}with{C}_M^{i,j}:={\Omega }^{(0,i)}({\wedge }^j{V}^{\vee }\otimes F)=\Gamma (M,{\mathcal{A}}^{0,i}({\wedge }^j{V}^{\vee }\otimes F)),$$

$${\mathcal{C}}_{c,M}^k={\oplus }_{i-j=k}{C}_{c,M}^{i,j}\text{mbox}with{C}_{c,M}^{i,j}:=\{\alpha \in C_M^{i,j}|\alpha \text{has compact support}\}.$$

$$\overline{\partial }:C_M^{i,j}⟶C_M^{i+1,j}{\iota }_s:C_M^{i,j}⟶C_M^{i,j-1}$$

$${\overline{\partial }}_s:=\overline{\partial }+{\iota }_s.$$

$${\mathbb{H}}^k(M;V,F)=H^k({\mathcal{C}}_M^{\bullet }),$$

$${\mathbb{H}}_c^k(M;V,F)=H^k({\mathcal{C}}_{c,M}^{\bullet }).$$

$$\bar{s}:=\frac{(\ast ,s{)}_{h_V}}{(s,s{)}_{h_V}}\in \Gamma (U,{\mathcal{A}}^{0,0}(V^{\vee }\otimes F)).$$

$$\bar{s}\wedge :\Gamma (U,{\mathcal{A}}^{0,i}({\wedge }^j{V}^{\vee }\otimes F))\to \Gamma (U,{\mathcal{A}}^{0,i}({\wedge }^{j+1}{V}^{\vee }\otimes F)).$$

$$T_{\rho }:{\mathcal{C}}_M\to {\mathcal{C}}_{c,M}{T}_{\rho }(\alpha ):=\rho \alpha +(\overline{\partial }\rho ){\mathcal{T}}_s\frac{1}{1+[\overline{\partial },{\mathcal{T}}_s]}(j^{\ast }\alpha )$$

$$R_{\rho }:{\mathcal{C}}_M\to {\mathcal{C}}_M{R}_{\rho }(\alpha ):=(1-\rho ){\mathcal{T}}_s\frac{1}{1+[\overline{\partial },{\mathcal{T}}_s]}(j^{\ast }\alpha ).$$

$$\frac{1}{1+[\overline{\partial },{\mathcal{T}}_s]}:=\sum _{k=0}^{\infty }(-1{)}^k[\overline{\partial },{\mathcal{T}}_s{]}^k$$

$$[A,B]=AB-(-1{)}^{|A||B|}BA.$$

$$[{\iota }_s,{\mathcal{T}}_s]=1\text{on}{\mathcal{C}}_U.$$

$$[P,[\overline{\partial },{\mathcal{T}}_s]]=0$$

$$[{\overline{\partial }}_s,R_{\rho }]$$

$${\mathbb{H}}_c^k(M;V,F)\cong {\mathbb{H}}^k(M;V,F).$$

$$\mathfrak{B}:={\oplus }_{i,j,k,l}{\Omega }^{(i,j)}({\wedge }^{k}V\otimes {\wedge }^l{V}^{\vee })$$

$$\overline{\partial }⟨\alpha ,\beta ⟩=⟨\overline{\partial }\alpha ,\beta ⟩+(-1{)}^{♯\alpha }⟨\alpha ,\overline{\partial }\beta ⟩.$$

$$u┘:{\Omega }^{(p,q)}({\wedge }^l{V}^{\vee })⟶{\Omega }^{(p+i,q+j)}({\wedge }^{k-l}V)$$

$$⟨u┘\theta ,{\nu }^{\ast }⟩=(-1{)}^{(i+j)l+(p+q)♯u+\frac{l(l-1)}{2}}⟨u,\theta \wedge {\nu }^{\ast }⟩,\forall {\nu }^{\ast }\in A^0({\wedge }^{k-l}{V}^{\vee }).$$

Full text

1. Introduction

The Serre duality theorem is a fundamental result in complex manifold, which establishes a duality between the cohomology of a complex manifold and the cohomology of with compact supports, provided that the $\overline{\partial}$ operator has closed range in appropriate degrees. In this paper we extend the Serre duality to the cohomology of Landau-Ginzburg models.

Let $V$ be a holomorphic bundle over a complex manifold (usually noncompact) $M$ with $\mathrm{rank}\,V=\dim M=n$, and $s$ be a holomorphic section of $V$ with compact zero loci $Z:=(s^{-1}(0))$. Let $V^{\vee}$ be the dual bundle of $V$, then $s$ induced the following Koszul complex

[TABLE]

where $\iota_{s}$ is the contraction operator induced by $s$.

Let $F$ be another holomorphic bundle over $M$, we have the following complex from (1.1)

[TABLE]

Denote by $\mathbb{H}^{\bullet}(M;V,F)$ the hypercohomology associated to the above complex. We will, somewhat abusively, write $\iota_{s}\otimes 1$ as $\iota_{s}$. Because (1.2) is exact outside the compact set $Z$, the cohomology $\mathbb{H}^{\bullet}(M;V,F)$ is finite dimensional over $\mathbb{C}$. The study of this type cohomology origins to the mathematical interpretation of the Landau-Ginzburg models, which had been widely studied in the following papers [1, 2, 3, 5, 7, 8, 9].

Let $\psi\in\Gamma(M,\det V\otimes\det\Omega_{M})$ be a holomorphic section, where $\Omega_{M}$ is the holomorphic cotangent bundle of $M$. There is a canonical pairing

[TABLE]

see (3.24). Then we have the following duality theorem.

Theorem 1.1**.**

Let $V,~{}F$ be holomorphic bundles over the complex manifold $M$ with $\mathrm{rank}\,V=\dim M=n$, and $s$ be a holomorphic section of $V$ with compact zero loci $Z=s^{-1}(0)$. Assume that $\psi\in\Gamma(M,\det V\otimes\det\Omega_{M})$ is nowhere vanishing. Then the above pairing $(-,-)_{\psi}$ is non-degenerate. Thus for $-n\leq k\leq n$,

[TABLE]

This is a generalization of the non-degenerate theorem of [5, Theorem A] and [10, Theorem 1.2].

Let $V=T_{M}$ be the holomorphic tangent bundle of the compact complex manifold $M$, and $F$ be a holomorphic bundle over $M$. Let $s$ be the zero section of $T_{M}$ and $\psi=c\in\Gamma(M,{\cal O}_{M})\cong\Gamma(M,\det T_{M}\otimes\det\Omega_{M})$ be a nonzero constant, then we recover the classical Serre duality theorem.

Corollary 1.2**.**

Let $F$ be a holomorphic bundle over a compact complex manifold $M$, then

[TABLE]

Let $V=\Omega_{M}$ be the holomorphic cotangent bundle of a small open ball $M=\{z\in\mathbb{C}^{n}||z|<\epsilon\}$, and $F={\cal O}_{M}$. Let $s=df$ be a holomorphic section of $V$, where $f$ is a holomorphic function on $M$. Let $\psi=dz_{1}\wedge\cdots\wedge dz_{n}\otimes e_{1}\wedge\cdots\wedge e_{n}$, where $\{z_{i}\}$ is the coordinate of $\mathbb{C}^{n}$ and $\{e_{i}\}$ is the holomorphic frame of $\Omega_{M}$. Assume that $s=df=f_{1}e_{1}+\cdots+f_{n}e_{n}$ and $Z=s^{-1}(0)=0$, then using Theorem 1.1 and the proof of [10, Theorem 1.3] we have

[TABLE]

For $g,h\in\Gamma(M,{\cal O}_{M})$, let $\psi^{\prime}=gh\psi$. By (3), we have

[TABLE]

where $\operatorname{Res}\frac{\psi^{\prime}}{s}$ is the virtual residue associated to $\psi^{\prime}$ and $s$, the symbol $\lfloor\frac{n+3}{2}\rfloor$ is the greatest integer less than or equal to $\frac{n+3}{2}$. The virtual residue, which had been constructed by Chang and the author in [3], coincides with the Grothendieck residue up to a sign. Therefore $\operatorname{Res}\frac{\psi^{\prime}}{s}$ equals to the Grothendieck residue $\mathrm{res}_{s}(g,h)=\int_{|f_{i}|=\epsilon_{i}}\frac{ghdz_{1}\wedge\cdots\wedge dz_{n}}{f_{1}\cdots f_{n}}$ up to a sign, see formula (4.6). Thus we recover the following local duality theorem, see [6, Page 659].

Corollary 1.3**.**

Let $V=\Omega_{M}$ be the holomorphic cotangent bundle of a small open ball $M=\{z\in\mathbb{C}^{n}||z|<\epsilon\}$, and $F={\cal O}_{M}$. Let $s=df$ be a holomorphic section of $V$, where $f$ is a holomorphic function on $M$. Let $\psi=dz_{1}\wedge\cdots\wedge dz_{n}\otimes e_{1}\wedge\cdots\wedge e_{n}$, where $\{z_{i}\}$ is the coordinate of $\mathbb{C}^{n}$ and $\{e_{i}\}$ is the holomorphic frame of $\Omega_{M}$. Assume that $s=df=f_{1}e_{1}+\cdots+f_{n}e_{n}$ and $Z=s^{-1}(0)=0$. Then

[TABLE]

is non-degenerate.

Acknowledgment: The author thanks C. I. Lazaroiu and Wanmin Liu for inspired discussion. He also thanks the IBS Center for Geometry and Physics for hospitality during his visit on May, 2018. This work was supported by the Start-up Fund of Hunan University.

2. Cohomology with compact support

In this section, we study different types of hypercohomology associate to the exact sequence (1.2). It is similar to Section 2 in [10]. As before let $V,~{}F$ be holomorphic bundles over a complex manifold $M$ with $\text{rank}\,V=\dim M=n$, and $s$ is a holomorphic section of $V$ with compact zero loci $Z=s^{-1}(0)$. Let $V^{\vee}$ be the dual bundle of $V$.

Let ${\cal A}^{i,j}(\wedge^{l}V^{\vee}\otimes F)$ be the sheaf of smooth $(i,j)$ forms on $M$ with value in $\wedge^{l}V^{\vee}\otimes F$. Let $\Omega^{(i,j)}(\wedge^{l}V^{\vee}\otimes F):=\Gamma(M,{\cal A}^{i,j}(\wedge^{l}V^{\vee}\otimes F))$ and assign its element $\alpha$ to have degree $\sharp\alpha=i+j-l$. Let

[TABLE]

Then

[TABLE]

is a graded commutative algebra with the (wedge) product uniquely extending wedge products in $\Omega^{\bullet},\wedge^{\bullet}V^{\vee}$ and mutual tensor products. Denote

[TABLE]

and

[TABLE]

Let ${\cal C}_{M}:=\oplus_{k}{\cal C}^{k}_{M}$ and ${\cal C}_{c,M}:=\oplus_{k}{\cal C}^{k}_{c,M}$. For $\alpha\in{\cal C}_{M}$ we denote $\alpha_{i,j}$ to be its component in $C^{i,j}_{M}$. Clearly, ${\cal C}_{M}$ is a bi-graded $C^{\infty}(M)$-module. Under the operations

[TABLE]

the space $C^{\bullet,\bullet}_{M}$ becomes a double complex and $C^{\bullet,\bullet}_{c,M}$ is a subcomplex. We shall study the cohomology of ${\cal C}^{\bullet}_{M}$ and ${\cal C}^{\bullet}_{c,M}$ with respect to the following coboundary operator

[TABLE]

One checks $\overline{\partial}_{s}^{2}=0$ using Leibniz rule of $\overline{\partial}$ and $\overline{\partial}s=0$. Denote by

[TABLE]

and

[TABLE]

Fix a Hermitian metric $h_{V}$ on $V$. For a nonzero $s$ on $U:=M\setminus Z$, we can form the following smooth section

[TABLE]

It associates a map

[TABLE]

To distinguish it in later calculation, we denote ${\cal T}_{s}:=\bar{s}\wedge:C^{\bullet,\bullet}_{U}\longrightarrow C^{\bullet,\bullet+1}_{U}$ , where $C^{\bullet,\bullet}_{U}:=\Gamma(U,{\cal A}^{0,\bullet}(\wedge^{\bullet}V^{\vee}\otimes F))$.

The injection $j:U\to M$ induces the restriction $j^{*}:C^{\bullet,\bullet}_{M}\to C^{\bullet,\bullet}_{U}$. Let $\rho$ be a smooth cut-off function on $M$ such that $\rho|_{U_{1}}\equiv 1$ and $\rho|_{M\setminus U_{2}}\equiv 0$ for some relatively compact open neighborhoods $U_{1}\subset\overline{U}_{1}\subset U_{2}$ of $Z$ in $M$.

We define the degree of an operator to be its change on the total degree of elements in ${\cal C}_{M}({\cal C}_{U})$. Then $\overline{\partial}$ and ${\cal T}_{s}$ are of degree $1$ and $-1$ respectively, and $[\overline{\partial},{\cal T}_{s}]=\overline{\partial}{\cal T}_{s}+{\cal T}_{s}\overline{\partial}$ is of degree [math]. Consider two operators introduced in [3, (3.1), (3.2)] or [12, page 11]

[TABLE]

and

[TABLE]

Here as an operator

[TABLE]

is well-defined since $[\overline{\partial},{\cal T}_{s}]^{k}(\alpha)=0$ whenever $k>n$. Clearly $T_{\rho}$ is of degree zero and $R_{\rho}$ is of degree by $-1$. Also $R_{\rho}({\cal C}_{c,M})\subset{\cal C}_{c,M}$ by definition.

Lemma 2.1**.**

$[\overline{\partial}_{s},R_{\rho}]=1-T_{\rho}$ * as operators on ${\cal C}_{M}$.*

Proof.

It is direct to check that 111 As a notation convention, we always denote $[,]$ for the graded commutator, that is for operators $A,B$ of degree $|A|$ and $|B|$, the bracket is given by

$[A,B]=AB-(-1)^{|A||B|}BA.$

[TABLE]

Moreover,

[TABLE]

for $P$ being $\iota_{s},\overline{\partial}$ or ${\cal T}_{s}$. Therefore, we have

[TABLE]

∎

Proposition 2.2**.**

The embedding $({\cal C}_{c,M},\overline{\partial}_{s})\rightarrow({\cal C}_{M},\overline{\partial}_{s})$ is a quasi-isomorphism. Thus for $-n\leq k\leq n$,

[TABLE]

Proof.

By Lemma 2.1 $\operatorname{H}^{\ast}({\cal C}_{M}/{\cal C}_{c,M},\overline{\partial}_{s})\equiv 0,$ and thus the proposition follows. ∎

3. Non-dengeneracy

First we recall the definition of operators on wedge products of vector bundles over $M$, see [3, Appendix]. Let $\Omega^{(i,j)}(\wedge^{k}V\otimes\wedge^{l}V^{\vee}):=\Gamma(M,{\cal A}^{i,j}(\wedge^{k}V\otimes\wedge^{l}V^{\vee}))$ be the smooth differential forms valued in $\wedge^{k}V\otimes\wedge^{l}V^{\vee}$. Let

[TABLE]

be a graded commutative algebra extending the wedge products of $\Omega^{\bullet},\wedge^{\bullet}V$ and $\wedge^{\bullet}V^{\vee}$. The degree of $\alpha\in\Omega^{(i,j)}(\wedge^{k}V\otimes\wedge^{l}V^{\vee})$ is $\sharp\alpha:=i+j+k-l$. We briefly denote $A^{0}(\wedge^{k}V\otimes\wedge^{l}V^{\vee})=\Omega^{(0,0)}(\wedge^{k}V\otimes\wedge^{l}V^{\vee})$.

Set $\kappa:\mathfrak{B}\to\Omega^{\bullet}$ which sends $\omega(e\otimes e^{\prime})$(for $\omega\in\Omega^{(i,j)},e\in\wedge^{k}V,e^{\prime}\in\wedge^{l}V^{\vee})$ to $\omega\langle e,e^{\prime}\rangle$, where $\langle\cdot,\cdot\rangle$ is the dual pairing between $\wedge^{k}V$ and $\wedge^{k}V^{\vee}$, and $\langle e,e^{\prime}\rangle=0$ when $k\neq l$. We further extend the pairing by setting $\langle\alpha,\beta\rangle:=\kappa(\alpha\beta)$ for $\alpha,\beta\in\mathfrak{B}$. It is direct to verify

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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

[TABLE]

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 $\gamma\in A^{0}(V^{\vee})$, we define

[TABLE]

where for $\nu\in\Omega^{(i,j)}(\wedge^{k}V)$, the $\iota_{\gamma}(\nu)$ is determined by

[TABLE]

We have the following identities.

Lemma 3.1** ([3]).**

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

[TABLE]

Lemma 3.2** ([3]).**

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

[TABLE]

Denote by ${\mathscr{D}}^{p,q}(\wedge^{l}V\otimes F^{\vee})$ the space of $\wedge^{l}V\otimes F^{\vee}$-valued $(p,q)$-current, which is the dual of the space $\Omega^{n-p,n-q}_{c}(\wedge^{l}V^{\vee}\otimes F)$. There is a naturally pairing

[TABLE]

where

[TABLE]

Denote

[TABLE]

The coboundary map $\delta_{s}$ of ${\cal D}^{\bullet}_{M}$ is defined as follows

[TABLE]

By (3.1) and (3.4),

[TABLE]

Thus $({\cal D}^{\bullet}_{M},(-1)^{\bullet+1}\delta_{s})$ is the dual complex of $({\cal C}_{c,M}^{\bullet},\overline{\partial}_{s})$. Let

[TABLE]

Because the complex $({\cal D}^{\bullet}_{M},(-1)^{\bullet+1}\delta_{s})$ is quasi-isomorphic to the complex $({\cal D}^{\bullet}_{M},\delta_{s})$, the pairing (3.7) induces a pairing on the hypercohomologies which we denote by

[TABLE]

Theorem 3.3**.**

Let $V,~{}F$ be holomorphic bundles over the complex manifold $M$ with $\mathrm{rank}\,V=\dim M=n$ and $s$ be a holomorphic section of $V$ with compact zero loci $Z=s^{-1}(0)$. Then the above pairing (3.11) is non-degenerate. Thus for $-n\leq k\leq n$

[TABLE]

Proof.

Because $\mathbb{H}_{c}^{k}(M;V,F)$ is finite dimensional and the complex $({\cal D}^{\bullet}_{M},(-1)^{\bullet+1}\delta_{s})$ is the dual complex of $({\cal C}_{c,M}^{\bullet},\overline{\partial}_{s})$, by applying [11, Theorem 1.6] and [11, Corollary 1.7], we obtain the theorem. ∎

Remark 3.4**.**

By the Dolbeault-Grothendieck Lemma [4, 3.29] for current, $\mathcal{H}^{\bullet}(M;V,F^{\vee})$ (up to a shift on degree) is the hypercohomology of the following complex,

[TABLE]

which is quasi-isomorphic to the complex

[TABLE]

Let $(\overline{{\cal D}}^{\bullet}_{M},\widetilde{\delta}_{s})$ be the complex with

[TABLE]

and the coboundary map $\widetilde{\delta}_{s}$ is defined as follows

[TABLE]

By the Dolbeault-Grothendieck Lemma [4, 3.29] for current, the complex $(\overline{{\cal D}}^{\bullet}_{M},\widetilde{\delta}_{s})$ is the Dolbeault resolution of the following complex

[TABLE]

Denote by

[TABLE]

the cohomology of the complex $(\overline{{\cal D}}^{\bullet}_{M},\widetilde{\delta}_{s})$.

Let $\psi\in\Gamma(M,\det V\otimes\det\Omega_{M})$ be a holomorphic section which is nowhere vanishing. It induces a bundle isomorphism

[TABLE]

Thus

[TABLE]

By Lemma 3.1, Lemma 3.2 and (3.8), the map

[TABLE]

is a complex isomorphism. Therefore for $-n\leq k\leq n$

[TABLE]

The complex (3.15) is quasi-isomorphic to

[TABLE]

and we denote this quasi-isomorphism by $\Phi$. It induces the following hypercohomology isomorphism

[TABLE]

Thus we have

[TABLE]

Combining (3.11) and (3.21), there is a pairing

[TABLE]

where $(\alpha,\beta)^{\prime}_{\psi}:=(\psi\lrcorner(H^{\bullet}(\Phi)(\alpha)),\beta)_{N}$.

From (3.21) and Theorem 3.3, we obtain the following statement.

Corollary 3.5**.**

Let $V,~{}F$ be holomorphic bundles over the complex manifold $M$ with $\mathrm{rank}\,V=\dim M=n$ and $s$ be a holomorphic section of $V$ with compact zero loci $Z=s^{-1}(0)$. Assume that $\psi\in\Gamma(M,\det V\otimes\det\Omega_{M})$ is a holomorphic section which is nowhere vanishing. Then the pairing $(-,-)^{\prime}_{\psi}$ is non-degenerate. Thus for $-n\leq k\leq n$

[TABLE]

We define the following pairing

[TABLE]

such that $(\alpha,\beta)_{\psi}:=([\alpha],[\beta])^{\prime}_{\psi}$, where $[\alpha]$ and $[\beta]$ are the image of $\alpha,\beta$ under the isomorphic $\mathbb{H}^{\bullet}(M;V,F)\cong\mathbb{H}_{c}^{\bullet}(M;V,F)$ and $\mathbb{H}^{\bullet}(M;V,F^{\vee})\cong\mathbb{H}^{\bullet}_{c}(M;V,F^{\vee})$. The pairing (3.24) is well defined because it does not dependent on the compact support representation. Therefore

Corollary 3.6**.**

Under the conditions as in Corollary 3.5, the pairing $(-,-)_{\psi}$ is non-degenerate. Thus for $-n\leq k\leq n$

[TABLE]

Let $V=\Omega_{M}$ be the holomorphic cotangent bundle of a small open ball $M=\{z\in\mathbb{C}^{n}||z|<\epsilon\}$, and $F={\cal O}_{M}$. Let $s=df$ be a holomorphic section of $V$, where $f$ is a holomorphic function on $M$. Let $\psi=dz_{1}\wedge\cdots\wedge dz_{n}\otimes e_{1}\wedge\cdots\wedge e_{n}$, where $\{z_{i}\}$ is the coordinate of $\mathbb{C}^{n}$ and $\{e_{i}\}$ is the holomorphic frame of $\Omega_{M}$. Assume that $s=df=f_{1}e_{1}+\cdots+f_{n}e_{n}$ and $Z=s^{-1}(0)=0$, then using Corollary 3.6 and the proof of [10, Theorem 1.3]

[TABLE]

On the other hand $s$ induced the following complex

[TABLE]

As in the definition of (3.10), let $\mathcal{H}^{\bullet}(M;V,{\cal O}_{M})$ be the hypercohomology of (3.26), and $\mathcal{H}_{c}^{\bullet}(M;V,{\cal O}_{M})$ be its hypercohomology with compact support. By [3, Proposition 3.2]

[TABLE]

for $-n\leq k\leq n$.

Because $\psi=dz_{1}\wedge\cdots\wedge dz_{n}\otimes e_{1}\wedge\cdots\wedge e_{n}$ is nowhere vanishing. It induces a bundle isomorphism

[TABLE]

So is the Dolbeault resolution of the two complex (1.1) and (3.26).

For $g,~{}h\in\Gamma(M,{\cal O}_{M})$, let $[g],~{}[h]$ be the images of $g,~{}h$ under the isomorphism $\mathbb{H}^{0}(M;V,F)\cong\mathbb{H}_{c}^{0}(M;V,F)$. By (3.27), (3.22) and (3.24) the pairing

[TABLE]

Let $\rho$ be a smooth cut-off function on $M$ such that $\rho|_{U_{1}}\equiv 1$ and $\rho|_{M\setminus U_{2}}\equiv 0$ for some relatively compact open neighborhoods $U_{1}\subset\overline{U}_{1}\subset U_{2}$ of [math] in $M$. By Lemma 2.1, we have

[TABLE]

Therefore

[TABLE]

For the nonzero $s$ on $U:=M\setminus Z$, the smooth section $\bar{s}:=\frac{(*,s)_{h_{V}}}{(s,s)_{h_{V}}}\in\Gamma(U,{\cal A}^{0,0}(V^{\vee}))$ induces a contraction

[TABLE]

Denote

[TABLE]

and

[TABLE]

Let $\widetilde{{\cal C}}_{M}:=\oplus_{k}\widetilde{{\cal C}}^{k}_{M}$ and $\widetilde{{\cal C}}_{c,M}:=\oplus_{k}\widetilde{{\cal C}}^{k}_{c,M}$. Let $j:U\to M$ be the injection. We can form the following operator by using the contraction $\iota_{\bar{s}}$

[TABLE]

Because $\psi$ is holomorphic, by Lemma 3.1

[TABLE]

Denote $\psi^{\prime}=gh\psi$, and applying [3, Proposition 3.3] to $\psi^{\prime}$ and $s=df$, we have

[TABLE]

where $\operatorname{Res}\frac{\psi^{\prime}}{s}$ is the virtual residue associated to $\psi^{\prime}$ and $s$, it coincides with the Grothendieck residue $\mathrm{res}_{s}(g,h)=\int_{|f_{i}|=\epsilon_{i}}\frac{ghdz_{1}\wedge\cdots\wedge dz_{n}}{f_{1}\cdots f_{n}}$ up to a sign, see formula (4.6). Thus we recover the local duality theorem, see [6, Page 659].

Corollary 3.7**.**

Let $V=\Omega_{M}$ be the holomorphic cotangent bundle of a small open ball $M=\{z\in\mathbb{C}^{n}||z|<\epsilon\}$, and $F={\cal O}_{M}$. Let $s=df$ be a holomorphic section of $V$, where $f$ is a holomorphic function on $M$. Let $\psi=dz_{1}\wedge\cdots\wedge dz_{n}\otimes e_{1}\wedge\cdots\wedge e_{n}$, where $\{z_{i}\}$ is the coordinate of $\mathbb{C}^{n}$ and $\{e_{i}\}$ is the holomorphic frame of $\Omega_{M}$. Assume that $s=df=f_{1}e_{1}+\cdots+f_{n}e_{n}$ and $Z=s^{-1}(0)=0$. Then

[TABLE]

is non-degenerate.

4. Appendix

In this appendix we recall the construction of the virtual residue given by Chang and the author in [3], and prove the relation between the virtual residue and the Grothendieck residue when the zero loci is zero-dimensional.

Let $V$ be a holomorphic bundle over a compact complex manifold $M$ with $\operatorname{rank}V=\dim M=n$. Let $s$ be a holomorphic section of $V$, and $Z=s^{-1}(0)$ be the compact zero loci.

Let $U:=M\setminus Z$, and let $V_{U}$ be the restriction of $V$ over $U$. Since $s$ is nowhere zero over $U$, the following Koszul sequence is exact over $U$

[TABLE]

The exact Koszul sequence induces a homomorphism

[TABLE]

One also has a canonical Dolbeault isomorphism

[TABLE]

Applying (4.1) and (4.2) to the holomorphic section $\psi\in\Gamma(M,K_{M}\otimes\det{V})$, and using that every $(n,n-1)$ form is $\partial$-closed, one obtains a (unique) De-Rham cohomology class

[TABLE]

Then the virtual residue is defined as

[TABLE]

where $N$ is a real $(2n-1)$-dimensional piecewise smooth compact subset of $M$ that surrounds $Z$, in the sense that $N=\partial T$ for some compact domain $T\subset M$, which contains $Z$ and is homotopically equivalent to $Z$.

When $M=\{z\in\mathbb{C}^{n}||z|<\epsilon\}$ is a small open ball, and $V=\Omega_{M}$ with the standard Hermitian metric $h_{V}$. Let $F={\cal O}_{M}$ and $s=df$, where $f$ is a holomorphic function on $M$. Let $\{z_{i}\}$ be the coordinate of $\mathbb{C}^{n}$, and $\{e_{i}\}$ be the holomorphic frame of $\Omega_{M}$. Assume that $s=df=f_{1}e_{1}+\cdots+f_{n}e_{n}$ and $Z=s^{-1}(0)=0$. Let $\bar{s}=\langle s,s\rangle_{h_{V}}^{-1}\sum_{i=1}^{n}\bar{f}_{i}e_{i}^{*}$, where $e_{i}^{*}$ is the dual basis of $V^{\vee}$. Then we have the following equalities on $U$

[TABLE]

and

[TABLE]

Let $g,~{}h$ be holomorphic functions on $M$. Then $\psi=ghdz_{1}\wedge\cdots\wedge dz_{n}\otimes\ e_{1}\wedge\cdots\wedge e_{n}$ is a holomorphic section of $\Gamma(M,K_{M}\otimes\det V)$. Therefore

[TABLE]

Let $N$ be a small sphere around [math], the virtual residue

[TABLE]

By Lemma in [6, Page 651] and the definition of $\mathrm{res}_{s}(g,h)$ in [6, Page 659], we have

[TABLE]