1 Introduction
Let
$G(\text{}\underline{x})\in \mathbb{C}[x]$
be a homogeneous polynomial of degree
$e\geqslant 1$
, where
$\text{}\underline{x}=(x_{0},\ldots ,x_{n})$
. Then
$G(\text{}\underline{x})$
defines a projective hypersurface
$X_{G}$
in
$\mathbb{P}^{n}$
. When
$X_{G}$
is smooth, in [Reference Park and Park10], the third author and his collaborator developed enhanced homotopy theory for the primitive middle-dimensional cohomology and period integrals of
$X_{G}$
by analysing the de Rham cohomology of the hypersurface complement
$U_{G}:=\mathbb{P}^{n}\setminus X_{G}$
. The main goal of this article is twofold. The first goal is to extend the result of [Reference Park and Park10] to a possibly singular projective Calabi–Yau hypersurfaceFootnote
1
. More precisely speaking, we construct a DGBV algebraFootnote
2
${\mathcal{A}}_{U}$
, which is a ring containing a polynomial subring over which it is a finitely generated module and computes the cohomology
$\bigoplus _{k=0}^{n}H_{\text{dR}}^{k}(U_{G}/\mathbb{C})$
of
$U_{G}$
.
The second goal is to compare our DGBV algebra
${\mathcal{A}}_{U}$
with the DGBV algebra
$\mathbf{t}$
for
$M=X_{G}(\mathbb{C})$
constructed in [Reference Barannikov and Kontsevich1] in the context of the homological mirror symmetry conjecture for
$M$
. The construction of the DGBV algebra
$\mathbf{t}$
is complex analytic based on Lie algebras of polyvector fields and
$\mathbf{t}$
computes the total cohomology
$H^{\bullet }(M)=\bigoplus _{k=0}^{2n-2}H^{k}(M,\mathbb{C})$
of
$M$
. If
$\tilde{M}$
is a symplectic manifold which is the mirror dual of
$M$
, then a version of the homological mirror symmetry conjecture asserts that the formal Frobenius manifold structure derived from
$\mathbf{t}$
in [Reference Barannikov and Kontsevich1] coincides with the formal Frobenius manifold structure on
$H_{\bullet }(\tilde{M},\unicode[STIX]{x1D6FD})$
constructed via Gromov–Witten classes of
$\tilde{M}$
.
Now we state two main theorems. Let us assume that
$X_{G}$
is Calabi–Yau, i.e.
$n+1=e$
throughout the article. But we do not assume that
$X_{G}$
is smooth. We introduce an extra variable
$y$
and consider the polynomial ring
$A:=\mathbb{C}[\text{}\underline{x}][y]$
. We also introduce more variables
$\unicode[STIX]{x1D702}_{i}$
corresponding to
$x_{i}$
for
$i=0,\ldots ,n$
. Then we take the symmetric tensor product of
$A$
and the alternating algebra
$\bigwedge (\text{}\underline{\unicode[STIX]{x1D702}}):=\bigwedge (\mathbb{C}\cdot \unicode[STIX]{x1D702}_{1}\oplus \cdots \oplus \mathbb{C}\cdot \unicode[STIX]{x1D702}_{n})$
generated by
$\unicode[STIX]{x1D702}_{1},\ldots ,\unicode[STIX]{x1D702}_{n}$
to define
$$\begin{eqnarray}{\mathcal{A}}_{G}:=A\otimes _{\mathbb{C}}\bigwedge (\text{}\underline{\unicode[STIX]{x1D702}})=\mathbb{C}[y,x_{0},\ldots ,x_{n}][\unicode[STIX]{x1D702}_{0},\ldots ,\unicode[STIX]{x1D702}_{n}].\end{eqnarray}$$
Let us introduce a grading on
${\mathcal{A}}_{G}$
, denoted by
$|\cdot |$
, and two differentials
$Q_{G}$
and
$K_{G}$
in order to make it a cochain complex:
$$\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}\displaystyle |x_{i}|=0,\qquad |y|=0,\qquad |\unicode[STIX]{x1D702}_{i}|=-1,~\forall i=0,\ldots ,n,\\ \displaystyle Q_{G}:=\displaystyle \mathop{\sum }_{i=0}^{n}y\frac{\unicode[STIX]{x2202}G}{\unicode[STIX]{x2202}x_{i}}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}_{i}},\\ \displaystyle K_{G}:=\displaystyle \mathop{\sum }_{i=0}^{n}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{i}}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}_{i}}+\displaystyle \mathop{\sum }_{i=0}^{n}y\frac{\unicode[STIX]{x2202}G}{\unicode[STIX]{x2202}x_{i}}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}_{i}}.\end{array} & & \displaystyle\end{eqnarray}$$
For any linear operator
$K:{\mathcal{A}}\rightarrow {\mathcal{A}}$
of degree 1, define
$$\begin{eqnarray}l_{2}^{K}(a,b):=(-1)^{|a|}(K(a\cdot b)-K(a)\cdot b-(-1)^{|a|}a\cdot K(b)),\quad a,b\in {\mathcal{A}}.\end{eqnarray}$$
Let
${\mathcal{A}}_{U}={\mathcal{A}}_{G,0}$
be the charge-zero subring of
${\mathcal{A}}_{G}$
, where we refer to §2.1 for the definition of the charge. Let
$H_{i}(U_{G},\mathbb{Z})$
be the singular homology group of
$U_{G}$
and
$$\begin{eqnarray}\mathbb{H}^{i}:=H_{\text{dR}}^{i}(U_{G}/\mathbb{C})\end{eqnarray}$$
be the de Rham cohomology group of
$U_{G}$
(with coefficients in
$\mathbb{C}$
) for each
$i=0,1,\ldots ,n$
. With notation
$Q_{U}=Q_{G}$
and
$K_{U}=K_{G}$
, the first main result can be written as follows.
Theorem 1.1. The quadruple
$\mathbf{BV}_{\!U}:=({\mathcal{A}}_{U},Q_{U},K_{U},l_{2}^{K_{U}})$
is a DGBV algebraFootnote
3
containing a polynomial subring over which it is a finitely generated module with the following properties.
(a) We have a decomposition
${\mathcal{A}}_{U}=\bigoplus _{-n-1\leqslant m\leqslant 0}{\mathcal{A}}_{U}^{m}$
and there are canonical isomorphisms for$$\begin{eqnarray}\displaystyle & \displaystyle J:H({\mathcal{A}}_{U},K_{U})=\bigoplus _{p\in \mathbb{Z}}\text{H}^{p}({\mathcal{A}}_{U},K_{U})~\overset{\simeq }{\rightarrow }~\mathbb{H}=\bigoplus _{m\in \mathbb{Z}}\mathbb{H}^{m}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle J^{-p}:\text{H}^{-p}({\mathcal{A}}_{U},K_{U})~\overset{\simeq }{\rightarrow }~\mathbb{H}^{n-p} & \displaystyle \nonumber\end{eqnarray}$$
$0\leqslant p\leqslant n$
.(b) The map
$J^{-p}$
satisfies
$J^{-p}(F^{s}H^{-p}({\mathcal{A}}_{U},K_{U}))\supseteq F^{s}\mathbb{H}^{n-p}$
for any integer
$s$
, where
$F^{s}\mathbb{H}^{n-p}$
is the decreasing Hodge filtrationFootnote
4
on
$\mathbb{H}^{i}$
.
Let
$\mathbf{PH}=\bigoplus _{k\geqslant 0}\mathbf{PH}^{k}$
, where
$\mathbf{PH}^{k}$
is the
$k$
th primitive cohomology of
$X_{G}$
with coefficients in
$\mathbb{C}$
. According to Griffiths’ theorem in [Reference Griffiths8], if
$X_{G}$
is smooth, there is a residue isomorphism
$\text{Res}:\mathbb{H}^{k}\rightarrow \mathbf{PH}^{k-1}$
for every
$k\geqslant 1$
. If
$X_{G}$
is not smooth, the knowledge on (the existence of) the residue morphism (between the hypersurface complement cohomology and the primitive cohomology) and its properties seems to be limited. As far as the authors know, some results are known about the residue morphism when the dimension of the singularity locus
$G^{-1}(0)_{\operatorname{sing}}$
in
$\mathbb{C}^{n+1}$
is 1: see [Reference Dimca6, Corollary (3.11), Ch. 6] for such results.
Remark 1.2. (a) For a comparison with the case of smooth projective hypersurfaces considered in [Reference Park and Park10], see §2.4. The above result is a partial generalization of [Reference Park and Park10], which deals with smooth projective hypersurfaces, to possibly singular projective Calabi–Yau hypersurfaces.
(b) Since
$({\mathcal{A}}_{U},\cdot ,Q_{U})$
is a commutative differential graded algebra (CDGA),
$H_{Q_{G}}^{0}({\mathcal{A}}_{U})_{(0)}$
has an induced ring structure. It is easy to see that the homogeneous coordinate ring of
$X_{G}$
is isomorphic to the weight-zero part of the cohomology ring
$H_{Q_{G}}^{0}({\mathcal{A}}_{U})_{(0)}$
. Since
$({\mathcal{A}}_{U},K_{U})$
can be viewed as a quantization of
$({\mathcal{A}}_{U},Q_{U})$
, the singular cohomology
$\mathbb{H}$
may be regarded as a quantization of the homogenous coordinate ring of
$X_{G}$
. For more details on this physical point of view, see [Reference Park and Park10, §1.4].
Remark 1.3. In the context of the homological mirror symmetry conjecture, there are two known constructions of DGBV algebras on the
$B$
-sideFootnote
5
: one is due to Barannikov and Kontsevich [Reference Barannikov and Kontsevich1] based on the extended formal deformations of compact Calabi–Yau manifolds, whose physical model is the nonlinear sigma model and BCOV (Bershadsky–Cecotti–Ooguri–Vafa) theory. The other is due to Li et al [Reference Li, Li and Saito9, Reference Saito11] based on primitive forms and versal deformations of an isolated singularity, whose physical model is the Landau–Ginzburg model. Our DGBV algebra
$\mathbf{BV}_{\!U}:=({\mathcal{A}}_{U},Q_{U},K_{U},l_{2}^{K_{U}})$
is still on the
$B$
-side but it is based on the gauged linear sigma model [Reference Witten12] and is different from the above two models and, in some sense, “interpolates” between the above two (see [Reference Cox and Katz2, Appendix B.5] for details). The difference with Barannikov and Kontsevich’s construction will be explained in Remark 1.5 after the second main theorem (Theorem 1.4). The difference with [Reference Li, Li and Saito9] lies in the property of the relevant potentials. More specifically, the DGBV algebra which appeared in [Reference Li, Li and Saito9, §2.2] is based on a mixture of Calabi–Yau models and Landau–Ginzburg models, and its underlying algebra is the algebra of functions on a Stein domain
${\mathcal{S}}$
inside
$\mathbb{C}^{N}$
and a potential function
$f:{\mathcal{S}}\rightarrow \mathbb{C}$
whose critical locus is finite. But the critical locus of our potential
$yG(\text{}\underline{x})$
is essentially the equation of motion space, which is, in particular, not finite. Therefore, the two constructions are not the same at the beginning. Of course, one could apply the theory of [Reference Li, Li and Saito9] to the potential
$G(\text{}\underline{x})$
when
$X_{G}$
is smooth (note that if
$X_{G}$
is not smooth, the critical locus is not finite), but the cohomology of the resulting DGBV algebra does not give the primitive middle-dimensional cohomology of
$X_{G}$
, which is the most interesting Hodge-theoretic invariant of
$X_{G}$
.
Now assume that
$X$
is smooth. Let
$M=X(\mathbb{C})$
. Then
$M$
is a compact Calabi–Yau (Kähler) manifold of dimension
$m:=n-1$
. Let
$T_{M}$
(respectively,
$\overline{T}_{M}$
) be the holomorphic (respectively, antiholomorphic) tangent bundle over
$M$
. Let
$T_{M}^{\ast }$
(respectively,
$\overline{T}_{M}^{\ast }$
) be the holomorphic (respectively, antiholomorphic) cotangent bundle over
$M$
. Barannikov and Kontsevich considered the differential graded Lie algebra
$$\begin{eqnarray}\mathbf{t}=\bigoplus _{k}\mathbf{t}^{k},\qquad \mathbf{t}^{k}:=\bigoplus _{q-p=k}\unicode[STIX]{x1D6E4}(M,\wedge ^{q}\overline{T}_{M}^{\ast }\otimes \wedge ^{p}T_{M})\end{eqnarray}$$
endowed with the differential
$\bar{\unicode[STIX]{x2202}}$
, and the Lie bracket
$[\cdot ,\cdot ]$
that comes from the cup-product on
$\bar{\unicode[STIX]{x2202}}$
-forms and the standard Schouten–Nijenhuys bracket on holomorphic polyvector fields. Since
$M$
is Calabi–Yau, there exists an everywhere non-vanishing holomorphic volume form
$\unicode[STIX]{x1D6FA}\in \unicode[STIX]{x1D6E4}(M,\wedge ^{m}T_{M}^{\ast })$
, which is defined up to a constant. Let us fix a choice of
$\unicode[STIX]{x1D6FA}$
. Define the differential
$D$
on
$\mathbf{t}$
by the formula
$$\begin{eqnarray}D(\unicode[STIX]{x1D6FE})\vdash \unicode[STIX]{x1D6FA}=\unicode[STIX]{x2202}(\unicode[STIX]{x1D6FE}\vdash \unicode[STIX]{x1D6FA}),\quad \unicode[STIX]{x1D6FE}\in \mathbf{t},\end{eqnarray}$$
where
$\vdash$
means the contraction. Note that
$\bar{\unicode[STIX]{x2202}}$
and
$D$
have degree
$1$
:
$$\begin{eqnarray}\bar{\unicode[STIX]{x2202}}:\mathbf{t}^{m}\rightarrow \mathbf{t}^{m+1},\qquad D:\mathbf{t}^{m}\rightarrow \mathbf{t}^{m+1},\quad m\in \mathbb{Z}.\end{eqnarray}$$
In [Reference Barannikov and Kontsevich1], Barannikov and Kontsevich showed that the quadruple
$(\mathbf{t},\bar{\unicode[STIX]{x2202}},\bar{\unicode[STIX]{x2202}}+D,[\cdot ,\cdot ])$
is a DGBV algebra such that
$H^{i}(\mathbf{t},\bar{\unicode[STIX]{x2202}}+D)\,\simeq \,H^{i+m}(M,\mathbb{C})$
for
$i=-m,\ldots ,m$
. Their motivation for constructing such a DGBV algebra is to find a formal Frobenius manifold structure on the
$B$
-side of the mirror symmetry conjecture. Our next result is a relationship between our DGBV algebra
$({\mathcal{A}}_{U},Q_{U},K_{U},l_{2}^{K_{U}})$
and
$(\mathbf{t},\bar{\unicode[STIX]{x2202}},\bar{\unicode[STIX]{x2202}}+D,[\cdot ,\cdot ])$
. Let
$H_{pr}^{k}(\mathbf{t},\bar{\unicode[STIX]{x2202}}+D)$
be the inverse image of the primitive cohomology
$\mathbf{PH}^{i+m}\subseteq H^{i+m}(M,\mathbb{C})$
under the above isomorphism.
Theorem 1.4. Let
$X_{G}$
be a smooth projective hypersurface in
$\mathbb{P}^{n}$
defined by
$G(\text{}\underline{x})$
. There is a cochain map
$\unicode[STIX]{x1D6E9}:({\mathcal{A}}_{U},K_{U}=Q_{U}+\unicode[STIX]{x1D6E5})\rightarrow (\mathbf{t},\bar{\unicode[STIX]{x2202}}+D)$
, i.e.
$\unicode[STIX]{x1D6E9}$
preserves the degree and
$(\bar{\unicode[STIX]{x2202}}+D)\circ \unicode[STIX]{x1D6E9}=\unicode[STIX]{x1D6E9}\circ K_{U}$
. Moreover,
$\unicode[STIX]{x1D6E9}$
induces an isomorphism
$H^{k}({\mathcal{A}}_{U},K_{U})\simeq H_{pr}^{k}(\mathbf{t},\bar{\unicode[STIX]{x2202}}+D)$
for
$k=-m,-m+1,\ldots ,0$
.
In particular, if
$m$
is odd, then the primitive cohomology is the same as the cohomology and
$\unicode[STIX]{x1D6E9}$
induces an isomorphism
$H^{k}({\mathcal{A}}_{U},K_{U})\simeq H^{k}(\mathbf{t},\bar{\unicode[STIX]{x2202}}+D)\simeq H^{k+m}(M,\mathbb{C})$
for
$k=-m,-m+1,\ldots ,0$
. If
$k=1,2,\ldots ,m$
, then
$H^{k}({\mathcal{A}}_{U},K_{U})=0$
, but
$H_{pr}^{k}(\mathbf{t},\bar{\unicode[STIX]{x2202}}+D)\simeq H^{2m-k}(\mathbf{t},\bar{\unicode[STIX]{x2202}}+D)$
by Poincaré duality. Note that if
$v=y^{k}u(\text{}\underline{x})\in {\mathcal{A}}_{U}^{0}$
with
$k\leqslant n-1=\dim _{\mathbb{C}}(M)$
(Macaulay’s theorem implies that
$y^{k}u(\text{}\underline{x})\in \text{Im}~(Q_{U})\cap {\mathcal{A}}_{U}^{0}$
if
$k>n-1$
), then
$\unicode[STIX]{x1D6E9}(v)\in \unicode[STIX]{x1D6E4}(M,\wedge ^{k}\overline{T}_{M}^{\ast }\otimes \wedge ^{k}T_{M})$
.
Remark 1.5. It can be checked that
$$\begin{eqnarray}\bar{\unicode[STIX]{x2202}}\circ \unicode[STIX]{x1D6E9}\neq \unicode[STIX]{x1D6E9}\circ Q_{U},\qquad D\circ \unicode[STIX]{x1D6E9}\neq \unicode[STIX]{x1D6E9}\circ \unicode[STIX]{x1D6E5},\end{eqnarray}$$
because
$\bar{\unicode[STIX]{x2202}}$
and
$D$
satisfy the so-called
$D\bar{\unicode[STIX]{x2202}}$
-lemma,
$$\begin{eqnarray}\text{Ker}~\bar{\unicode[STIX]{x2202}}\cap \text{Ker}~D\cap (\text{Im}~\bar{\unicode[STIX]{x2202}}\oplus \text{Im}~D)=\text{Im}~(\bar{\unicode[STIX]{x2202}}\circ D),\end{eqnarray}$$
but
$\unicode[STIX]{x1D6E5}$
and
$Q$
do not satisfy the corresponding
$\unicode[STIX]{x1D6E5}Q$
-lemma. The identity element
$1\in {\mathcal{A}}_{U}^{0}$
belongs to
$\text{Ker}~Q\cap \text{Ker}~\unicode[STIX]{x1D6E5}\cap \text{Im}~(\unicode[STIX]{x1D6E5})$
, but 1 cannot belong to
$\text{Im}~(Q\circ \unicode[STIX]{x1D6E5})$
due to degree reasons. Thus, the map
$\unicode[STIX]{x1D6E9}$
does not send
$Q_{U}$
and
$\unicode[STIX]{x1D6E5}$
to
$\bar{\unicode[STIX]{x2202}}$
and
$D$
, respectively. In particular, the DGBV algebra structure of
${\mathcal{A}}_{U}$
is not isomorphic to
$\mathbf{t}$
.
Now we briefly explain the contents of each section. In §2, we give a construction of the DGBV algebra
$\mathbf{BV}_{\!U}$
for
$\mathbb{H}$
and describe its properties. Its main purpose is to prove Theorem 1.1. Section 2.1 is devoted to describing
$\mathbf{BV}_{\!U}$
in detail. In §2.2, we prove part (a) of Theorem 1.1. In §2.3, we prove part (b) of Theorem 1.1. In §2.4, we compare our construction here with the smooth projective hypersurface case considered in [Reference Park and Park10]. Finally, §3 is devoted to a proof of Theorem 1.4.
2 A DGBV algebra attached to projective Calabi–Yau hypersurface
2.1 A Dwork-type DGBV algebra
We start by providing the definition of DGBV algebras.
Definition 2.1. Let
$({\mathcal{A}},\cdot )$
be a unital
$\mathbb{Z}$
-graded super-commutative and associative
$\mathbb{C}$
-algebra. Let
$[\bullet ,\bullet ]:{\mathcal{A}}\otimes {\mathcal{A}}\rightarrow {\mathcal{A}}$
be a bilinear map of degree 1.
(a) A pair
$({\mathcal{A}},[\cdot ,\cdot ])$
is called a Gerstenhaber algebra if (4)for any homogeneous elements$$\begin{eqnarray}\displaystyle \begin{array}{@{}rcl@{}}[a,b]\ & =\ & -(-1)^{(|a|-1)(|b|-1)}[b,a],\\ ~[a,[b,c]]\ & =\ & [[a,b],c]+(-1)^{(|a|-1)(|b|-1)}[b,[a,c]],\\ ~[a,b\cdot c]\ & =\ & [a,b]\cdot c+(-1)^{(|a|-1)\cdot |b|}b\cdot [a,c]\end{array} & & \displaystyle\end{eqnarray}$$
$a,b,c\in {\mathcal{A}}_{X}.$
(b) A triple
$({\mathcal{A}},K,l_{2}^{K}(\cdot ,\cdot ))$
is called a GBV (Gerstenhaber–Batalin–Vilkovisky) algebraFootnote
6
if for any homogeneous elements$$\begin{eqnarray}K^{2}=0,\qquad l_{2}^{K}(a,b\cdot c)=l_{2}^{K}(a,b)\cdot c+(-1)^{(|a|-1)\cdot |b|}b\cdot l_{2}^{K}(a,c)\end{eqnarray}$$
$a,b,c\in {\mathcal{A}}_{X}.$
(c) A quadruple
$({\mathcal{A}},Q,K,l_{2}^{K}(\cdot ,\cdot ))$
, where
$Q:{\mathcal{A}}\rightarrow {\mathcal{A}}$
is a linear map of degree 1 and
$({\mathcal{A}},K,l_{2}^{K}(\cdot ,\cdot ))$
is a GBV algebra, is a DGBV algebra if $$\begin{eqnarray}Q^{2}=QK+KQ=0,\qquad Q(a\cdot b)=Q(a)\cdot b+(-1)^{|a|}a\cdot Q(b).\end{eqnarray}$$
We briefly explain that our cochain complex
$({\mathcal{A}}_{G},K_{G})$
in the introduction originates from the Chevalley–Eilenberg complex for some abelian Lie algebra representation. Let
$A=\mathbb{C}[y][\text{}\underline{x}]$
and consider the Dwork potential
$$\begin{eqnarray}S(y,\text{}\underline{x}):=y\cdot G(\text{}\underline{x}).\end{eqnarray}$$
Let
$\mathfrak{g}$
be an abelian Lie algebra of dimension
$n+1$
. Let
$\unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D6FC}_{2},\ldots ,\unicode[STIX]{x1D6FC}_{n+1}$
be a
$\mathbb{C}$
-basis of
$\mathfrak{g}$
. We associate a Lie algebra representation
$\unicode[STIX]{x1D70C}_{G}$
on
$A$
of
$\mathfrak{g}$
as follows:
$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{i}:=\unicode[STIX]{x1D70C}_{G}(\unicode[STIX]{x1D6FC}_{i}):=\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{i}}+\frac{\unicode[STIX]{x2202}S(y,\text{}\underline{x})}{\unicode[STIX]{x2202}x_{i}}\quad \text{for }i=0,1,\ldots ,n.\end{eqnarray}$$
We extend this
$\mathbb{C}$
-linearly to get a map
$\unicode[STIX]{x1D70C}_{G}:\mathfrak{g}\rightarrow \operatorname{End}_{\mathbb{C}}(A)$
. This is clearly a Lie algebra representation of
$\mathfrak{g}$
. Then we consider the dual Chevalley–Eilenberg complex
$({\mathcal{A}}_{\unicode[STIX]{x1D70C}_{G}},K_{\unicode[STIX]{x1D70C}_{G}})$
associated to
$\unicode[STIX]{x1D70C}_{G}$
as in [Reference Park and Park10], which computes the Lie algebra homology of
$\unicode[STIX]{x1D70C}_{G}$
, i.e.
$H_{K_{\unicode[STIX]{x1D70C}_{G}}}^{i}({\mathcal{A}}_{\unicode[STIX]{x1D70C}_{G}})\simeq H_{n+1-i}(\mathfrak{g},A)$
for
$i\in \mathbb{Z}$
. The underlying vector space of the dual Chevalley–Eilenberg complex is
$A\otimes \bigwedge \mathfrak{g}$
, where
$\bigwedge \mathfrak{g}$
is the alternating algebra generated by
$\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{n+1}$
. Then, with the identification
$\unicode[STIX]{x1D702}_{i}=\unicode[STIX]{x1D6FC}_{i+1}$
, where
$\unicode[STIX]{x1D702}_{i}$
is considered to have degree
$-1$
, it is easy to check that
$({\mathcal{A}}_{\unicode[STIX]{x1D70C}_{G}},K_{\unicode[STIX]{x1D70C}_{G}})$
is actually the cochain complex
$({\mathcal{A}}_{G},K_{G})$
. We recall that
$$\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}\displaystyle {\mathcal{A}}_{G}:=\mathbb{C}[y,x_{0},\ldots ,x_{n}][\unicode[STIX]{x1D702}_{0},\ldots ,\unicode[STIX]{x1D702}_{n}],\\ \displaystyle K_{G}=K_{U}:=\displaystyle \mathop{\sum }_{i=0}^{n}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{i}}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}_{i}}+\displaystyle \mathop{\sum }_{i=0}^{n}y\frac{\unicode[STIX]{x2202}G}{\unicode[STIX]{x2202}x_{i}}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}_{i}},\\ \displaystyle Q_{G}=Q_{U}:=K_{G}-\unicode[STIX]{x1D6E5},\qquad \unicode[STIX]{x1D6E5}:=\mathop{\sum }_{i=0}^{n}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{i}}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}_{i}}.\end{array} & & \displaystyle\end{eqnarray}$$
Then it is easy to check that
$Q_{G}$
and
$K_{G}$
are of degree 1 and
$K_{G}^{2}=Q_{G}^{2}=0$
, and that
${\mathcal{A}}_{G}$
becomes a
$\mathbb{Z}$
-graded super-commutative algebra, i.e.
$x\cdot y=(-1)^{|x||y|}y\cdot x$
for homogeneous elements
$x,y\in {\mathcal{A}}_{G}$
such that
${\mathcal{A}}_{G}^{0}=A=\mathbb{C}[\text{}\underline{x}][y]$
and
${\mathcal{A}}_{G}^{-k}:=\bigoplus _{0\leqslant i_{1}<\cdots <i_{k}\leqslant n}A\cdot \unicode[STIX]{x1D702}_{i_{1}}\cdots \unicode[STIX]{x1D702}_{i_{k}}$
for
$k=1,\ldots ,n$
. Note that
$\unicode[STIX]{x1D702}_{i}\cdot \unicode[STIX]{x1D702}_{j}=-\unicode[STIX]{x1D702}_{j}\cdot \unicode[STIX]{x1D702}_{i}$
, which implies that
$\unicode[STIX]{x1D702}_{i}^{2}=0$
. Since
$(\unicode[STIX]{x2202}S(y,\text{}\underline{x})/\unicode[STIX]{x2202}x_{i})(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D702}_{i})$
is a differential operator of order 1, the differential
$Q_{G}$
is a derivation of the product of
${\mathcal{A}}$
. Thus,
$({\mathcal{A}},\cdot ,Q_{G})$
is a CDGA. But
$K_{G}$
is not a derivation of the product, because the differential operator
$(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x_{i})(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D702}_{i})$
has order 2.
We define two more additive gradings on
${\mathcal{A}}_{G}$
with respect to the multiplication, called charge
$ch\in \mathbb{Z}$
and weight
$wt\in \mathbb{Z}$
, by the following rules:
$$\begin{eqnarray}\displaystyle & \displaystyle ch(y_{})=-e,\qquad ch(x_{i})=1,\qquad ch(\unicode[STIX]{x1D702}_{i})=-1, & \displaystyle \nonumber\\ \displaystyle & \displaystyle wt(y_{})=1,\qquad wt(x_{i})=0,\qquad wt(\unicode[STIX]{x1D702}_{i})=1 & \displaystyle \nonumber\end{eqnarray}$$
for
$i=0,1,\ldots ,n$
. Write a decomposition as follows:
$$\begin{eqnarray}{\mathcal{A}}_{G}=\bigoplus _{ch,wt}{\mathcal{A}}_{G,ch,(wt)}=\bigoplus _{-n-1\leqslant j\leqslant 0}\bigoplus _{w\in \mathbb{Z}^{{\geqslant}0}}\bigoplus _{\unicode[STIX]{x1D706}\in \mathbb{Z}}{\mathcal{A}}_{G,\unicode[STIX]{x1D706},(w)}^{j}.\end{eqnarray}$$
Note that the construction of
$({\mathcal{A}}_{G},K_{G})$
using the polynomial
$yG(x)$
and charge and weight operators comes from Witten’s gauged linear sigma model in [Reference Witten12]. It turns out that the cochain complex
$({\mathcal{A}}_{G},K_{G})$
computes the cohomology of the affine hypersurface
$Y_{F}$
in
$\mathbb{A}^{n+1}$
defined by
$F(\text{}\underline{x}):=G(\text{}\underline{x})-1$
. Define, for all
$p\geqslant 0$
,
$$\begin{eqnarray}{\mathcal{A}}_{U}:=\bigoplus _{p=0}^{n+1}{\mathcal{A}}_{U}^{-p},\quad \text{where}~{\mathcal{A}}_{U}^{-p}:={\mathcal{A}}_{G,0}^{-p}.\end{eqnarray}$$
Because
$Q_{G}$
and
$K_{G}$
preserve the charge, the charge-zero parts
$({\mathcal{A}}_{U},Q_{U})$
and
$({\mathcal{A}}_{U},K_{U})$
are also cochain complexes, where
$Q_{U}=Q_{G}$
and
$K_{U}=K_{G}$
. The
$wt$
grading gives a (decreasing) filtered subcomplex
$(F^{\bullet }{\mathcal{A}}_{U},K_{U})$
of
$({\mathcal{A}}_{U},K_{U})$
defined by
$$\begin{eqnarray}F^{0}{\mathcal{A}}_{U}={\mathcal{A}}_{U},\qquad F^{i}{\mathcal{A}}_{U}^{}=\bigoplus _{k\leqslant n-1-i}{\mathcal{A}}_{U,(k)}^{}=\bigoplus _{k\leqslant n-1-i}{\mathcal{A}}_{G,0,(k)}^{},\quad i\geqslant 1.\end{eqnarray}$$
Recall that
${\mathcal{A}}_{U}$
is the charge-zero subalgebra of
${\mathcal{A}}_{G}$
. Then
$({\mathcal{A}}_{U},K_{U})$
is a cochain complex with a binary super-commutative product. The following proposition gives a proof that
$\mathbf{BV}_{\!U}$
in Theorem 1.1 is a DGBV algebra containing a polynomial subring
${\mathcal{A}}_{U}^{0}$
over which it is a finitely generated module.
Proposition 2.2. The quadruple
$\mathbf{BV}_{\!U}:=({\mathcal{A}}_{U},Q_{U},K_{U},l_{2}^{K_{U}})$
is a DGBV algebra, which is generated by
$y,x_{0},\ldots ,x_{n},\unicode[STIX]{x1D702}_{0},\ldots ,\unicode[STIX]{x1D702}_{n}$
as
$\mathbb{C}$
-algebras.
Proof. This can be deduced from straightforward (but lengthy) computations, which are left to the reader. In fact, one can check that the dual Chevalley–Eilenberg complex attached to a representation of any abelian Lie algebra consisting of differential operators of order
${\leqslant}1$
becomes a DGBV algebra for fairly general reasons; for example, see [Reference Park and Park10, §§2.2 and 2.3]. This fact is intimately related to the BV-BRST (Batalin–Vilkovisky–Becchi–Rouet–Stora–Tyutin) formalism in the mathematical physics literature.◻
2.2 Computation of
$K$
-cohomology
The goal of this subsection is to prove part (b) of Theorem 1.1, i.e. the cohomology of
$({\mathcal{A}}_{U},K_{U})$
is isomorphic to the de Rham cohomology of
$U_{G}$
. For this purpose, we define a
$\mathbb{C}$
-linear map
$\unicode[STIX]{x1D6FD}^{-p}:{\mathcal{A}}_{U}^{-p}\rightarrow \unicode[STIX]{x1D6FA}^{n-p}(U_{G})$
, for each
$0\leqslant p\leqslant n$
, by assigning
$$\begin{eqnarray}\displaystyle y^{k-1}u(\text{}\underline{x})\unicode[STIX]{x1D702}_{i_{1}}\cdots \unicode[STIX]{x1D702}_{j_{p}} & \mapsto & \displaystyle (-1)^{j_{1}+\cdots +j_{p}}(-1)^{k-1}(k-1)!\nonumber\\ \displaystyle & & \displaystyle \times \,\displaystyle \frac{u(\text{}\underline{x})}{G(\text{}\underline{x})^{k}}E(dx_{0}\cdots \widehat{dx_{j_{1}}}\cdots \widehat{dx_{j_{p}}}\cdots dx_{n})\end{eqnarray}$$
and extending it
$\mathbb{C}$
-linearly, where
$E(\unicode[STIX]{x1D714}):=\langle \sum _{i=0}^{n}x_{i}(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x_{i}),\unicode[STIX]{x1D714}\rangle$
is the contraction with the Euler vector field. Then we will show in Theorem 2.3 that
$$\begin{eqnarray}\unicode[STIX]{x1D6FD}^{-p+1}\circ K_{U}=d\circ \unicode[STIX]{x1D6FD}^{-p}\end{eqnarray}$$
and so
$\unicode[STIX]{x1D6FD}^{-p}$
induces a map on cohomologies. In other words, we have the following commutative diagram:
Because
$U_{G}$
is a smooth affine variety (regardless of whether
$G$
is smooth or not in
$\mathbf{P}^{n}$
), the Grothendieck theorem implies that
$$\begin{eqnarray}H_{\text{dR}}^{k}(U_{G}/\mathbb{C})\simeq H^{k}(\unicode[STIX]{x1D6FA}^{\bullet }(U_{G}),d).\end{eqnarray}$$
In fact, the affine coordinate ring of
$U_{G}$
is
${\mathcal{A}}_{U}^{0}/(y\cdot G(\text{}\underline{x})-1)$
.
Theorem 2.3. For every
$0\leqslant p\leqslant n$
, the map
$\unicode[STIX]{x1D6FD}^{-p}$
satisfies
$\unicode[STIX]{x1D6FD}^{-p+1}\circ K_{U}=d\circ \unicode[STIX]{x1D6FD}^{-p}$
and induces an isomorphism
$$\begin{eqnarray}\bar{\unicode[STIX]{x1D6FD}}^{-p}:H^{-p}({\mathcal{A}}_{U},K_{U})\cong H^{n-p}(\unicode[STIX]{x1D6FA}^{\bullet }(U_{G}),d),\end{eqnarray}$$
so that we have
$H^{-p}({\mathcal{A}}_{U},K_{U})\simeq \mathbb{H}^{n-p}:=H_{\text{dR}}^{n-p}(U_{G}/\mathbb{C})$
.
Proof. We prove this by decomposing
$\unicode[STIX]{x1D6FD}^{-p}$
into the composition of three cochain maps
$\unicode[STIX]{x1D6FD}_{3}^{-p}\circ \unicode[STIX]{x1D6FD}_{2}^{-p}\circ \unicode[STIX]{x1D6FD}_{1}^{-p}$
and showing that they induce all
$\mathbb{C}$
-linear isomorphisms on relevant cohomologies. The map
$\unicode[STIX]{x1D6FD}_{1}^{-p}$
comes from some kind of Poincaré duality between Lie algebra homology and cohomology for the abelian Lie algebra
$\mathfrak{g}$
, which is a cochain map to “a twisted de Rham complex”,
$\unicode[STIX]{x1D6FD}_{2}^{-p}$
is just book-keeping for the correct degrees and
$\unicode[STIX]{x1D6FD}_{3}^{-p}$
comes from a spectral sequence in [Reference Dimca5] for the Milnor fibration of the homogeneous polynomial
$G(\text{}\underline{x})$
.
We start with the construction of
$\unicode[STIX]{x1D6FD}_{1}^{-p}$
. We define an additive degree on
$A\otimes \bigwedge (d\text{}\underline{x})$
, where
$\bigwedge (d\text{}\underline{x})$
is the ideal generated by
$dx_{0},\ldots ,dx_{n}$
of the de Rham algebra of the affine space
$\mathbb{C}^{n+1}$
, by
$$\begin{eqnarray}|y|=-e,\qquad |u(\text{}\underline{x})|=\deg u,\qquad |dx_{J}|=|dx_{j_{1}}\cdots dx_{j_{p}}|=|J|=p.\end{eqnarray}$$
We denote the homogeneous component of degree
$n+1$
by
$(A\otimes \bigwedge (d\text{}\underline{x}))_{n+1}$
. Then we define a
$\mathbb{C}$
-linear map
$\unicode[STIX]{x1D6FD}_{1}^{-p}:{\mathcal{A}}_{U}^{-p}\rightarrow (A\otimes \bigwedge (d\text{}\underline{x}))_{n+1}^{n+1-p}$
by the assignment
$$\begin{eqnarray}\displaystyle y^{k-1}u(\text{}\underline{x})\unicode[STIX]{x1D702}_{J}\mapsto (-1)^{j_{1}+\cdots +j_{p}+p}y^{k-1}u(\text{}\underline{x})\,dx_{0}\cdots \widehat{dx_{J}}\cdots dx_{n} & & \displaystyle \nonumber\end{eqnarray}$$
for
$J=\{j_{1},\ldots ,j_{p}\}$
. Then it is a simple computation that
$(d_{\text{}\underline{x}}+ydG)\circ \unicode[STIX]{x1D6FD}_{1}^{-p}=-\unicode[STIX]{x1D6FD}_{1}^{-p+1}\circ K_{U}$
:
$$\begin{eqnarray}\displaystyle & & \displaystyle ((d_{\text{}\underline{x}}+ydG)\circ \unicode[STIX]{x1D6FD}_{1}^{-p})(y^{k-1}u(\text{}\underline{x})\unicode[STIX]{x1D702}_{J})\nonumber\\ \displaystyle & & \displaystyle \qquad =\mathop{\sum }_{l=1}^{p}(-1)^{j_{1}+\cdots +j_{p}+p+(j_{l}-l+1)}y^{k-1}\nonumber\\ \displaystyle & & \displaystyle \qquad \quad \times \,\biggl(\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x_{j_{l}}}+y\frac{\unicode[STIX]{x2202}G}{\unicode[STIX]{x2202}x_{j_{l}}}u\biggr)dx_{0}\cdots \widehat{dx_{J-\{j_{l}\}}}\cdots dx_{n}\nonumber\\ \displaystyle & & \displaystyle \qquad =-\mathop{\sum }_{l=1}^{p}(-1)^{j_{1}+\cdots +j_{p}+(p-1)+(j_{l}-l+1)}y^{k-1}\nonumber\\ \displaystyle & & \displaystyle \qquad \quad \times \,\biggl(\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x_{j_{l}}}+y\frac{\unicode[STIX]{x2202}G}{\unicode[STIX]{x2202}x_{j_{l}}}u\biggr)dx_{0}\cdots \widehat{dx_{J-\{j_{l}\}}}\cdots dx_{n}\nonumber\\ \displaystyle & & \displaystyle \qquad =-\unicode[STIX]{x1D6FD}_{1}^{-p+1}\biggl(\mathop{\sum }_{l=1}^{p}(-1)^{l-1}\biggl(\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{j_{l}}}+y\frac{\unicode[STIX]{x2202}G}{\unicode[STIX]{x2202}x_{j_{l}}}\biggr)y^{k-1}u(\text{}\underline{x})\unicode[STIX]{x1D702}_{J-\{j_{l}\}}\biggr)\nonumber\\ \displaystyle & & \displaystyle \qquad =-(\unicode[STIX]{x1D6FD}_{1}^{-p+1}\circ K_{U})(y^{k-1}u(\text{}\underline{x})\unicode[STIX]{x1D702}_{J}).\nonumber\end{eqnarray}$$
Since
$\unicode[STIX]{x1D6FD}_{1}^{-p}$
is a
$\mathbb{C}$
-linear map which commutes with the differentials and sends a
$\mathbb{C}$
-basis to a
$\mathbb{C}$
-basis, it is obviously a twisted cochain isomorphism
$$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{1}=\bigoplus _{p}\unicode[STIX]{x1D6FD}_{1}^{-p}:\biggl(\bigoplus _{p=0}^{n}{\mathcal{A}}_{U}^{-p},K_{U}\biggr)\rightarrow \biggl(\biggl(A\otimes \bigwedge (d\text{}\underline{x})\biggr)_{n+1},-(d_{\text{}\underline{x}}+ydG)\biggr).\end{eqnarray}$$
Secondly, we consider the exterior algebra
$\unicode[STIX]{x1D6FA}_{\mathbb{C}^{n+1}}$
of regular differential forms on the affine space
$\mathbb{C}^{n+1}$
with the twisted differential
$D_{G}$
and define an additive grading by
$$\begin{eqnarray}D_{G}(\unicode[STIX]{x1D714}):=-d\unicode[STIX]{x1D714}+\displaystyle \frac{|\unicode[STIX]{x1D714}|}{e}dG\wedge \unicode[STIX]{x1D714},\qquad |\unicode[STIX]{x1D714}|=|u(\text{}\underline{x})dx_{J}|=\deg u+|J|\end{eqnarray}$$
for
$\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}_{\mathbb{C}^{n+1}}$
. We denote the homogeneous component of grading
$ke$
by
$\unicode[STIX]{x1D6FA}_{\mathbb{C}^{n+1},ke}$
. Then we define a
$\mathbb{C}$
-linear map
$\unicode[STIX]{x1D6FD}_{2}^{-p}:(A\otimes \bigwedge (d\text{}\underline{x}))_{n+1}^{n+1-p}\rightarrow \bigoplus _{k\geqslant 1}\unicode[STIX]{x1D6FA}_{\mathbb{C}^{n+1},ke}^{n+1-p}$
by assigning
$$\begin{eqnarray}\displaystyle y^{k-1}\unicode[STIX]{x1D714}\mapsto (-1)^{k-1}(k-1)!\unicode[STIX]{x1D714}. & & \displaystyle \nonumber\end{eqnarray}$$
It is easy to check that
$\unicode[STIX]{x1D6FD}_{2}$
is a cochain map, i.e.
$D_{G}\circ \unicode[STIX]{x1D6FD}_{2}=\unicode[STIX]{x1D6FD}_{2}\circ (-d_{\text{}\underline{x}}-ydG)$
:
$$\begin{eqnarray}\displaystyle (D_{G}\circ \unicode[STIX]{x1D6FD}_{2}(y^{k-1}\unicode[STIX]{x1D714})) & = & \displaystyle (-1)^{k-1}(k-1)!(-d\unicode[STIX]{x1D714}+kdG\wedge \unicode[STIX]{x1D714})\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D6FD}_{2}(-y^{k-1}d\unicode[STIX]{x1D714}-y^{k}dG\wedge \unicode[STIX]{x1D714})\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D6FD}_{2}\circ (-d_{\text{}\underline{x}}-ydG)(y^{k-1}\unicode[STIX]{x1D714}).\nonumber\end{eqnarray}$$
Conversely, we can define the inverse cochain map
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}_{2}^{-p}:\bigoplus _{k\geqslant 1}\unicode[STIX]{x1D6FA}_{\mathbb{C}^{n+1},ke}^{n+1-p} & \rightarrow & \displaystyle \biggl(A\otimes \bigwedge (d\text{}\underline{x})\biggr)_{n+1}^{n+1-p}\nonumber\\ \displaystyle \unicode[STIX]{x1D714} & \mapsto & \displaystyle \displaystyle \frac{1}{(-1)^{k-1}(k-1)!}y^{k-1}\unicode[STIX]{x1D714}\nonumber\end{eqnarray}$$
for
$\unicode[STIX]{x1D714}$
with
$|\unicode[STIX]{x1D714}|=ke$
and conclude that
$((A\otimes \bigwedge (d\text{}\underline{x}))_{n+1},-(d_{\text{}\underline{x}}+ydG))$
and
$(\bigoplus _{k\geqslant 1}\unicode[STIX]{x1D6FA}_{\mathbb{C}^{n+1},ke},D_{G})$
are cochain isomorphic. Finally, we define a
$\mathbb{C}$
-linear mapFootnote
7
$$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{3}:\biggl(\bigoplus _{k\geqslant 1}\unicode[STIX]{x1D6FA}_{\mathbb{C}^{n+1},ke},D_{G}\biggr)\rightarrow (\unicode[STIX]{x1D6FA}(U_{G}),d)\end{eqnarray}$$
by
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FD}_{3}(\unicode[STIX]{x1D714}):=\displaystyle \frac{E(\unicode[STIX]{x1D714})}{G(\text{}\underline{x})^{k}}\quad \text{for }\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}_{\mathbb{C}^{n+1},ke} & & \displaystyle \nonumber\end{eqnarray}$$
and denote
$\unicode[STIX]{x1D6FD}_{3}(\unicode[STIX]{x1D714})$
by
$\unicode[STIX]{x1D6FD}_{3}^{-p}(\unicode[STIX]{x1D714})$
when
$\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}_{\mathbb{C}^{n+1}}^{n+1-p}$
. If we apply [Reference Dimca5, Theorem 1.8] for
$a=0$
, where
$\bigoplus _{k\geqslant 1}\unicode[STIX]{x1D6FA}_{\mathbb{C}^{n+1},~ke}^{n+1-p}$
corresponds to
$B_{0}^{n-p}$
,
$\unicode[STIX]{x1D6FA}^{n-p}(U_{G})$
corresponds to
$A_{0}^{n-p}$
and
$\unicode[STIX]{x1D6FD}_{3}$
corresponds to
$\bar{\unicode[STIX]{x1D6FF}}$
, then we see that
$\unicode[STIX]{x1D6FD}_{3}^{-p}$
is a cochain map (up to sign) and induces an isomorphism
$\bar{\unicode[STIX]{x1D6FD}}_{3}^{-p}:H^{n+1-p}(\bigoplus _{k\geqslant 1}\unicode[STIX]{x1D6FA}_{\mathbb{C}^{n+1},ke},D_{G})\,\overset{{\sim}}{\rightarrow }\,H^{n-p}(\unicode[STIX]{x1D6FA}(U_{G}),d)$
for each
$0\leqslant p\leqslant n$
. For the reader’s convenience, we include the detailed computation for the claim
$d\circ \unicode[STIX]{x1D6FD}_{3}=\unicode[STIX]{x1D6FD}_{3}\circ D_{G}$
:
$$\begin{eqnarray}\displaystyle (d\circ \unicode[STIX]{x1D6FD}_{3})(\unicode[STIX]{x1D714}) & = & \displaystyle d\biggl(\frac{E(\unicode[STIX]{x1D714})}{G^{k}}\biggr)\nonumber\\ \displaystyle & = & \displaystyle -\frac{E(Gd\unicode[STIX]{x1D714}-(|\unicode[STIX]{x1D714}|/e)dG\wedge \unicode[STIX]{x1D714})}{G^{k+1}}\nonumber\\ \displaystyle & = & \displaystyle (\unicode[STIX]{x1D6FD}_{3}\circ D_{G})(\unicode[STIX]{x1D714})\nonumber\end{eqnarray}$$
for
$\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}_{\mathbb{C}^{n+1},ke}$
. Here we refer to [Reference Dimca6, p. 181] for the second equality, which is based on the well-known formula
$E(d\unicode[STIX]{x1D714})+d(E(\unicode[STIX]{x1D714}))=|\unicode[STIX]{x1D714}|\unicode[STIX]{x1D714}$
for a homogeneous differential form
$\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}_{\mathbb{C}^{n+1}}$
, and used that
$E$
is a
$A$
-linear graded derivation,
$d\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}_{\mathbb{C}^{n+1},ke}$
and
$dG\wedge \unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}_{\mathbb{C}^{n+1},(k+1)e}$
for the third equality.
It is straightforward to check that
$\unicode[STIX]{x1D6FD}^{-p}=\unicode[STIX]{x1D6FD}_{3}^{-p}\circ \unicode[STIX]{x1D6FD}_{2}^{-p}\circ \unicode[STIX]{x1D6FD}_{1}^{-p}$
and hence
$\unicode[STIX]{x1D6FD}$
is a cochain map and
$\bar{\unicode[STIX]{x1D6FD}}$
is an isomorphism. We can summarize the situation in the following diagram:
Since one can directly check that
$H^{-n-1}({\mathcal{A}}_{U},K_{U})=0$
, part (b) of Theorem 1.1 now follows.
Remark 2.4. Even if
$X_{G}$
is not Calabi–Yau, the same argument shows that the cohomology of the cochain complex
$$\begin{eqnarray}({\mathcal{A}}_{U},K_{U}):=({\mathcal{A}}_{G,e-(n+1)},K_{G})\end{eqnarray}$$
consisting of elements of charge
$e-(n+1)$
in
${\mathcal{A}}_{G}$
is still isomorphic to
$\bigoplus _{k=0}^{n}H_{\text{dR}}^{k}(U_{G}/\mathbb{C})$
. But
$({\mathcal{A}}_{U},K_{U})$
does not inherit the ring structure of
${\mathcal{A}}_{G}$
in the non-Calabi–Yau case and so we cannot make a DGBV algebra in this way.
2.3 The Hodge filtration
We prove part (c) of Theorem 1.1. Since
$Q_{G}$
preserves
$wt$
and
$\unicode[STIX]{x1D6E5}$
decreases
$wt$
by 1, the following definitions:
$$\begin{eqnarray}F^{0}{\mathcal{A}}_{U}={\mathcal{A}}_{U},\qquad F^{i}{\mathcal{A}}_{U}^{}=\bigoplus _{k\leqslant n-1-i}{\mathcal{A}}_{U,(k)}^{}=\bigoplus _{k\leqslant n-1-i}{\mathcal{A}}_{G,0,(k)}^{},\quad i\geqslant 1\end{eqnarray}$$
give a (decreasing) filtered subcomplex
$(F^{\bullet }{\mathcal{A}}_{U},K_{U})$
of
$({\mathcal{A}}_{U},K_{U})$
. It follows that the cohomology
$H^{s}=H^{s}({\mathcal{A}}_{G},K_{G})$
has the induced decreasing filtration
$$\begin{eqnarray}F_{wt}^{\bullet }H^{s}:=H^{s}(F^{\bullet }{\mathcal{A}}_{G},K_{G}),\quad s\in \mathbb{Z}.\end{eqnarray}$$
Let
$F^{s}\mathbb{H}^{k}$
be the decreasing Hodge filtration on
$\mathbb{H}^{k}$
, which is part of the mixed Hodge structure on
$\mathbb{H}^{k}=H_{\text{dR}}^{k}(U_{G}/\mathbb{C})$
constructed by Deligne [Reference Deligne3].
Proposition 2.5. For each
$p=0,1,\ldots ,n$
, the isomorphism
$J^{-p}:H^{-p}=H^{-p}({\mathcal{A}}_{X},K_{X})\rightarrow \mathbb{H}^{n-p}$
satisfies
$J^{-p}(F_{wt}^{s}H^{-p})\supseteq F^{s}\mathbb{H}^{n-p}$
for any integer
$s$
.
Proof. This is a straightforward reinterpretation of the main theorem of [Reference Deligne and Dimca4], for which we also refer to [Reference Dimca6, Theorem (1.31), Ch. 6], since our filtration
$F_{wt}^{s}H^{-p}$
matches exactly with the polar filtration
$P^{s}\mathbb{H}^{n-p}$
under the map
$J^{-p}$
. ◻
Remark 2.6. It is known that
$J^{-p}(F_{wt}^{s}H^{-p})=F^{s}\mathbb{H}^{n-p}$
when
$X_{G}$
is smooth in
$\mathbb{P}^{n}$
.
2.4 Comparison with the smooth case
In this section, we compare our DGBV
$\mathbf{BV}_{\!U}$
with the DGBV algebra for a smooth projective hypersurface in
$\mathbf{P}^{n}$
that appeared in [Reference Park and Park10]. The construction in [Reference Park and Park10] works regardless of the Calabi–Yau property of
$X_{G}$
. Let us briefly review the construction in [Reference Park and Park10]. The only difference from our construction is that the authors introduced
$\unicode[STIX]{x1D702}_{-1}$
corresponding to
$y$
. Then we take the symmetric tensor product of
$A$
and the alternating algebra
$\displaystyle \bigwedge (\text{}\underline{\unicode[STIX]{x1D702}}):=\bigwedge (\mathbb{C}\cdot \unicode[STIX]{x1D702}_{-1}\oplus \mathbb{C}\cdot \unicode[STIX]{x1D702}_{0}\oplus \cdots \oplus \mathbb{C}\cdot \unicode[STIX]{x1D702}_{n})$
generated by
$\unicode[STIX]{x1D702}_{-1},\unicode[STIX]{x1D702}_{0},\ldots ,\unicode[STIX]{x1D702}_{n}$
to define
$$\begin{eqnarray}{\mathcal{A}}_{X}:=A\otimes _{\mathbb{C}}\bigwedge (\text{}\underline{\unicode[STIX]{x1D702}})=\mathbb{C}[y,x_{0},\ldots ,x_{n}][\unicode[STIX]{x1D702}_{-1},\unicode[STIX]{x1D702}_{0},\ldots ,\unicode[STIX]{x1D702}_{n}].\end{eqnarray}$$
Let us introduce a grading on
${\mathcal{A}}_{X}$
:
$$\begin{eqnarray}\displaystyle |x_{i}|=0,\qquad |y|=0,\qquad |\unicode[STIX]{x1D702}_{i}|=-1\quad \forall i=-1,\ldots ,n. & & \displaystyle \nonumber\end{eqnarray}$$
The differentials on
${\mathcal{A}}_{X}$
are defined as follows:
$$\begin{eqnarray}\displaystyle & \displaystyle K_{X}=\unicode[STIX]{x1D6E5}+Q_{X},\quad \unicode[STIX]{x1D6E5}=\displaystyle \mathop{\sum }_{i=0}^{n}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{i}}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}_{i}}+\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}_{-1}}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle Q_{X}=\mathop{\sum }_{i=0}^{n}y\frac{\unicode[STIX]{x2202}G}{\unicode[STIX]{x2202}x_{i}}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}_{i}}+G(\text{}\underline{x})\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}_{-1}}. & \displaystyle \nonumber\end{eqnarray}$$
Compare this definition with (1). If
$X_{G}$
is smooth in
$\mathbf{P}^{n}$
, it was proved in [Reference Park and Park10] that
$({\mathcal{A}}_{X},Q_{X},K_{X},l_{2}^{K_{X}})$
is a DGBV algebra and
$$\begin{eqnarray}H^{0}({\mathcal{A}}_{X},K_{X})\simeq \mathbb{H}^{n}=H_{\text{dR}}^{n}(U_{G}/\mathbb{C}).\end{eqnarray}$$
Note that the explicit computation of
$H^{k}({\mathcal{A}}_{X},K_{X})$
, for
$k=-1,-2,\ldots ,$
was also given (see [Reference Park and Park10, Proposition 4.14]), but
$H^{k}({\mathcal{A}}_{X},K_{X})$
is not isomorphic to
$\mathbb{H}^{n+k}$
as it is in our DGBV algebra
$\mathbf{BV}_{\!U}$
.
Now we assume that
$X_{G}$
is Calabi–Yau and smooth. Then Theorem 2.3 and (12) imply that one should have an isomorphism
$$\begin{eqnarray}H^{0}({\mathcal{A}}_{X},K_{X})\simeq H^{0}({\mathcal{A}}_{U},K_{U}).\end{eqnarray}$$
A direct way to see the above isomorphism is to use the factsFootnote 8
$$\begin{eqnarray}H^{0}({\mathcal{A}}_{X},K_{X})\simeq H^{0}({\mathcal{A}}_{X,0},Q_{X})\quad \text{and}\quad H^{0}({\mathcal{A}}_{U},K_{U})\simeq H^{0}({\mathcal{A}}_{U},Q_{U}),\end{eqnarray}$$
where we extend the definition of charge on
${\mathcal{A}}_{G}$
to
${\mathcal{A}}_{X}$
by setting
$ch(\unicode[STIX]{x1D702}_{-1})=e$
. Note that
$$\begin{eqnarray}\displaystyle H^{0}({\mathcal{A}}_{X,0},Q_{X})={\mathcal{A}}_{U}^{0}/\text{Im}(Q_{X})\cap {\mathcal{A}}_{U}^{0} & & \displaystyle \nonumber\end{eqnarray}$$
and
$$\begin{eqnarray}\displaystyle H^{0}({\mathcal{A}}_{U},Q_{U})={\mathcal{A}}_{U}^{0}/\text{Im}(Q_{U})\cap {\mathcal{A}}_{U}^{0}. & & \displaystyle \nonumber\end{eqnarray}$$
One can easily check that
$\text{Im}(Q_{X})\cap {\mathcal{A}}_{U}^{0}=\text{Im}(Q_{U})\cap {\mathcal{A}}_{U}^{0}$
, which implies (13).
3 Comparison of DGBV algebras
The goal of this section is to prove Theorem 1.4. Let
$M=X(\mathbb{C})$
, where
$X$
is assumed to be smooth from now on. Recall the quadruple
$(\mathbf{t},\bar{\unicode[STIX]{x2202}},\bar{\unicode[STIX]{x2202}}+D,[\cdot ,\cdot ])$
from (2) in §1. Note that
$$\begin{eqnarray}0\rightarrow \mathbf{t}^{-m}\overset{\bar{\unicode[STIX]{x2202}},D}{\longrightarrow }\mathbf{t}^{-m+1}\overset{\bar{\unicode[STIX]{x2202}},D}{\longrightarrow }\cdots \overset{\bar{\unicode[STIX]{x2202}},D}{\longrightarrow }\mathbf{t}^{0}\overset{\bar{\unicode[STIX]{x2202}},D}{\longrightarrow }\cdots \overset{\bar{\unicode[STIX]{x2202}},D}{\longrightarrow }\mathbf{t}^{m-1}\overset{\bar{\unicode[STIX]{x2202}},D}{\longrightarrow }\mathbf{t}^{m}\rightarrow 0.\end{eqnarray}$$
The wedge product induces a
$\mathbb{Z}$
-graded associative and super-commutative algebra structure
$\bullet$
on
$\mathbf{t}=\bigoplus _{-m\leqslant k\leqslant m}\mathbf{t}^{k}$
:
$$\begin{eqnarray}a\bullet b:=(-1)^{q_{2}p_{1}}(u_{1}\wedge u_{2})\otimes (v_{1}\wedge v_{2}),\qquad a\wedge b=(-1)^{k_{1}k_{2}}b\wedge a\end{eqnarray}$$
for any homogeneous elements
$a=u_{1}\otimes v_{1}\in \mathbf{t}^{k_{1}}=\mathbf{t}^{q_{1}-p_{1}}$
and
$b=u_{2}\otimes v_{2}\in \mathbf{t}^{k_{2}}=\mathbf{t}^{q_{2}-p_{2}}$
, where
$u_{i}\in \unicode[STIX]{x1D6E4}(M,\wedge ^{q_{i}}\overline{T}_{M}^{\ast })$
and
$v_{i}\in \unicode[STIX]{x1D6E4}(M,\wedge ^{p_{i}}T_{M})$
. Recall the
$l_{2}$
-descendant of
$D$
with respect to the product
$\cdot$
:
$$\begin{eqnarray}l_{2}^{D}(a,b):=(-1)^{|a|}\cdot (D(ab)-D(a)\bullet b-(-1)^{|a|}a\bullet D(b)),\quad a,b\in \mathbf{t}.\end{eqnarray}$$
Then the Bogomoov–Tian–Todorov lemma implies that
$l_{2}^{D}$
is identified with the standard Schouten–Nijenhuys bracket
$[\cdot ,\cdot ]$
on
$\mathbf{t}$
. In [Reference Barannikov and Kontsevich1], it was proved that the quadruple
$(\mathbf{t},\bullet ,\bar{\unicode[STIX]{x2202}},\bar{\unicode[STIX]{x2202}}+D,l_{2}^{D})$
is a DGBV algebra. Here we repeat its characterizing properties.
(a)
$(\mathbf{t},\bullet )$
is a unital
$\mathbb{Z}$
-graded super-commutative and associative
$\mathbb{C}$
-algebra.(b)
$D$
is a differential operator of order 2 and
$(\mathbf{t},\bar{\unicode[STIX]{x2202}}+D,l_{2}^{D})$
is a (shifted) differential graded Lie algebra (DGLA).(c) The triple
$(\mathbf{t},\bullet ,l_{2}^{D})$
is a Gerstenhaber algebra.(d)
$(\mathbf{t},\bullet ,\bar{\unicode[STIX]{x2202}})$
is a commutative differential graded algebra (CDGA) such that
$\bar{\unicode[STIX]{x2202}}^{2}=(\bar{\unicode[STIX]{x2202}}+D)^{2}=\bar{\unicode[STIX]{x2202}}D+D\bar{\unicode[STIX]{x2202}}=0$
; $$\begin{eqnarray}\bar{\unicode[STIX]{x2202}}(a\cdot b)=\bar{\unicode[STIX]{x2202}}(a)\bullet b+(-1)^{|a|}a\bullet \bar{\unicode[STIX]{x2202}}(b),\quad a,b,c\in \mathbf{t}.\end{eqnarray}$$
Let
$U=U_{G}(\mathbb{C})$
. Let us start a proof of Theorem 1.4. We briefly recall the residue map
$\text{Res}$
. Let
$T$
be a sufficiently small closed tubular neighbourhood of
$M$
in
$\mathbf{P}^{n}(\mathbb{C})$
. Let
$S$
be the boundary of
$T$
. Let
$\unicode[STIX]{x1D70C}:T\rightarrow M$
be the (topological) projection mapFootnote
9
. Then
$\unicode[STIX]{x1D70C}$
restricted to
$S$
is an
$S^{1}$
-bundle over
$M$
, where
$S^{1}$
is a circle. If
$j_{S}:S{\hookrightarrow}U$
is the inclusion map, then the residue map
$$\begin{eqnarray}\text{Res}:\unicode[STIX]{x1D6FA}^{k}(U)\rightarrow \unicode[STIX]{x1D6FA}^{k-1}(M),\quad k=1,\ldots ,n,\end{eqnarray}$$
is defined by
$$\begin{eqnarray}\text{Res}:=\unicode[STIX]{x1D70C}_{\ast }\circ j_{S}^{\ast },\end{eqnarray}$$
where
$j_{S}^{\ast }$
is the natural pull-back of differential forms and
$\unicode[STIX]{x1D70C}_{\ast }$
is integration along the fibres of
$\unicode[STIX]{x1D70C}:S\rightarrow M$
. This residue map is well defined on the level of differential forms; Dimca considered this phenomenon surprising (see [Reference Dimca7, p. 96] for details). Moreover, it commutes with exterior differentials, i.e.
$$\begin{eqnarray}\text{Res}\circ d=d\circ \text{Res},\end{eqnarray}$$
and thus
$\text{Res}$
induces a map on the level of cohomology classes. In the seminal papers [Reference Griffiths8], Griffiths showed that
$\text{Res}$
factors through an isomorphism
$H_{\text{dR}}^{k}(U/\mathbb{C})\rightarrow \mathbf{PH}^{k-1}$
, where
$\mathbf{PH}^{k-1}$
is the primitive part of
$H^{k-1}(M,\mathbb{C})$
.
For a given choice of the holomorphic volume form
$\unicode[STIX]{x1D6FA}$
, since the contraction operator
$$\begin{eqnarray}\unicode[STIX]{x1D6E4}(M,\wedge ^{r}T_{M})\rightarrow \unicode[STIX]{x1D6E4}(M,\wedge ^{m-r}T_{M}^{\ast }),\quad X\mapsto X\vdash \unicode[STIX]{x1D6FA}\end{eqnarray}$$
is an isomorphism of
$\mathbb{C}$
-vector spaces, the induced contraction operator
$$\begin{eqnarray}g_{\unicode[STIX]{x1D6FA}}:\mathbf{t}^{k}\rightarrow \unicode[STIX]{x1D6FA}^{k+m}(M),\quad \unicode[STIX]{x1D6FE}\mapsto \unicode[STIX]{x1D6FE}\vdash \unicode[STIX]{x1D6FA}\end{eqnarray}$$
is an isomorphism of
$\mathbb{C}$
-vector spaces. It is clear that this map
$g_{\unicode[STIX]{x1D6FA}}$
sends
$\bar{\unicode[STIX]{x2202}}$
and
$D$
to
$\bar{\unicode[STIX]{x2202}}$
and
$\unicode[STIX]{x2202}$
, respectively, from the definition (3). Define
$f_{\unicode[STIX]{x1D6FA}}$
to be the inverse of
$g_{\unicode[STIX]{x1D6FA}}$
:
$$\begin{eqnarray}f_{\unicode[STIX]{x1D6FA}}=g_{\unicode[STIX]{x1D6FA}}^{-1}:\unicode[STIX]{x1D6FA}^{k}(M)\rightarrow \mathbf{t}^{k-m}.\end{eqnarray}$$
Now we define a map
$\unicode[STIX]{x1D6E9}$
:
$$\begin{eqnarray}\unicode[STIX]{x1D6E9}:=f_{\unicode[STIX]{x1D6FA}}\circ \text{Res}\circ \unicode[STIX]{x1D6FD}:{\mathcal{A}}_{U}^{k}\rightarrow \mathbf{t}^{k},\quad k=-m,-m+1,\ldots ,0.\end{eqnarray}$$
By construction, we have
$$\begin{eqnarray}(f_{\unicode[STIX]{x1D6FA}}\circ \text{Res})\circ d=(\bar{\unicode[STIX]{x2202}}+D)\circ (f_{\unicode[STIX]{x1D6FA}}\circ \text{Res})\quad \text{ and }\quad \unicode[STIX]{x1D6FD}\circ K_{U}=d\circ \unicode[STIX]{x1D6FD},\end{eqnarray}$$
which, in turn, implies that
$$\begin{eqnarray}(\bar{\unicode[STIX]{x2202}}+D)\circ \unicode[STIX]{x1D6E9}=\unicode[STIX]{x1D6E9}\circ K_{U}.\end{eqnarray}$$
In summary, the cochain map
$\unicode[STIX]{x1D6E9}:({\mathcal{A}}_{U},K_{U})\rightarrow (\mathbf{t},\bar{\unicode[STIX]{x2202}}+D)$
preserves degree and
$(\bar{\unicode[STIX]{x2202}}+D)\circ \unicode[STIX]{x1D6E9}=\unicode[STIX]{x1D6E9}\circ K_{U}$
. Moreover, it is clear that
$\unicode[STIX]{x1D6E9}$
induces an isomorphism
$H^{k}({\mathcal{A}}_{U},K_{U})\simeq H_{pr}^{k}(\mathbf{t},\bar{\unicode[STIX]{x2202}}+D)$
for
$k=-m,-m+1,\ldots ,0$
from Theorem 2.3, the definition of
$f_{\unicode[STIX]{x1D6FA}}$
and the property (15) of
$\text{Res}$
. This finishes the proof of Theorem 1.4.
Acknowledgements
The authors thank the anonymous referee for several valuable comments that improved the quality of the paper. The work of Jeehoon Park was supported by the Samsung Science & Technology Foundation (SSTF-BA1502).
