Mirror symmetry and automorphisms
Alessandro Chiodo Elana Kalashnikov
math.AG


A Guide to the Bott Index and Localizer Index
    Terry A. Loring
    
=============================================






 
We show that there is an extra dimension to the mirror duality
discovered in the early nineties by Greene–Plesser and Berglund–Hübsch. Their duality matches cohomology classes  of  
two  Calabi–Yau orbifolds. When both orbifolds are equipped with an automorphism s of the same order, our mirror duality involves the weight of the action of s^* 
on cohomology. In particular it matches  the respective s-fixed loci,
which are not Calabi–Yau in general.
When applied to K3 surfaces with non-symplectic automorphism s of odd prime order, this 
provides a proof that Berglund–Hübsch mirror symmetry implies K3 lattice mirror symmetry replacing earlier case-by-case treatments.








 











§ INTRODUCTION
 
The earliest formulation of mirror symmetry relates pairs of d-dimensional Calabi–Yau manifolds X, X^∨ with mirror Hodge diamonds: 

    h^p,q(X)=h^d-p,q(X^∨).

In the early 1990s, physicists Greene, Morrison, and Plesser found many such mirror pairs <cit.>, starting with a Calabi–Yau (and Fermat) hypersurface in projective space and constructing a mirror, which is a resolution of the quotient of the same hypersurface by a finite group. In 1992, this construction was generalized by Berglund–Hübsch <cit.>, starting with a Calabi–Yau given as a quotient of a more general hypersurface in weighted projective spaces by a finite group. The hypersurface is a Calabi–Yau orbifold defined as the zero locus of a quasi-homogenous polynomial W=∑_i=0^n ∏_j=0^n x_j^m_ij such that W is non-degenerate and “invertible” (i.e. with as many variables as monomials). After quotienting out by a finite group H of diagonal symmetries within (n+1;) one obtains 
the orbifold Σ_W,H.
 The mirror Σ_W^∨,H^∨ is another such quotient of a hypersurface modulo a finite group.  The  hypersurface is given by the polynomial W^∨, defined by transposing the matrix of the exponents E=[m_ij] of W. The group H^∨ is a subgroup of (n+1;) Cartier dual to H and 
preserving W^∨, see (<ref>). Then, the mirror 
duality can be stated in terms of orbifold Chen–Ruan cohomology as 

    H_^p,q(Σ_W,H;)=H_^d-p,q(Σ_W^∨,H^∨;),

which implies the same relation in ordinary cohomology whenever there exists crepant resolutions.








The striking mirror relation above becomes elementary when 
we look at it through the lenses of singularity theory or, in 
physics terminology, the Landau–Ginzburg (LG) model. 
This happens because mirror symmetry holds for LG models without any 
Calabi–Yau condition. 
In this paper we present this change of perspective through the LG model via 
the crepant resolution of a singularity, see Section <ref>. 
This not only allows us to simplify previous proofs of 
LG/CY correspondence by the first author 
with Ruan <cit.>; it also yields a new statement 
of mirror symmetry 





relating the fixed loci of powers of 
an isomorphism s of Σ, the Hodge decomposition, and the 
weights the representation s^* in cohomology. 


Let W=x_0^k+f(x_1,…,x_n) be a non-degenerate,  quasi-homogenous, invertible polynomial. 
Let us consider again the automorphisms groups H⊆ W and its dual H^∨∈ W within
(n+1;). The 
Calabi–Yau orbifolds Σ_W,H, Σ_W^∨,H^∨ are equipped with the 
action by the group _k of kth roots of unity spanned by 
s x_0 ↦ e^2π i/k x_0. For i in 
the group of characters /k=(_k;_m) we 
consider the weight-i term of cohomology 

    H^*(  ,)_i={x| s^*x=i(s)x}.

The first statement is that 
the s-invariant cohomology mirrors the “moving” cohomology: the sum of 
all cycles of nonvanishing weight.

Theorem A (see Thm <ref>, part 1). 
Consider the mirror pair sΣ_W,H→Σ_W,H 
and sΣ_W^∨,H^∨→Σ_W^∨,H^∨. We have

    H^p,q_orb(Σ_W,H;)_0≅⊕_i=1^k-1 H^d-p,q_orb(Σ_W^∨,H^∨;)_i,

where d=n-1 is the dimension of Σ_W,H.


The locus of  geometric points of  Σ_W,H which are fixed by s 
also exhibits a mirror phenomenon. Since Σ_W,H is a stack, let us provide a definition for this s-fixed locus. 
For s a finite order automorphism acting on a smooth Deligne–Mumford orbifold,
we consider the graph of Γ_s𝔛→𝔛×𝔛 
and its intersection with the graph of the identity (the diagonal morphism)

    𝔛s,  𝔛×𝔛,  𝕀×𝔛,

(we write s and 𝕀 instead of the respective graphs).  
We recall that orbifold cohomology is simply the (age-shifted) cohomology of this product for
s=𝕀. The s-orbifold cohomology is defined as the age-shifted cohomology of the above 
fibred product in general (see Defn. <ref>). 
This is a bi-graded vector space and, if the coarse space X of X admits a crepant resolution X where s lifts, there is a bidegree-preserving isomorphism
H_s^*( X;)≅ H_s^*( X;),
where the right hand side is the age-shifted cohomology of the s-fixed locus in X, see Prop. <ref>.



Under the same conditions on W and H as above, set Σ=Σ_W,H and Σ^∨=Σ_W^∨,H^∨. If the order k of s is not prime, then s acts non-trivially on the fixed locus of powers of s. The s-moving cohomology of the fixed locus of powers of s mirrors the same on Σ^∨, interweaving the weight and the exponent of the power of s. 

Theorem B (see Thm <ref>, part 3).
Let b,t≠ 0. Then, we have 

    H^p,q_s^b(Σ)(b/k)_t≅ 
    H^d-p,q_s^-t(Σ^∨)(k-t/k)_-b,

where d=n-2, 
the largest dimension of the components of the s-fixed locus.


Finally, also the fixed cohomology of each power s^j  exhibits a mirror phenomenon, but only after adding certain moving cycles in Σ. Namely, the cycles we add are
all those whose weight differs from  0 (i.e. moving cycles) and from j (the exponent of s). 
We denote this group 
by H^p,q_𝕀,j(Σ), see (<ref>). 

Theorem C (see Thm <ref>, part 2).

For 0<j<k, we have

    [H^p,q_s^j(Σ)(jk)]^s ⊕H^p,q_𝕀,j(Σ)≅[H^d-p,q_s^j(Σ^∨)(jk)]^s ⊕H_𝕀,j^d-p,q(Σ^∨).

The correcting terms H^* disappear 
when k=2 (for k=2, we have s^j=s and there 
is no positive weight except 1). This shows how the statement above specialises 
to the construction of Borcea–Voisin mirror pairs (see <cit.>).


In dimension 2, and after resolving, these results are about mirror symmetry for K3 surfaces with non-symplectic automorphisms. Suppose X and X^∨ are crepant resolutions of Σ_W,H and Σ_W^∨,H^∨ where W is a polynomial in 4 variables. The above mirror theorems imply that the topological invariants of the fixed locus of the K3 surface X controls that of X^∨; we refer to Corollary <ref> for simple formulae on the number of fixed points and the genera of the fixed curves. 
The automorphism s also gives the K3 surface a lattice polarization: H^2(X,)^s. There is another version of mirror symmetry for lattice polarized K3 surfaces, arising from the work of Nikulin <cit.>, Dolgachev <cit.>, Voisin <cit.>, and Borcea <cit.>. When the order of s is odd and prime, this lattice is characterised by the invariants (r,a): the rank and the discriminant. Families of lattice polarized K3 surfaces come in mirror pairs, and in the odd prime case this mirror symmetry takes a lattice with invariants (r,a) to (20-r,a). 
The following corollary is a theorem of Comparin, Lyons, Priddis, and Suggs <cit.> proven by case-by-case analysis. Here, it is shown directly from 
the above statements (see Thm. <ref>).

Corollary (<cit.>). 
Let p be prime and different from 2. Let Σ_W,H and Σ_W^∨,H^∨ be mirror K3 orbifolds with order-p automorphisms s, s^∨, and let Σ and Σ^∨ be crepant resolutions with automorphisms also denoted s, s^∨. Then  Σ and Σ^∨ are mirror as lattice polarized K3 surfaces. 





 §.§ Relation to previous work

This paper generalises the results of <cit.>. There, only involutions were considered;  here the mirror theorems apply to automorphism of any order. There, Theorems A and C are simpler (invariant classes mirror anti-invariant classes in Theorem A and no extra terms appear in Theorem C). Theorem B does not apply in the involution case. In the above Corollary, we do not consider the order-2 case treated in <cit.>; in the present paper this allows us to deduce the lattice mirror symmetry statement of <cit.> in full. 

Section <ref> restates and recasts the proof of 
 mirror symmetry through LG models and the correspondence
between cohomology and LG models in terms of resolutions of singularities (see Theorem <ref>). 
This may be regarded as the outcome of the work of many authors, we refer to 
<cit.>,  <cit.>,  <cit.>
 <cit.>, <cit.>, <cit.> and <cit.> and <cit.> 
 validating over the years the  
 approach of the physicists Intriligator–Vafa <cit.> and Witten <cit.>.
It is also worth mentioning that the main object of our study, a polynomial W=x_0^k + f(x_1,...,x_n) with the cyclic symmetry group of kth roots of unity acting on x_0, was used in Varchenko's proof of semicontinuity of 
Steenbrink's spectra of singularities (<cit.> and <cit.>). 
We hope that this may lead to further
explanations of mirror symmetry in the framework of singularity theory. 
In particular, our setup only concerns hypersurfaces in weighted projective space, it would be interesting to see if it extends to other contexts 
where mirror constructions are known. 

Finally, it is worth mentioning that the work of Comparin, Lyons, Priddis, and Suggs <cit.>, earlier work of Artebani, Boissière and Sarti <cit.> and more generally Nikulin's classification <cit.> yield 
several tables summarising 
explicit treatments of K3 surfaces via resolution of singularities. 
Much of these data are now 
embodied into the s-weighted Hodge numbers of Theorems A, B, and C. We provide some 
examples for this in the tables at the end of <ref>.







 §.§ Structure of the paper
 
Section <ref> states notation and terminology. 
Section <ref> presents the Berglund–Hübsch mirror symmetry construction.
Section <ref>  sets up our generalisation of 
orbifold cohomology sensing the s-fixed locus: s-orbifold cohomology.
Section <ref> illustrates and reproves the transition to Landau-Ginzburg models which is crucial in the proof. In particular it  provides a  straightforward description of the LG/CY correspondence from 
the crepant resolution conjecture without using the combinatorial model of <cit.>.
Section <ref> is the technical heart of the paper; it proves 
the main theorem (Theorem <ref>) on the LG side. 
Section <ref> translates the result from the 
LG side  to the CY side. It contains Theorem <ref>
proving the statements A, B, and C  and 
Theorem <ref> specialising to K3 surfaces.



 §.§ Aknowledgements

We are grateful to Behrang Noohi, whose explanations 
clarified inertia stacks to us. 
We are grateful to Baohua Fu, Lie Fu, Dan Israel, and Takehiko Yasuda for many 
helpful conversations. We thank Davide Cesare Veniani, with
whom we started studying involutions of Calabi–Yau orbifolds, for continuing to share his insights and
expertise. 



§ TERMINOLOGY
 
Deligne–Mumford orbifolds are smooth separated Deligne–Mumford stacks
with a dense open subset isomorphic to an algebraic variety. 


 §.§ Conventions

We work with schemes and stacks over the complex numbers. All schemes are Noetherian and separated.
By linear algebraic group we mean a closed subgroup of 
_m() for some m.
We often need to identify 
a stack locally. 
In order to avoid repeated mention of 
étale localization or strict Henselizations, we often use the expression 
“the 
local picture of the stack X at the geometric point x∈ X is the same as U at u∈ U”. 
By this we mean that the strict Henselization of X at x is the same of that 
of U at u. 
Often it is enough to say that there is an étale neighbourhood X' of x 
 and an isomorphism X'→ U' with an étale neighbourhood U' of u.
We refer to <cit.> for a definition of the 
strict Henselization 
and to <cit.> for further discussion (see in particular the “algebra-to-analysis translation”, where strict Henselizations 
are described analytically as the germ of X at x). 






 §.§ Notation
 We list here notation that occurs throughout the entire paper. 





 


V^K    is the invariant subspace of a vector space V linearized by a finite group K.

( w)     is the quotient stack [(^n∖ 0)/_m], where _m acts with weights w.

Z(f)     is the variety defined as 
 zero locus of f∈[x_1,…, x_n].

H(a,b)    is the bigraded vector space with shifted grading: [H(a,b)]^p,q=H^p+a,q+b.

 




 We add the subscript 
( w) when we refer to the zero locus in ( w) of a polynomial f which is w-weighted homogeneous. In this way we have

    Z_( w)(f)=[U/_m],      with U=Z(f)⊂^n∖ 0.




We often write H(a) for H(a,a).
 


We only consider cohomology with  coefficients; therefore, we sometimes write H^*(X;) as H^*(X).



Given an automorphism α of 𝔛, 
we write Γ_α for 
the graph 𝔛→𝔛×𝔛. However, to simplify 
formulæ, 
we often abuse notation 
and use α for the graph Γ_ as well as the automorphism. In this way, in subscripts, the diagonal 
Δ𝔛→𝔛×𝔛 will be often written as 𝕀_𝔛 or simply 𝕀.





§ SETUP

We recall the general setup of non-degenerate polynomials P where
the theory of Jacobi rings applies. Then we introduce polynomials 
of the special form 
    W(x_0,x_1,…,x_n)=x_0^k+f(x_1,…, x_n)

for n>0.



 §.§ Non-degenerate polynomials

We consider quasi-homogeneous polynomials P of degree d and 
of weights w_0,…,w_n

    P(^w_0x_0,…,^w_nx_n)=^dP(x_0,…,x_n),

for all ∈.
We assume that that the polynomial P is non-degenerate; i.e.
the choice of weights and 
degree is unique and  the partial derivatives of W vanish 
simultaneously only at the origin. 
We consider the zero locus 

    Σ_P=Z_( w)(P)⊂( w)
 
which is, by non-degeneracy, a smooth hypersurface within 
the weighted projective stack ( w)=
[(^n+1∖ 0)/_m] with _m acting 
with weights  w_0,…,w_n.
The polynomial is of Calabi-Yau type if 

    ∑_i=0^nw_i=d.

This implies that the canonical bundle of Σ_P
is trivial; we refer to Σ_P as a 
Calabi–Yau orbifold.



Because P is non-degenerate, 
the group of its diagonal automorphisms

    _P={(_0,…,_n)| P(_0x_0,…, _nx_n)=P(x_0,…,x_n)}

 is finite. Indeed, the n× h matrix E=(m_i,j) of the exponents of 
P=∑_i=1^h c_i ∏_j=0^n x_j^m_i,j
is left invertible as a consequence of the uniqueness of the 
vector (w_i/d)_i=0^n=E^-1 1.
Since we are working over , we adopt the notation 

    [a_0,…,a_n]=(exp(2π a_i))_i=0^n

for a_i∈∩ [0,1[.
The age of the diagonal matrix above is 

    age[a_0,…,a_n]=∑_i=0^na_i.

The distinguished diagonal symmetry 

    j_P=[w_0/d,…,w_n/d],

usually denoted by j, spans the intersection _P∩_m, where 
_m is the group of automorphisms of 
the form (^w_0,…,^w_n).
The automorphism j_P is the monodromy operator of the fibration 
defined by 
W restricted to the complement in ^n+1 of the 
zero locus Z(P); we will denote by M_P the generic Milnor fibre

    @R=.5pc
    M_P[rr][dd]       ^n+1∖ Z(P)[dd]^P
       □   
    
    t≠0 [rr]      ^×.


For any subgroup H of _W containing j_P we consider the 
Deligne–Mumford stacks 
    Σ_P,H=[Σ_P/H_0],     M_P,H=[M_P/H];

where H_0=H/(H∩_m)=H/⟨ j⟩ and 
acts faithfully on Σ_P. The orbifold Σ_P,H 
is a smooth codimension-1 substack of [( w)/H_0]

    Σ_P,H⊂ [( w)/H_0],

and has trivial canonical bundle as soon as P is Calabi–Yau and 
H lies in 

    _P:=_P∩(n+1;).
  




 §.§ Polynomials with automorphism

More specifically, we focus on 
polynomials of Calabi–Yau type of the form 

    W(x_0,x_1,…,x_n)=(x_0)^k+f(x_1,…,x_n)

We have 
_W=_k×_f, 
where, using again the choice exp(2π i/k), the first factor is regarded here 
as /k, canonically 
generated by 
the order-k automorphism 

    s=[1/k,0,…,0].

We have a /k-action on the stack Σ_W,H 

    𝗆/k×Σ_W,H→Σ_W,H.

We have  j_W=s· j_f; where j_f is regarded as an element of 
_f. 

Instead of H⊆ W∩(n+1;) 
containing j_W, we can equivalently work with subgroups  
K ⊂_f satisfying 

    (j_f)^k∈ K⊆_f,

(we recover H by considering the subgroup of W spanned by  j_W and K).
More generally we consider the subgroup of _W  

    K[j_W,s]=∑_a,b=0^k-1 (j_W)^a(s)^bK,
 with its natural  
 (1/k/)-gradings 
 
    d_j=a/k,     d_s=b/k.

By (<ref>), exp(2π d_s(g)) is the determinant of an 
element g∈ K[j_W,s]. 



§ INERTIA
  
We consider a finite group G acting on a Deligne–Mumford orbifold X 

    𝗆 G× X → X.

We consider the G-inertia stack I_G( X)
fitting in the following fibre diagram

    @R=.5pc
    I_G( X)[rr][dd]        X [dd]_Δ
       □   
    
    G× X [rr]_(𝗆,𝗉𝗋_2)       X× X

When G is a trivial group, I X is the ordinary inertia of X. 
There is a locally constant function 
    a I X→

which assigns to each geometric point (x∈ X, g∈(x))∈ I X
the rational number (g) (see  <cit.>
and <cit.>). 


In this way we have 

    I_G( X)=(G× X)×_(𝗆,𝗉𝗋_2), 𝔛×𝔛, Δ X=_g∈ G I_g( X)

where I_g( X) is the g-inertia orbifold 

    I_g( X):=𝔛g,  𝔛×𝔛,  𝕀×𝔛.
 

The G-inertia stack of X
fits in the fibre diagram

    @R=.5pc
    I_G( X)[rr] [dd]^𝗉       X [dd]
       □   
    
     I[ X/ G][rr]        [ X/G]

and we may  regard 𝗉 as a G-torsor by pullback of X→ [ X/G]. 

The G-action on _g∈ G I_g X 
is given by conjugation on the indices and by F_h  I_g  X →  
I_hgh^-1 X 
on the components, where F_h acts by the effect 
of h on the first factor of (<ref>)
and by the identity on the second. 



 §.§ A g-orbifolded cohomology

The cohomology of the g-inertia stack coincides with the cohomology of 
the g-fixed locus when X is representable. 
In the spirit of orbifold cohomology we define 
 g-orbifold cohomology groups 
which are invariant under K-equivalence. 

The g-orbifold cohomology is the cohomology of (<ref>)
shifted by the locally constant function “age” given by 

    a_g I_g( X)→ I[ X/G].

We assume that X is smooth, so that I_g( X) is smooth and 
all coarse spaces are quasi-smooth; in particular cohomology groups admit a Hodge
decomposition. 
Starting from a Hodge
decomposition of weight n, for any r∈, we can produce a new 
decomposition of weight n-2r via 
H(r)^p,q=H^p+r,p+r. We will denote by  
(r) the isomorphism induced by the identity at the level 
of the vector spaces; it identifies the Hodge decomposition 
of weight n with the Hodge decomposition of weight n-2r.
 
 
 We can now provide the definition of g-orbifolded cohomology. 


For any g∈ G the g-orbifold cohomology is defined as

    H_g^*( X;)=H^*(I_g( X);)(- a_g).



We point out the slight abuse of notation:  is not constant in 
general, but, since it is locally constant, the shift operates independently on each cohomology group arising from each connected component. A  precise notation should read 

    H^p,q(  ;)(𝔞)=⊕_r∈_≥ 0H^p,q(𝔞^-1(r);)(-r).



For g=𝕀=1_G, the above definition coincides with Chen–Ruan orbifold cohomology

    H_𝕀^*( X;)=H_CR^*( X;).

In this paper, we often consider the relative version of orbifold Chen–Ruan cohomology; indeed 
when Z is a substack of X then I( Z) is a substack of I( X) 
and we set 

    H_𝕀^*( X, Z;)=H^*(I( X),I( Z);)(- a_𝕀),

where a_𝕀 is the age function on I( X).

Yasuda <cit.> proves the invariance of 
the Hodge decomposition of Chen–Ruan cohomology 
of smooth Deligne–Mumford stacks 𝔛 and 𝔜 
whenever
there exists a smooth and proper Deligne–Mumuford stack Z with birational 
morphisms Z→ X and Z→ Y 
with ω_ Z/ X≅ω_ Z/ X. 
In particular, for Gorenstein orbifolds 
 Chen–Ruan cohomology coincides with 
the cohomology of any crepant resolution of the 
coarse space. 
Furthermore, we have the following proposition.

Let G by a finite group acting on a Gorenstein orbifold X. Let us assume that
the coarse space X of X admits a crepant resolution X where we 
can lift the G-action 
induced by X on X. Then, for any g∈ G we have 
a bidegree-preserving isomorphism 

    H_g^*( X;)≅ H_g^*( X;).

In particular, the isomorphism identifies H_g^*( X;) with 
H^*( X_g;)(- a_g),
where a_g is the composite X_g→ [ X/G] and of the age function 
[ X/G]→. 
 

The stack X and its resolution X are K-equivalent.
In order to see this, we  consider the 
 Z= X× _X  X and the associated reduced stack. 
Then, there exists a proper birational morphism Z' → Z such that Z' is smooth. 
This is explained in Sect. 4.5, 2, of Yasuda's paper <cit.> 
(this is essentially due to Villamayor papers <cit.> and <cit.> showing  the existence of 
resolutions compatible with smooth, in particular étale, morphisms). Actually, 
in his recent generalization <cit.>, Yasuda 
proves that it suffices to consider the reduction and the 
normalization of Z', without
 any resolution. This happens because his new statements allows us to
extend the definition 
of orbifold cohomology
to singular or wild (in positive characteristic) 
Deligne–Mumford stacks. 


Now we consider the abelian group H=⟨ g⟩. Then 
A'=[ X/H] and 
A”=[ X/H] are K-equivalent by the same argument. Indeed the action of H 
descends compatibly to 
the coarse space X and we can consider the stack A=[X/H] and 
the morphisms A'→ A and A”→ A. Then, the reduced stack associated
to the fibred product 
A'×_ A A” can be resolved and yields a smooth Deligne–Mumford 
stack Z mapping to A' and A”. As above, the fact that 
the canonical bundles of 
X and X are the pullback of ω_X is enough to show that Z→ A'=[ X/H] and Z→ A'=[ X/H] is a K-equivalence.



The desired claim follows because the 
cohomology of I_g X and that of I_g X appear as 
summands of the Chen–Ruan cohomology groups of [ X/H] and of [ X/H]. 
Indeed they arise as the cohomology groups of the 
sectors attached to g whose cohomology are 
the g-invariant classes of I_g( X) and I_g( X). 
Since g operates trivially on these sectors, we can regard these contributions as H^*(I_g( X); )  and H^*(I_g( X);). We should further mention 
that we obtain an identification at the level of the age-shifted g-orbifolded cohomology H_g^*(  ;) due to the fact that 
the age  is a rational function factoring through the 
age function of [ X/H] and of [ X/H].


The proposition above only claims the existence of an isomorphism. In special cases 
in dimension 2 we have proven the existence of an explicit isomorphism, see <cit.>.  

In special cases where 
a_g is constant, the above theorem 
allows us to relate the g-orbifold cohomology to the cohomology
of the g-fixed locus of the resolution via a constant shift by a_g.
The following example generalises the case of 
anti-symplectic involutions of orbifold K3 surfaces considered in <cit.> 
(this case occurs below for k=2).

Consider a proper, 
smooth, Gorenstein, Deligne–Mumford orbifold X of dimension 2 satisfying the
 Calabi–Yau condition ω≅. We refer to this as a K3 orbifold because 
 there exists a minimal resolution X which is a K3 surface. 
 Consider the volume form Ω of X, which descends on X. 
 We assume that g is an order-k automorphism of X whose induced action on  Ω 
 is multiplication by e^2 π i (k-1)/k. Then, 
 g naturally lifts to the minimal resolution X; furthermore, locally at each fixed point of X,
 the action of g can be written as 1/k[a,b] with a+b=k-1 (this happens because 
 the case a+b=2k-1 is impossible). In this way the age shift 𝔞_g at the fixed loci always equals 1-1/k 

    H_g^*( X;)=H^*( X_g ;)(1/k-1).








§ LANDAU–GINZBURG STATE SPACE
 
The expression “Landau–Ginzburg” comes from 
physics and 
is often used for -valued functions defined 
on vector spaces possibly equipped with the action of a group. More generally
the definition is 
extended to vector bundles on a stack. 
In this paper we only use it for the above setup P [^n+1/H]→,
where P is a non-degenerate polynomial and j∈ H⊆_P. 
Indeed this may be regarded as a -valued function defined on 
a rank-(n+1) vector bundle on the stack BH=[/H]. 
We show how this geometric setup is 
naturally connected to Σ_W,H via K-equivalence. 



 §.§ K-equivalence

Consider the rank-(n+1) vector bundle 

    =_(d)(-w_0)⊕…⊕_(d)(-w_n)=[^n+1/⟨ j⟩];

its coarse space X=^n+1/⟨ j⟩, and the smooth Deligne–Mumford stack  

    𝕃=_( w)(-d)→ X,

total space  of the line bundle of degree -d on ( w). The 
stacks  and  are the  two GIT quotients of ×^n+1 modulo _m operating with weights (-d,w_0,…,w_n+1).
Notice that  without the origin coincides with the line bundle 
 without the zero section: ^× =^×. 

We assume that P is of Calabi–Yau type in the sense of (<ref>). 
Then, the canonical bundle of  descends to X and its pullback 
to  coincides with ω_. 
Following the same argument as above, by Yasuda <cit.>,
we have 

    Φ H^p,q_g(;) H^p,q_g(;)

for any p,q∈ and for any g∈_W. 

The isomorphism H^d_g(;)→ H^d_g(;) 
is not canonical; notice, however, that 
we can at least impose a compatibility with respect to the 
restrictions 
H^d_g(;)→ H^d_g(;) and 
H^d_g(;) → H^d_g(;)  
for =[M_P/⟨ j ⟩] included in ^× = ^×⊆,.
This happens because the fundamental classes of the 
inertia stacks I_g(), I_g() and I_g() 
attached to the same automorphism 
β=g·(^-d,^w_1,…,^w_n) with
 ∈∪_j=1^m _w_j can be identified
 since their bidegree equal ((β),(β)) by construction. In this way,
 we can require that (<ref>)
 respects the canonical identification between the fundamental 
 classes of I_g() and I_g() and this is enough to 
 insure that the following diagram commutes

    @C=1.1pc…[r] 
       H^d_g(,;)[r][d]    H^d_g(;)[d]_Φ[r]    
    H^d_g(;)[d]_=[r]    H^d+1_g(,;)[r][d]   …
    …[r]   H^d_g(,;)[r]    H^d_g(;) [r]    
    H^d_g(;)[r]    H^d+1_g(,;)[r]   …

 and yields a bidegree-preserving isomorphism 
 
    H^d_g(,;)≅ H^d_g(,;).

 Note that  can be regarded as 
 the generic fibre of P→ as well as the 
 generic fibre of P→. 
 If we consider any group H⊆_P containing j
 we can apply the above claim to _H=[/H_0], 
 _H=[/H_0] and _P,H
 =[/H_0]. We get 
 
    H^p,q_g(_H,_P,H;)≅ H^p,q_g(_H,_P,H;)

 for any p,q and for any g∈_W. 
 
   The left hand side is naturally identified via the Thom isomorphism to the Chen–Ruan cohomology of 
  Σ_P,H up to a (-1)-shift whereas the left hand side is naturally identified to 
  an orbifold version of the Jacobi ring known as the FJRW 
  or Landau–Ginzburg state space.
 We detail these two aspects in the next two sections.




 §.§ Thom isomorphism

Consider P→
and its generic fibre =P^-1(t) for t≠ 0.
We have an isomorphism of Hodge structures

    H^*(, ;)≅ H^*(Σ_P;)(-1).

This happens because the left hand side can be regarded after retraction as 

    H^*(( w),( w)∖Σ_P;)
 which is isomorphic 
to the (-1)-shifted cohomology of Σ_P by the Thom isomorphism. 


Equation (<ref>) suggests that the orbifold cohomology H^p,q_𝕀(_H,_P,H;) is 
related to the orbifold cohomology of Σ_P. 
However, the argument above does not yield an isomorphism respecting the 
orbifold cohomology bidegree. 
This happens because 

    H^*(_H,_P,H;)≅ H^*(( w),( w)∖Σ_P;)≅ H^*(Σ_P;)(-1)

may fail in orbifold cohomology. 
However the first author and Nagel proved that  
Equation (<ref>) holds in orbifold cohomology without changes even 
when the Thom isomorphism
H_𝕀^*(( w),( w)∖Σ_P;)≅ H_𝕀^*(Σ_P;)(-1)
does not. We regard 

    H_𝕀^*(, ;)≅ H_𝕀^*(Σ_P;)(-1)

as the correct formulation of Thom isomorphism in orbifold cohomology. We refer to 
the theorem below.

For the benefit of the reader we illustrate the study 
of H_𝕀^*(( w),( w)∖Σ_P;) and 
of H_𝕀^*(Σ_P;)(-1)
sector-by-sector. We 
distinguish two cases. 
For ordinary sectors (such as the untwisted sector) 
Σ_P,g is a codimension-1 
hypersurface in ( w_g). Then, there is an identification 
H^*(( w_g),( w_g)∖Σ_P,g;)≅ H^*(Σ_P,g;)(-1) and the age shift of ( w_g) coincides with that of Σ_P,g since 
g acts trivially on the normal bundle N_Σ_P,g/( w_g)). 
On the other hand, it may  happen 
that ( w_g) and Σ_P,g coincide as we illustrate in Example <ref>. 
In these cases we have 
H^*(( w_g),( w_g)∖Σ_P,g;)=
H^*(( w_g);)≅ H^*(Σ_P,g;)
without Tate shift (-1).
Furthermore the difference between the age shift of the sector 
( w_g) and that of Σ_P,g is strictly positive: it equals the age 
q∈ ]0,1[ of the character 
operating via g on N_Σ_P,g/( w_g)). 
In these cases we have  
H^*(( w_g),( w_g)∖Σ_P,g;)≅ H^*(Σ_P,g;)(-q). It is now possible to observe that we have 

    H^*_𝕀(_g,_g∖Σ_P,g;)≅ H_𝕀^*(Σ_P,g;)(-q-(1-q))=
    H_𝕀^*(Σ_P,g;)(-1)

as desired. The result is proven in 
<cit.> for all complete intersections. We get 
 
For any H⊆_P containing j_P and g∈_P and 
for any p,q∈, we have 

    H^p,q_g(_H, _P,H;)≅ H^p,q_g(Σ_P,H;)(-1).



The following example is added here in the sake of clarity, 
but plays no essential role in the rest of the text; 
it illustrates in a simple way the issue arising for 
a Calabi–Yau embedded in a nonGorenstein ( w).

Consider the hypersurface Σ defined by x^3+xy=0 in (1,2). It consists of 
two points, the orbit p=(-x^2,x) and the point (x=0)=(2). 
We have I(Σ)=Σ_0⊔(2)_1 and I((1,2))=(1,2)_0⊔(2)_1, 
where the label j=0,1 denotes the root of unity exp(2π  j/2) 
acting as (,^2) with nonempty fixed locus. The orbifold cohomology of Σ is 
^3 concentrated in degree (0,0) (the age-shift does not intervene
for a 0-dimensional stack). Since the 
Thom isomorphism holds after a (-1)-shift, we compare 
^3(-1) to 
    H^*_𝕀((1,2),(1,2)∖Σ)=
    ^2(-1)⊕ H^*((2))(-1/2)≅^2(-1)⊕(-1/2).
 
On the other hand, we set  via P=x^3+xy as above; then ^3(-1) matches  

    H^*_𝕀(,)=
    ^2(-1)⊕ H^*((2))(- 1)≅^3(-1).
 
We refer to <cit.>.




 
The ambient cohomology of Σ_P,H is Poincaré dual to the image of the 
the homology of 
 I_g(Σ_P,H) within the homology of I_g(( w)). By the identification above 
we may regard it also as 
the image of 
the morphism 

    H^k_g(_H, _P,H;)→ H^k_g(_H;).

We can also consider the primitive cohomology 
of I_g(Σ_P,H) whose direct image vanish  
in I_g(( w)). Than the kernel of the 
morphism (<ref>) above matches 
the primitive cohomology of Σ_P,H in 
H^*(Σ_P,H;)(1)

    H_g,amb^*(Σ_W,H;)(-1)   =(H^k_g(_H, _P,H;)→ H^k_g(_H;)),
    
    H_g,prim^*(Σ_W,H;)(-1)   =(H^k_g(_H, _P,H;)→ H^k_g(_H;)).

We now turn to the LG side, where 
the image of the 
analogue morphism allows us to describe the so called “narrow” and “broad” sectors. 






 §.§ Jacobi ring
 The 
Jacobi ring 
    P = dx_0∧…∧ dx_n[x_0,…, x_n]/ 
    (∂_0P, …, ∂_n P),

regarded as 
a -vector space, has 
dimension ∏_j d-w_j/w_j (due to the non-degeneracy of the polynomial P) and 
is isomorphic to H^*(^n,M_P;).
The natural monodromy action of _d=⟨ j⟩ 
from (<ref>), and more generally 
the action of any (_0,…, _n)∈_P,

    (_0,…, _n)·(∏_j=0^n z_j^b_j-1⋀_j=0^n dz_j)
    = ∏_j=0^n  _j^b_j(∏_j=0^n z_j^b_j-1⋀_j=0^ndz_j),

allows us to write 

    [(f)^j_f]_p,q=H^p,q(;P^-1(t)),

where the subscript p,q denotes the elements 

    (∏_j=0^n z_j^b_j-1⋀_j=0^n dz_j)  with   (p,q)=(n-∑_j b_jw_j/d ,∑_j b_jw_j/d).

The above claim is due to Steenbrink <cit.>
in the present weighted homogenous setup, see also 
<cit.>. 


The action of _P on (P) is well defined because 
any automorphism (_0,…,_n) operates on 
each monomial in ∂_i P by multiplication by _i^-1. 
This happens because (_0,…,_n) fixes 
x_i∂_iP since it fixes P. 

Furthermore, the grading (p,q) is well defined 
simply because (x_i)=w_i/d yields 
a -grading on [x_0,…, x_n], 
which descends to a -grading of (P) 
because the Jacobi ideal (∂_0P, …, ∂_n P) 
is homogeneous (each monomial in ∂_i P 
has degree d-w_i/d).  



This calls for the following definition.

For a quasi-homogenous polynomial P of degree d and weight w_0,…,w_n and for any H⊆_P containing j_P,
the g-orbifolded  
Landau–Ginzburg state space is 

    ℋ_P,H,g=
    ⊕_h∈ g H( P_h)^H(-(h)),

where, for any diagonal symmetry h∈ H, we consider 
the Jacobi ring 
P_h,
where P_h is the  restriction of 
P to the ring of polynomials in the 
h-fixed variables. 

Notice that, as a consequence of the non-degeneracy of 
P, the restriction P_g 
is still a non-degenerate polynomial.


When g is the identity 
we recover the FJRW state space ℋ_P,H.








We immediately have  

    ℋ _P,H,g^p,q =H_g^p,q(_H;_P,H).


The elements h for which no variables are fixed 
yield a summand (P_h)=; these are special elements in 
the FJRW state space; they span the subspace of the 
so called narrow classes. In FJRW theory, 
the remaining summands are referred to as 
broad classes. 
In complete analogy with  
ambient cohomology, we can identify the narrow and broad classes to 
the image and the kernel of 
the morphism 

    H^k_g(_H, _P,H;)→ H^k_g(_H;).

We have

    _P,H,g^*,nar=(H^k_g(_H, _P,H;)→ H^k_g(_H;)),
    _P,H,g^*,broad=(H^k_g(_H, _P,H;)→ H^k_g(_H;)).






 §.§ Landau–Ginzburg/Calabi–Yau correspondence

The above equations (<ref>), (<ref>), and (<ref>) add up to a simple 
proof of the so called Landau–Ginzburg/Calabi–Yau correspondence based on Yasuda's theorem and K-equivalence (insured by the Calabi–Yau condition). 

For any non-degenerate 
quasi-homogeneous polynomial P of Calabi–Yau type,
for any group H⊆_P containing j_P, and for any g∈_P,
we have 

    Φ H_g^p,q(Σ_P,H;)(-1)
    _P,H,g^p,q.


Since the above isomorphism follows from 
K-equivalence, it is not explicitly given. 
In  <cit.> we provide an explicit automorphism. 
In <cit.> we generalize it to complete 
intersections.
Notice that, by a slight abuse of notation, we adopted the same notation for 
the above isomorphism as for Φ H_g^*(;)→ H_g^*(;) from 
(<ref>).













§ UNPROJECTED MIRROR SYMMETRY
 


 §.§ Mirror duality

The mirror construction due to Berglund and Hübsch <cit.> is elementary. 
It applies to non-degenerate polynomials P of invertible type, i.e.
having as many monomials as variables. Up to rescaling the variables 
these polynomials are entirely encoded by the 
exponent matrix E=(m_i,j), and are paired to a second polynomial of the same type

    P(x_0,…, x_n)=∑_i=0^n ∏_j=0^n x_j^m_i,j,     
    P^∨(x_0,…, x_n)=∑_i=0^n∏_j=0^n  x_j^m_j,i,

by transposing the matrix of exponents E. 


The square matrix E is invertible 
because the vector (w_i/d)_i=0^n is uniquely determined as a 
consequence of the 
non-degeneracy condition.
The inverse matrix E^-1=(m^i,j) allows a simple description of _P: 
the columns express the symmetries 
    ρ_j=[m^0,j,m^2,j,…,m^n,j]
 spanning 
_P. It is also easy to see that 
the columns of E express all the relations ∑_i m_i,jρ_i
among these generators. Naturally, the rows of E^-1 provide an expression 
for the symmetries ρ_i generating _P^∨ under the relations 
provided by the rows ∑_j m_i,jρ_j of E. 
In particular we have a canonical isomorphism 

    _P^∨ = (_P)^*,

where (G)^* denotes the Cartier dual (G,_m). 
The identification matches the symmetry [q_0,…, q_n] to 
the homomorphism mapping ρ_i to  
exp(2π q_i) ∈/. 
Based on this identification, for any subset S⊆_P, 
we set S^∨⊆_P^∨ as follows 

    S^∨= {∈ (_P)^* | |_S=0}.

 This is a duality exchanging subgroups of _P and 
subgroups of _P^∨; for any group H⊆_P,
we can write 

    H^∨=(_P↠(H;_m)).

It exchanges 
J_P with _P^∨.
It reverses the inclusions. 


We define the unprojected state space 

    U_P=⊕_h∈_P(P_h)(-(h))

by summing over all diagonal symmetries and without taking any invariant. 
For each summand (P_h) 
there exists an _P^∨-grading 
defined by 

    ℓ_h(P_h)   →_P^∨
    ∏_j x_j^b_j-1⋀_j dx_j   ↦∏_j ρ_j^b_j,

where all the products run over the set F_h of labels 
of variables fixed by h.
The map is well defined because each monomial in ∂_iP_h 
maps to J_P^∨(ρ_i)^-1. This happens because each monomial appearing
in ∂_iP_h dx_i 
maps to the same automorphism as each monomial appearing in P.
Furthermore, the automorphism obtained in this way is the identity
by the relation ∑_j m_i,jρ_j
discussed above. 
We can finally conclude that ∂_iP_h ⋀_j dx_j 
maps to the same automorphism as ⋀_j≠ i dx_j: namely 
∏_j≠ iρ_j=
J_P^∨(ρ_i)^-1.
In this way the unprojected state space admits a double decomposition

    U_P=⊕_h∈_P(P_h)(-(h))=
    ⊕_h∈_P ⊕_k∈_P^∨ 
    U_h^k(P),

where U_h^k(P) is the k-graded component of (P_h)(-(h)).
We write 

    U_H^K(P)=⊕_h∈ H ⊕_k∈ K 
    U_h^k(P),

for any set H⊆_P and K⊆_P^∨.
When a subscript H or a supscript K is omitted we assume 
that H or K equal _P or _P^∨.
When no ambiguity may occur, we omit the polynomial P in the notation.

The vector space 
U_H^K is the K^∨-invariant subspace of U_H

    U_H^K(P)=[U_H(P)]^K^∨.

In particular we have 

    _P,H,g= U_gH^H^∨(P).



This happens because, for any form f in (P_h)(-(h))⊆ U_P
the following equivalence holds. We have ℓ_h(f)∈ K if and only if 
f is invariant with respect to K^∨. This is just another way to phrase 
the definition of K^∨. 


 
For any h∈_P and k∈_P^∨, 
we have an explicit isomorphism 

    U_h^k(P)≅ U_k^h(P^∨),

yielding an explicit isomorphism 

    M_P U_P    ⟶ U_P^∨

mapping (p,q)-classes to (n+1-p,q) classes. 

We illustrate the isomorphism explicitly in the special case where 
P=x^k. It is elementary and it  plays a crucial role in this paper. 

Let P=x^k. Then _P equals _k (because we fix a primitive kth root ξ). 
For 1∈_k=_k, we have P_1=P, so 
    (P_1)=dx[x]/(x^k-1)= ∑_h=1^k-1 U_1^ξ^h

with U_1^ξ^h spanned by x^h-1dx.
Furthermore, for i≠ 0, we have P_ξ^i=0, so 
    (P_x^i)==U_ξ^i^1  (for i=1,…, k-1).
 By mapping x^i-1dx∈(P_1) to the 
generator 1_ξ^i of (P_x^i) we have defined a map matching (1-i/k,i/k)-classes 
to  (i/k,i/k)-classes.


If P is a polynomial which 
can be expressed as the sum of two invertible and non-degenerate 
polynomials P' and P” involving disjoint sets of variables we clearly have 
_P=_P”×_P”. This and the  theorem  above
imply the following crucial properties of the mirror map M_P. 


  Thom–Sebastiani.
If P is a polynomial which 
can be expressed as the sum of two invertible and non-degenerate 
polynomials 

    P=P'(x'_0,…,x'_n_1)+P”(x”_0,…,x”_n_2)
 
involving two disjoint sets of variables,
then we have 
    M_P=M_P'⊗ M_P”.


  Group actions.
For any H, K ⊆_P, the restriction of M_P
yields an isomorphism  

    M_P  U_H(P)^K ≅ U_H^∨(P^∨)^K^∨.


This mirror construction is due to Berglund and Hübsch <cit.>. The result presented here 
appeared first in this form in Krawitz <cit.>; 
we should also refer  to Berglund and Henningson <cit.> for 
the group duality, to Kreuzer and Skarke <cit.> for a systematic study,
and to Borisov <cit.> for a  reinterpretation 
of the setup and further generalizations in terms of vertex algebræ.

There are many consequences of the existence of M_P and of its 
properties with respect to group actions. We list a few of them, starting from 
the first, most transparent, application. 
It appeared in <cit.> and it should be regarded as 
a combination of the mirror map M_P of Krawitz and Borisov <cit.> and of 
the LG/CY isomorphism Φ of the first named author with Ruan <cit.>. In the present setup, it is extremely elementary to prove its main 
statement.

For any 
invertible, non-degenerate P of Calabi–Yau type and 
for any H⊆_P satisfying j_P∈ H⊆_P, we have
an isomorphism 

    H_𝕀^p,q(Σ_P,H;)≅ H_𝕀^n-1-p,q(Σ_P^∨,H^∨;)      (p,q∈).



Since M_P satisfies the above property with respect to group actions we have 
M_P(_P,H,𝕀)=_P^∨,H^∨,𝕀. In order to apply Theorem <ref>
we need the Calabi–Yau condition on P and the conditions H∋ j_P and H^∨∋ j_P^∨. The last equation is equivalent to 
H^∨⊆_P. The claim follows. 

The Thom–Sebastiani property 
applies to P'=x_0^k and P”=f adding up to 
    W=x_0^k+f(x_1,…, x_n).
 
The aim of this paper is to study the relation between the above
cohomological mirror symmetry and the the symmetry s=[1/k,0,…,0].

Let (ϕ,h) be a monomial element

    (ϕ,h)=(∏_j=0^n z_j^b_j-1⋀_j=0^n dz_j)

 in (P^∨_h)(-(h)). Let ξ=exp(2π /k). Then, 
s^*(ϕ,h)=ξ^i(ϕ,h) if and only if M_W(ϕ,h) is of the form 
(ϕ',h') with h'=[i/k,a_1,…,a_n]. In particular 
M_W maps invariant elements to non-invariant elements.

 This happens because s spans _x_0^k, whose dual group is 
trivial.  The claim follows by M_W=M_x_0^k⊗ M_f (see Example <ref>).




 §.§ Unprojected states and automorphisms
 
We study the behaviour of U_W with respect to s. 
We begin by restricting to a conveniently large state space _W within U_W.
 
Let us consider 
W=x_0^k+f(x_1,…,x_n) and, as in <ref>, a
subgroup K ⊂_f satisfying 

    (j_f)^k∈ K⊆_f.
 
Set 

    _K[j_W,s]^K(W)=[U_K[j_W,s](W)]^K=⊕_h∈ K[j_W,s](P_h)^K(-(h)).

If no ambiguity arises, when the polynomial W and the group K are fixed, we write simply . 

In the above setup we have three groups: K, K[j_W] and K[j_W,s]. Only K[j_W] satisfies the 
two conditions of mirror symmetry theorems: namely, it contains j_W and is contained in _W.
Its mirror group K[j_W]^∨ has the same properties. 
The following 
proposition (of immediate proof) describes how K and K[j_W,s] behave with respect to 
the the group duality.

Consider K∈_f satisfying 
(j_f)^k∈ K⊆_f. Then we have 
    (j_f^∨)^k∈ (K[j_W,s])^∨⊆_f^∨  and   K^∨=(K[j_W,s])^∨[j_W^∨,s].

Furthermore, we have a mirror isomorphism 

    M_W(_K[j_W,s]^K(W))^p,q=
    (_K^∨^K[j_W,s]^∨(W^∨))^n+1-p,q.


The unprojected state space projects to the sum of state spaces of the form 
_W,H,g
after taking j_W-invariant elements.

We have 

    ^j_W=⊕_b=0^k-1_W,K[j_W],s^b.

In particular, if W is Calabi–Yau, we have 

    ^j_W=⊕_b=0^k-1 H_s^b^*(Σ_W,K[j_W];),

where (b/k+p,b/k+q)-classes in H_s^b^*(Σ_W,K[j_W];)
match (b/k+p,b/k+q)-classes in _W,K[j_W],s^b
for any p,q∈.  





 §.§ The twist and the elevators

Throughout this section the polynomial W and the group K will be fixed; 
 we simplify the notation  and write 
    :=_K[j_W,s]^K(W),     
     j:=j_W,    ^∨:=_K^∨^K[j_W,s]^∨(W^∨),     j^∨=j_W^∨.

 We also write M for the mirror map M_W.

Note that the monomial element 
(ϕ,h)∈ with 
    ϕ=∏_i x_i^b_i⋀_i∈ I dx_i    and      
    I={i| h· x_i=x_i}
 is an 
eigenvector with respect to the diagonal symmetry
=[p_0,…,p_n]: the 
eigenvalue is 
exp(2π∑_j b_jp_j).
It is natural to 
attach to each (ϕ,g) and  the 
so-called 
-charge of the form ϕ defined on the
g-fixed space: 

    Q_ (ϕ,g)↦ Q_(ϕ,g)=
    ∑_j∈ J b_jp_j .





We decompose  as  

    = ⊕_a=0^k-1⊕_b=0^k-1⊕_g∈ j^as^b K( W_g)^K.

 and we can consider the following
/-valued gradings on 
the set of generators 

    (ϕ=∏_i∈ I x_i^b_i-1⋀_i∈ I dx_i,   g=[p_0,p_1,…,p_n]∈ j^as^bK)



  * the j-charge
Q_j=Q_j (ϕ,g)↦ Q_j(ϕ,g);

  * the j-degree 
d_j=a/k;

  * the s-charge 
Q_s=Q_s (ϕ,g)↦ Q_s(ϕ,g);

  * the s-degree
d_s=b/k.


We can now decompose 

    =⊕_0≤ a,b,c,d≤ k-1[| (d_j,d_s,Q_j,Q_s)=1/k(a,b,c,d)].

The following proposition further simplifies the decomposition.
[the moving subspace, the fixed subspace)]
For any element (ϕ,g) we have either 
(i) d_s=-d_j, or (ii) Q_s ≠ 0.


This happens because Q_s= 0 if and only if 
g· x_0=x_0, i.e. 
0∈ I. By definition of d_s and d_j we have 
g∈ j^kd_js^kd_sK and g· x_0=exp(2π (d_j+d_s)) x_0. 
We conclude that Q_s=0 if and only if 
d_s+d_j∈.

In other words  decomposes into an s-moving part ^m (Q_s≠0)
and an s-fixed part ^f (Q_s=0)

    =^m⊕^f=[| Q_s≠ 0]⊕ [| Q_s=0]

and the first summand  is [| d_j+d_s=0]; hence 
the three parameters d_j,Q_j,Q_s suffice for decomposing 
^m and 
the three parameters d_j,d_s,Q_j suffice for decomposing _F.
We write

    ^m   =
    ⊕_0≤ X,Y<k
    0< Z< k[| (d_j,Q_s-Q_j,Q_s)=1/k(X,Y,Z)]=⊕_0≤ X,Y< k
    0< Z<k^m_X,Y,Z
    ,
    ^f   =
    ⊕_0≤ X,Y< k
    0< Z< k[| (d_j,-Q_j,d_j+d_s)=1/k(X,Y,Z)]
    =⊕_0≤ X,Y< k
    0< z< k^f_X,Y,Z,

where the choice of the three parameters in {0,…, k-1} modulo k

    X=kd_j,  Y=k(Q_s-Q_j)=-kQ_j   in ^m
     k(Q_s-Q_j)    in ^f,   Z
    =kQ_s   in ^m
     k(d_j+d_s)    in ^f

is motivated by 
the following fact.
[twist]
For Z=1,…, k-1, we have an isomorphism

    τ^m_X,Y,Z   →^f_X,Y,Z
    
    (x_0^Z-1dx_0∧ϕ,g)   ↦ (ϕ,s^Zg)

transforming (p,q)-classes into  (p-1+2Z/k,q)-classes.
 

Indeed the above homomorphism  exchanges d_j+d_s with Q_s  
and 
preserves d_j and Q_s-Q_j.



The index Q_s-Q_j may be regarded as the (opposite of) the j-charge
of the form ϕ restricted to (x_0=0). 


There are natural isomorphisms matching ^f_X,Y,1≅^f_X,Y,2≅…≅^f_X,Y,k-1 and 
^m_X,Y,1≅^m_X,Y,2≅…≅^m_X,Y,k-1. We 
refer to them as “elevators”.
[elevators] 
For any 0≤ X,Y< k and 0<Z'<Z”<k
we have the isomorphisms

    e^m_Z',Z”^f_X,Y,Z'   →^f_X,Y,Z”               e^f_Z',Z”^f_X,Y,Z'   →^f_X,Y,Z”
    
    (ϕ, g)   ↦ (x_0^Z”-Z'ϕ, g)               (ϕ, g)   ↦ (ϕ, s^Z”-Z'g),

with e^m_Z',Z” and e^f_Z',Z” 
transforming (p,q)-classes into classes whose bidegrees equal
 (p-(Z”-Z')/k,q+(Z”-Z')/k) and (p+(Z”-Z')/k,q+(Z”-Z')/k), respectively.
For 0<Z'<Z”<k, we set e^m_Z”,Z'=(e^m_Z',Z”)^-1 and e^f_Z”,Z'=(e^f_Z',Z”)^-1.
  
Proposition <ref> specializes to the following statement. 
 
 The mirror isomorphism M yields isomorphisms
 
    M^f_X,Y,Z (^∨)^m_Y,X,Z,         M^m_X,Y,Z    (^∨)^f_Y,X,Z.

 
 
 
 Let us consider M as a morphism mapping U_W to U_W^∨. 
 For W=(x_0)^k+f we have W^∨=( x_0)^k+f^∨. Using (<ref>),
 every state of the form 
 (ϕ,g)∈ U_W can be regarded as an element of 
 
    (ϕ,g)∈ U^b_1_a_1⊗ U^b_2_a_2

 with a_1∈_(x_0)^k=/k, b_1∈_( x_0)^k=/k, 
 a_2∈_f and b_2∈_f^∨.
 Example <ref> shows that there are only two possibilities: (1) b_1=1 or (2) a_1=1. More precisely, in case (1) 
 (ϕ,g) is in ^f, it is fixed by s, b_1 is the trivial symmetry 1∈_(x_0)^k and a_1 is the nontrivial character corresponding to kd_s∈/k∖{0}. 
 In case (2) 
 (ϕ,g) is in ^m, it is not fixed by s, a_1 is trivial whereas b_1 is  the nontrivial character kQ_s ∈/k∖{0}. 
Since M exchanges a_1 and b_1 this proves that M exchanges ^m and ^f and 
preserves the coordinate Z which coincides with kd_s and kQ_s within ^m and ^f. 

Furthermore M maps U_a_2^b_2(f) to U_b_2^a_2(f^∨) with 
a_2∈_f[j_f] and b_2∈_f^∨[j_f^∨]. We recall that 
j_f^k∈ on both sides; therefore a_2 and b_2 are _k-characters. 
The claim (X,Y,Z)↦ (Y,X,Z) follows from 

    a_2=-kd_j,      b_2=-kQ_s+kQ_j,

where _k-characters are identified with elements of /k.
The first identity is immediate: a_2 is related to (ϕ,g)∈ U_W by a_2=g|_x_0=0.
The identity follows from (j|_x_0=0)=ξ_k^-1 by the Calabi–Yau condition. 
The second identity follows from the definition of

    ℓ_a_2(f_a_2)→(f^∨),     ∏_j x_j^b_j-1⋀_j dx_j↦∏_j ρ_j^b_j
from 
(<ref>): 
the determinant of ρ_j is ξ_d^w_j; hence 
(∏_j ρ_j^b_j) is identified with  the j_f-charge Q_j_f 
of the form ϕ restricted to 
(x_0=0). This yields an identification between b_2 and the _k-character
kQ_j-kQ_s.
 

 
 
 
 In view of the above proposition  
mirror symmetry operates as a plane symmetry exchanging the two blocks 
see Figure <ref>.

For Fermat potentials all the above discussion can be carried out more 
explicitly because the group elements a=∏_i=1^n ρ_i^a_i coincide 
with 1/d[a_1w_1,…,a_nw_n]. By adopting this notation, the 
space U_a^b(W) may be regarded as the one-dimensional space 
spanned by 

    (ϕ=∏_i=0^n x_i^b_i-1⋀_i∈ I dx_i,    a=∏_i=0^nρ_i^a_i)

where 

    a_i=0 ⇔ b_i≠ 0.

Mirror symmetry is simply an exchange of 
the w/d-valued vectors a and b. 
The bidegree (p,q) coincides with 
    (#( b)-∑_i=0^n b_i w_i/d
    + ∑_i=0^n a_i w_i/d, ∑_i=0^n b_i w_i/d+ ∑_i=0^n a_i w_i/d,)

where #( b) is the number of elements i such that b_i≠ 0.
Notice that Q_s is b_0/k and d_s+d_j is a_0/k; 
therefore, the equivalence in Proposition <ref> reads b_0=0 
is a special case of (<ref>). 
Furthermore we have 

    (X,Y,Z)= (a_0-| a|,b_0-| b|,b_0)   on the moving side,
    
    (a_0-| a|,b_0-| b|,a_0)   on the fixed side.

It is now clear that M exchanges the moving side with the fixed side, 
(X,Y,Z) with (Y,X,Z), and 
(p,q) with (n+1-p,q).




In view of Proposition <ref>,
we obtain the j-invariant contribution 
by setting Q_j=0. By (<ref>), this amounts to imposing Y=Z within 
^m and Y=0 within ^f. 
We get 

    (⊕_0≤ X< k
    0< Z< k^m_X,Z,Z)⊕(⊕_0≤ X< k
    0< Z< k^f_X,0,Z)


We get a picture of the j-invariant state space 
_W,K[j_W],𝕀 
by setting 
X=0 within 
^m and X=Z within ^f

    _W,K[j_W],𝕀=(⊕_0< t< k^m_0,t,t)⊕(⊕_0<t< k^f_t,0,t)

(we refer to Figure <ref>).
More generally,
the (d_s=b)-part of ^j_W is the 
state space _W,K[j_W],s^b 
(see Proposition <ref>).
By (<ref>), we obtain it by setting 
X=-b within 
^m and Z=X+b within ^f 

    _W,K[j_W],s^b=(⊕_0< t< k^m_-b,t,t)⊕(⊕_0≤ t< k
    t+b≠ 0^f_t,0,t+b).

Notice that the second summand only depends on  ^f_0,0,1
and ^f_1,0,1,…^f_k-1,0,k-1 since, for b≠ 0, it equals

    _W,K[j_W],s^b=
    (⊕_0< t< k^m_-b,t,t)⊕(e^f_1,b(^f_0,0,1)⊕⊕_0< t< k
    t+b≠ 0
    e^f_t,t+b(^f_t,0,t)),

with the convention (e^f_i,j)=(e^f_j,i)^-1 if j<i.
By Proposition <ref> the above data correspond to 
H_s^b^*(Σ_W,K[j_W];) under the Calabi–Yau condition. 

 Proposition <ref> relates it to the cohomology
of an s^b-fixed locus within a crepant resolution. 
Using this geometric picture, we can predict some vanishing conditions, which we 
prove in general, without relying on any Calabi–Yau condition in the next proposition.
The first guess is immediate: since
⟨ s⟩ 
operates trivially on an s-fixed locus, it is natural to expect that 
^m_-1,t,t vanishes for all t. 
More generally, since 
⟨ s^b⟩ 
operates trivially on an s^b-fixed locus, we expect that 
^m_-b,t,t vanishes  if tb≡ 0 mod . 
We prove that this holds true regardless of any Calabi–Yau 
condition or existence of crepant resolution. 

Let b∈{0,…, k-1}. We have 
    ^m_b,t,t= 0,

unless  
t is a multiple of k/(b,k) in k. 


We prove that 
    ^m_b,t,t≠ 0
 implies bt∈ k.
Recall that t/k equals Q_s-Q_j. For 
^m_b,t,t≠ 0, we can compute Q_s-Q_j explicitly using 
(ϕ,j^b g)∈^m_b,t,t with g∈_f and ϕ 
a g-invariant form 

    ϕ=x_0^kQ_s-1∏_k∈ I'x^b_k-1 dx_0∧⋀_k∈ I'dx_k,

with I'={k≥ 1| j^b g· x_k=x_k}⊆{1,…, N}. Using w_0/d=1/k,  we get 

    Q_s-Q_j=Q_s-kQ_s1/k-∑_k∈ I' b_kw_k/d=-∑_k∈ I' b_kw_k/d.

Let us write g as [0,p_1…, p_n]∈_f; then we have k∈ I' if and only if 

    bw_k/d + p_k∈.

Then k divides bt, because  

    bt/k = -b∑_k∈ I b_kw_k/d= -∑_k∈ I b_k bw_k/d=
    ∑_k∈ I b_k p_k∈,

where the last relation holds since ϕ is g-invariant. 
 


 §.§ Mirror symmetry on the Landau–Ginzburg side

In this section, we derive an interpretation of Proposition <ref>
in terms of the Landau–Ginzburg state space. This amounts to 
expressing both sides of the isomorphism 
^m_X,Y,Z≅^f_Y,X,Z in terms of j_W-invariant spaces.

Consider the j_W-invariant  summands 
    ^m_X,Y,Y⊂^m   and  ^f_X,0,Z⊂^f.
 Their mirrors
are  (^∨)^f_Y,X,Y and (^∨)^m_0,X,Z, and lie in 
the j_W-invariant part if and only if X=0 and X=Z. This happens if and 
only if we consider the mirror of [| Q_j=d_s=0] (imposing X=0 in ^m 
and X=Z in ^f is the same as requiring d_s=0). 


We obtain the first consequence of Proposition <ref>. Let 

    [_W,H,𝕀]^p,q_χ_s=i

be the eigenspace on which 
s operates as the character i∈/k. 
 For any H∈_W containing j_W, we have

    M[_W,H,𝕀]^p,q_χ_s=0⊕_i=1^k-1[_W^∨,H^∨,𝕀]^n+1-p,q_χ_s=i,

where H=K[j]. 

We now study
^f_0,0,i. 
The subspace  ^f_0,0,i mirrors ^m_0,0,i. 
By applying the twist τ from Proposition <ref>, 
we land again on ^f_0,0,i
which is a part of the j-invariant state space _W^∨,H^∨,s^i. 
 
Using the elevators of Proposition <ref>, we obtain 
a homomorphism 

    el_i:=⊕_t≠ 0,k-i e^f_t,t+i⊕_t ≠ 0, k-i^f_t,0,t[_ W,H,s^i]^s

whose cokernel coincides with _0,0,i^f. Notice that t+i is understood to be mod k. This means that the effect of the map on grading can be described as:

    (el_i)(ik)≅⊕_0<t<k-i^f_t,0,t⊕⊕_k-i<t<k^f_t,0,t(1).

We write el^∨_i for the same construction on the mirror. We conclude 

    M[_W,H,s^i(ik)^s/(el_i)(ik)]^p,q[_W^∨,H^∨,s^i(ik)^s/(el^∨_i)(ik)]^n-p,q,

where the bidegrees have been computed using (i/k)^p,q=^p+i/k,q+i/k, the fact that M_W transforms (p,q)-classes to (n+1-p,q)-classes, 
and the twist τ maps (p,q)-classes to (p-1+2i/k,q)-classes.

Note that using (<ref>) (recalling that d_j and d_s switch under mirror symmetry)  we can write 

    [(el^∨_i)(ik)]^n-p,q=
    ⊕_0<j<k-i[_W,H,𝕀(1,0)]^p,q_χ_s=j⊕⊕_k-i<j<k[_W,H,𝕀(0,1)]^p,q_χ_s=j.

Write _g for _W,H,g and ^∨_g for _W^∨,H^∨,g.  We obtain

    M [_s^i(ik)]_χ_s=0^p,q⊕⊕_j<k-i[_𝕀(1,0)]^p,q_χ_s=j⊕⊕_j>k-i[_𝕀(0,1)]^p,q_χ_s=j
    
    [^∨_s^i(ik)]_χ_s=0^n-p,q⊕⊕_j<k-i[^∨_𝕀(1,0)]^n-p,q_χ_s=j⊕⊕_j>k-i[^∨_𝕀(0,1)]^n-p,q_χ_s=j,

where 0<j<k.
Finally we focus on the moving part of 
_W,K[j_W],s^b which, by (<ref>) can be written 
as 
⊕_0< t< k^m_k-b,t,t. This is the 
decomposition of _W,K[j],s^b
into eigenspaces corresponding to the s-action operating 
as the character t∈/k. 
By applying the mirror map M_W, the twist τ^-1,
and the elevator e^f_t,k-b
 we get 

    ^m_k-b,t,t 
    (^∨)^f_t,k-b,t
    (^∨)^m_t,k-b,t
    (^∨)^m_t,k-b,k-b.

Therefore we have

    [_W,H,s^b(b/k)]^p,q_χ_s=t≅[_W^∨,H^∨,s^t(k-t/k)]^n-p,q_χ_s=k-b.

Notice that the map on the bidegrees is the composite of 

  * a shift
(p,q)↦ (p+b/k,q+b/k), 

  * mirror symmetry  
(p,q)↦ (n+1-p,q), 
  * τ^-1 yielding 
(p,q)↦ (p+1-2t/k,q), 

  * the elevator 
yielding (p,q)↦ (p+(b-(k-t))/k,q+(b+(k-t))/k)

  * a shift backwards (p,q)↦ (p-(k-t)/k,q-(k-t)/k),

inducing (p,q)↦ (n-p,q).

In the following statement we apply to (<ref>), 
(<ref>), and (<ref>) to the 
geometric interpretation (<ref>) 
of the Landau–Ginzburg state space 
in terms of relative cohomology of (_H,_W,H) provided in 
<ref>. 

Let W=x_0^k+f(x_1,…,x_n) be a
 quasi-homogeneous non-degenerate polynomial 
 of degree d and weights w_1,…,w_n. 
Assume j_W∈ H⊆_W 
(in particular ∑_j w_j is a positive multiple of d).
Then W descends to 
_H=[/H_0]→ and its generic fibre is _W,H.
Consider the automorphism  s=[1/k,0,…,0]_H→_H,
the orbifold cohomology groups H_𝕀^*(_H,_W,H)
and H_s^*(_H,_W,H).

For 0<i<k, define H^*_𝕀,i(_H,_W,H) to be the bigraded vector space

    ⊕_j<k-i[H^*_𝕀(_H,_W,H)(1,0)]_χ_s=j⊕⊕_j>k-i[H^*_𝕀(_H,_W,H)(0,1)]_χ_s=j;

here j ∈{1,…,k-1}.
This is the padding needed to state the mirror theorem. 


Let W=x_0^k+f(x_1,…,x_n) be a
 quasi-homogeneous, non-degenerate, invertible polynomial and H a group of symmetries 
 satisfying j_W∈ H⊆_W. 
As above, the polynomial W descends to 
_H=[/H_0]→ and its generic fibre is _W,H.

Then, for b and t≠ 0, 
we have 


  * H^p,q_𝕀(_H,_W,H)_χ_s=0≅⊕_i=1^k-1 H^n+1-p,q_𝕀(_H^∨,_W^∨,H^∨)_χ_s=i;

  * Let :=_W,H and ^∨:=_W^∨,H^∨. 
 
 For 0<i<k,  


    [H^p,q_s^i(_H,)(ik)]^s ⊕H^p,q_𝕀,i(_H,) ≅[H^n-p,q_s^i(_H^∨,^∨)(ik)]^s ⊕H_𝕀,i^n-p,q(_H^∨,^∨) ;




  * H^p,q_s^b(_H,_W,H)(b/k)_χ_s=t≅ 
H^n-p,q_s^-t(_H^∨,_W^∨,H^∨)
(k-t/k)_χ_s=-b.



Since H equals K[j_W] for a suitable K⊆_f containing j_W^k,
we can conclude via _P,H,g^p,q =H_g^p,q(_H;_P,H). 



§ GEOMETRIC MIRROR SYMMETRY

If W is of Calabi–Yau type,
via the Landau–Ginzburg/Calabi–Yau correspondence  of Theorem <ref>
based on Φ H^*(;)→ H^*(;), we 
provide an equivalent statement on the Calabi–Yau side. 

The existence of the isomorphism Φ is guaranteed by the Calabi–Yau condition 
(ensuring K-equivalence). As before, for 0<i<k, define H^*_𝕀,i(Σ_W,H) to be the bigraded vector space

    ⊕_j<k-i[H^*_𝕀(Σ_W,H)(1,0)]_χ_s=j⊕⊕_j>k-i[H^*_𝕀(Σ_W,H)(0,1)]_χ_s=j;

where again, j runs between 1 and k-1. 
Then we have the following statement.

Let W=x_0^k+f(x_1,…,x_n) be a
be a quasi-homogeneous, non-degenerate, 
 invertible, Calabi–Yau polynomial and H a group of symmetries 
 satisfying j_W∈ H⊆_W.  Let Σ=Σ_W,H and Σ^∨=Σ_W^∨,H^∨.
Then the following holds for b,t≠ 0.  


  * Let d=n-1. Then H^p,q_𝕀(Σ)_χ_s=0≅⊕_i=1^k-1 H^d-p,q_𝕀(Σ^∨)_χ_s=i;

  * Let d=n-2. For 0<i<k,

    [H^p,q_s^i(Σ)(ik)]^s ⊕H^p,q_𝕀,i(Σ)≅[H^d-p,q_s^i(Σ^∨)(ik)]^s ⊕H_𝕀,i^d-p,q(Σ^∨);


  * Let d=n-2. Then H^p,q_s^b(Σ)_χ_s=t(b/k)≅ 
H^d-p,q_s^-t(Σ^∨)(k-t/k)_χ_s=-b.



 This follows immediately from Theorem <ref> and the LG/CY correspondence. 

 In the theorem, d denotes the maximum of the dimensions of the components of the inertia stack considered in each case. 

For k=2, the second equation of the statement of Theorem <ref> can be stated as a mirror symmetry statement involving the cohomology 
groups H^*_s. 
Notice that the first statement says that Berglund–Hübsch mirror symmetry exchanges 
invariant  (p,q)-classes for Σ_W,H and  anti-invariant
(n-1-p,q)-classes of Σ_W^∨,H^∨ (and vice versa).
Finally the third statement is trivial because both sides vanish by Proposition <ref>. 
In this way we recover the main theorem of <cit.>.

Let W=x_0^2+f(x_1,…,x_n) be a
be a quasi-homogeneous, non-degenerate, 
 invertible, Calabi–Yau polynomial and H a group of symmetries 
 satisfying j_W∈ H⊆_W. 
Then,
we have 

    H^p.q_𝕀(Σ_W,H)^±   ≅ H^n-1-p,q_𝕀(Σ_W^∨,H^∨)^∓;
    H^p,q_s(Σ_W,H)(1/2)   ≅ 
    H^n-2-p,q_s(Σ_W^∨,H^∨)(1/2).




Let us consider  E=(x^6+y^3+z^2=0) 
within (1,2,3) with its order-6 symmetry s=[1/6,0,0]. 
In this case the “Calabi–Yau orbifold” is represented by an elliptic curve. 
The cohomology groups H_s^b^* describe the cohomology of the 
s^b-fixed loci E_b, shifted by (b/6,b/6).
Furthermore, the mirror of E coincides with E,  because the defining equation is of
Fermat type and J equals  (the order of  is w_xw_yw_z/ and equals the order  of J). 
This example allows us to test  as a state space 
computing the cohomology of E, and the cohomology of its fixed spaces 
satisfying E_1=E_2∩ E_3 and E_2=E_4. 
Since E is the elliptic curve with order-6 complex multiplication, E_1 is the origin and the fixed spaces  E_3(=E[2]) and E_2 are respectively a set of 4 points and 3 points interecting at the origin. Clearly E_3∖ E_1 is the unique 
order-3 orbit and 
E_2∖ E_1 is the unique order-2 orbit. 


The bth  row in the table below represents the ranks of contributions of [d_s=b/6] whereas the 
ath column represents the contributions to the state space of [d_j=a/6].
Notice that, by means of the elevators, all rows are identical except for the anti-diagonal 
entries of the form [d_j+d_s=0], which we underlined. 



(H_𝕀)=    2     1     0     0     0     1 

(H_s)=   0     1     0     0     0     0 

(H_s^2)=   0     1     0     0     1     1 

(H_s^3)=   0     1     0     2     0     1 

(H_s^4)=   0     1     1     0     0     1 

(H_s^5)=   0     0     0     0     0     1 




The 0th row is the 4-dimensional cohomology of the elliptic curve E organised in its 
2-dimensional primitive part (spanned by the forms dx∧ dy∧ dz
and x^4ydx∧ dy∧ dz)
and its 2-dimensional ambient part arising in the state space [d_s=0, d_j=a/6] for a=1 
and a=5
(j and j^5 correspond to the only narrow sectors of the state space, i.e. the 
only powers of [1/6,1/3,1/2] fixing only the origin). 
On the row corresponding to H^*_s there is a single contribution for d_j=1/5. This happens because E_1 is a point. Furthermore 
[d_j=5/6, d_s=1/6]=[d_j=1/6, d_s=5/6] vanish 
by Proposition <ref>. 
The remaining anti-diagonal terms are 
[d_j=4/6, d_s=2/6]=[d_j=2/6, d_s=4/6]=
⟨ x^2dx∧ dz⟩ and 
[d_j=3/3, d_s=3/3]=⟨ xydx∧ dy,x^3dx∧ dy⟩.


The above mirror symmetry statement (1) involves the first row and claims that 
all fixed cohomology classes appearing for d_j=1/6,…,5/6 match 
the classes of [d_j=0, d_s=0]; we already noticed that this identifies two 2-dimensional spaces of ambient and primitive cohomology. 
Statement (2), for i=1, says that the 1 dimensional space
[d_s=0, q_j=1,2,3,4] (spanned by the class dx∧ dy∧ dz) matches the cohomology class spanned by x^4 dx ∧ dy ∧ dz.


Statement (3) is a map M xydx∧ dy↦ x^2dx∧ dz∈[d_j=2/6, d_s=4/6]
and a map M x^3dx∧ dy↦ x^2dx∧ dz∈[d_j=4/6, d_s=2/6].
In this way 
    M[d_j=3/6, d_s=3/6][d_j=2/6, d_s=4/6]⊕[d_j=4/6, d_s=2/6].

In geometric terms mirror symmetry matches the order-2 orbit to the order-3 orbit. More 
precisely, the mirror statement (3) claims that there are as many eigenvectors 
of eigenvalue (χ_6)^2 and (χ_6)^4 in the cohomology of E_3 as 
eigenvectors 
of eigenvalue (χ_6)^3 in the cohomology of E_2 and of 
E_4.



We consider the genus-3 curve C defined by the degree-4 Fermat quartic 
x_1^4+x_2^4+x_3^4=0 in ^2. The 4-fold cover of ^2 ramified on C 
is a K3 surface defined as the vanishing locus of the polynomial W=x_0^4+x_1^4+x_2^4+x_3^4. In this example, the Calabi–Yau orbifold Σ_W  is again representable and we can treat the cohomologies H_𝕀^* and H_s^* as ordinary 
cohomologies of the K3 surface and of the ramification locus. As in the previous example, we display the cohomological data in a table. 
The bth  row in the table below represents the ranks of contributions of [d_s=b/4] whereas the 
ath column represents the contributions to the state space of [d_j=a/4].



 
(H_𝕀)=    [scale=0.4] at (1,2) 0;
at (1,1) 6+7+6;
at (-0.5,1) 1;
at (2.5,1) 1;
at (1,0) 0;
     [scale=0.4]at (1,2) 0;
at (1,1) 0;
at (0,1) 0;
at (2,1) 0;
at (1,0) 1;
     
[scale=0.4]at (1,2) 0;
at (1,1) 1;
at (0,1) 0;
at (2,1) 0;
at (1,0) 0;
     [scale=0.4]at (1,2) 1;
at (1,1) 0;
at (0,1) 0;
at (2,1) 0;
at (1,0) 0;
  

(H_s)=    [scale=0.4]at (0,0) 0;
at (0,1) 0;
at (-0.5,0.5) 3;
at (0.5,0.5) 3;
     [scale=0.4]at (0,0) 1;
at (0,1) 0;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;
     [scale=0.4]at (0,0) 0;
at (0,1) 1;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;
     [scale=0.4]at (0,0) 0;
at (0,1) 0;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;  

(H_s^2)=    [scale=0.4]at (0,0) 0;
at (0,1) 0;
at (-0.5,0.5) 3;
at (0.5,0.5) 3;
     [scale=0.4]at (0,0) 1;
at (0,1) 0;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;
     [scale=0.4]at (0,0) 0;
at (0,1) 0;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;
     [scale=0.4]at (0,0) 0;
at (0,1) 1;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;  

(H_s^3)=    [scale=0.4]at (0,0) 0;
at (0,1) 0;
at (-0.5,0.5) 3;
at (0.5,0.5) 3;
     [scale=0.4]at (0,0) 1;
at (0,1) 0;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;
     [scale=0.4]at (0,0) 1;
at (0,1) 0;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;
     [scale=0.4]at (0,0) 0;
at (0,1) 1;
at (-0.5,0.5) 0;
at (0.5,0.5) 0; 






The colors in the table refer to the weight of s: cohomology in red has character 1, blue has character 2, and green has character 3. 
Statement (2) involves, on one side, the cohomology of the curve C (H_s) and the moving cohomology of the K3 surface with weights 1 and 2.
The total cohomology on one side of Statement (2) is thus

    at (0,0) 1;
     at (0,1) 1;
     at (-0.5,0.5) 3+1;
     at (0.5,0.5) 3+13;
    .

We notice that the only -invariant broad cohomology 
classes in the entire unprojected state space U(W) are contained in 
U(W)_𝕀; this implies  U(W)_sg^=0. 
Hence H^*_prim,s vanishes. 
One can compute the mirror table as



 
(H_𝕀)=    [scale=0.4] at (1,2) 0;
at (1,1) 0+1+1;
at (-0.5,1) 1;
at (2.5,1) 1;
at (1,0) 0;
     [scale=0.4]at (1,2) 0;
at (1,1) 6;
at (0,1) 0;
at (2,1) 0;
at (1,0) 1;
     
[scale=0.4]at (1,2) 0;
at (1,1) 7;
at (0,1) 0;
at (2,1) 0;
at (1,0) 0;
     [scale=0.4]at (1,2) 1;
at (1,1) 6;
at (0,1) 0;
at (2,1) 0;
at (1,0) 0;
  

(H_s)=    [scale=0.4]at (0,0) 3;
at (0,1) 3;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;
     [scale=0.4]at (0,0) 1;
at (0,1) 6;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;
     [scale=0.4]at (0,0) 0;
at (0,1) 7;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;
     [scale=0.4]at (0,0) 0;
at (0,1) 0;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;  

(H_s^2)=    [scale=0.4]at (0,0) 3;
at (0,1) 3;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;
     [scale=0.4]at (0,0) 1;
at (0,1) 6;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;
     [scale=0.4]at (0,0) 0;
at (0,1) 0;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;
     [scale=0.4]at (0,0) 6;
at (0,1) 1;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;  

(H_s^3)=    [scale=0.4]at (0,0) 3;
at (0,1) 3;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;
     [scale=0.4]at (0,0) 0;
at (0,1) 0;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;
     [scale=0.4]at (0,0) 7;
at (0,1) 0;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;
     [scale=0.4]at (0,0) 6;
at (0,1) 1;
at (-0.5,0.5) 0;
at (0.5,0.5) 0; 






From this, we see that the mirror s-fixed locus is four projective curves and 12 isolated fixed points. The mirror Hodge diamond for Statement (2) is

    at (0,0) 4;
     at (0,1) 16;
     at (-0.5,0.5) 0+1;
     at (0.5,0.5) 0+1;
    .

Note that despite being an order 2 automorphism, the Hodge diamonds of the fixed loci of s^2 do not mirror each other. 

Clearly mirror symmetry should also yield 
a relation between the quantum invariants of the primitive classes 
of the curve and the orbifold quantum invariants of these sectors. 







The structure of the of the above example is shared by all K3 orbifolds of this type with order 4 automorphism. Combining the mirror theorem with the fact that s and s^3 have the same fixed locus (and hence cohomology of the same dimension), we can see that for any W=x_0^4+f(x_1,x_2,x_3) and group G, the table for Σ_W,G is given by 



 
(H_𝕀)=    [scale=0.4] at (1,2) 0;
at (1,1) a-1+b+a-1;
at (-1,1) 1;
at (3,1) 1;
at (1,0) 0;
     [scale=0.4]at (1,2) 0;
at (1,1) a^-1;
at (-0.5,1) 0;
at (2.5,1) 0;
at (1,0) 1;
     
[scale=0.4]at (1,2) 0;
at (1,1) b^;
at (-0.5,1) 0;
at (2.5,1) 0;
at (1,0) 0;
     [scale=0.4]at (1,2) 1;
at (1,1) a^-1;
at (-0.5,1) 0;
at (2.5,1) 0;
at (1,0) 0;
  

(H_s)=    [scale=0.4]at (0,0) g^;
at (0,1) g^;
at (-0.5,0.5) g;
at (0.5,0.5) g;
     [scale=0.4]at (0,0) 1;
at (0,1) a^-1;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;
     [scale=0.4]at (0,0) 0;
at (0,1) b^;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;
     [scale=0.4]at (0,0) 0;
at (0,1) 0;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;  

(H_s^2)=    [scale=0.4]at (0,0) g^;
at (0,1) g^;
at (-0.5,0.5) g;
at (0.5,0.5) g;
     [scale=0.4]at (0,0) 1;
at (0,1) a^-1;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;
     [scale=0.4]at (0,0) c;
at (0,1) c;
at (-0.5,0.5) c^;
at (0.5,0.5) c^;
     [scale=0.4]at (0,0) a^-1;
at (0,1) 1;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;  

(H_s^3)=    [scale=0.4]at (0,0) g^;
at (0,1) g^;
at (-0.5,0.5) g;
at (0.5,0.5) g;
     [scale=0.4]at (0,0) 0;
at (0,1) 0;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;
     [scale=0.4]at (0,0) b^;
at (0,1) 0;
at (-0.5,0.5) 0;
at (0.5,0.5) 0;
     [scale=0.4]at (0,0) a^-1;
at (0,1) 1;
at (-0.5,0.5) 0;
at (0.5,0.5) 0; 



.


The table for the Σ_W^∨,G^∨ is obtained from this table by replacing x ↦ x^∨. 

Using this table, we can find relationships between the topological invariants of the fixed loci of crepant resolutions of X:=Σ_W,G  and its mirror X^∨. Example <ref> shows that there is an isomorphism between the s^3-orbifold cohomology of X and the cohomology of the s fixed locus in the resolution X.  Recall that this is because for K3 surfaces, the age function is constant (of 3/4) on the s^3-orbifold cohomology of the resolution. By similar reasoning, the s^2-orbifold cohomology of X also has a constant age function (of 1/2). 

Now consider the following invariants for i=1,2:


  * f_i, the number of isolated fixed points of s^i;

  * g_i, the sum of the genera of the fixed curves of s^i;

  * N_i, the number of curves in the fixed locus of s^i. 

A superscript ∨ indicates the invariants of the mirror K3.  
 We have 


  * N_1=g_1^∨+1;

  * N_2+g_2+f_1=20-N_2^∨-g_2^∨-f_1^∨.



The table above implies that 2a+b+2a^∨+b^∨=24, and that g_1=g, N_1=g^∨+1, and f_1=a^∨+b^∨-2. The statements for the mirror invariants are obtained by x ↔ x^∨: for example, N_1^∨= g+1. 
 
Similarly, N_2-g_2^∨=(g^∨+c+a^∨)-(g^∨+c)=a^∨, which implies the statement. 


The same analysis also works for K3 surfaces with prime order automorphisms. Let W be a Calabi–Yau polynomial of the form W=x_0^p+f(x_1,x_2,x_3) for p prime. Then the Landau–Ginzburg state space breaks down as

 
(H_𝕀)=   
[scale=0.4]at (1,2) 0;
at (1,1) (p-1)a-2;
at (-0.7,1) 1;
at (2.7,1) 1;
at (1,0) 0;
     
[scale=0.4]at (1,2) 0;
at (1,1) a^- 1;
at (0,1) 0;
at (2,1) 0;
at (1,0) 1;
     ⋯    
[scale=0.4]at (1,2) 0;
at (1,1) a^;
at (0,1) 0;
at (2,1) 0;
at (1,0) 0;
     ⋯    
[scale=0.4]at (1,2) 1;
at (1,1) a^- 1;
at (0,1) 0;
at (2,1) 0;
at (1,0) 0;
 

(H_s)=    [scale=0.4]at (1,0) g^;
at (1,1) g^;
at (1.5,0.5) g;
at (0.5,0.5) g;
    
[scale=0.4]at (1,0) 1;
at (1,1) a^- 1;
at (1.5,0.5) 0;
at (0.5,0.5) 0;
     ⋯    
[scale=0.4]at (1,0) 0;
at (1,1) a^;
at (1.5,0.5) 0;
at (0.5,0.5) 0;
     ⋯    
[scale=0.4]at (1,0) 0;
at (1,1) 0;
at (1.5,0.5) 0;
at (0.5,0.5) 0;


(H_s^2)=   [scale=0.4]at (1,0) g^;
at (1,1) g^;
at (1.5,0.5) g;
at (0.5,0.5) g;
    
[scale=0.4]at (1,0) 1;
at (1,1) a^-1;
at (1.5,0.5) 0;
at (0.5,0.5) 0;
     
[scale=0.4]at (1,0) 0;
at (1,1) a^;
at (1.5,0.5) 0;
at (0.5,0.5) 0;
     ⋯    
[scale=0.4]at (1,0) 0;
at (1,1) 0;
at (1.5,0.5) 0;
at (0.5,0.5) 0;
    
[scale=0.4]at (1,0) a^-1;
at (1,1) 1;
at (1.5,0.5) 0;
at (0.5,0.5) 0;


7|c|⋮⋮

(H_s^p-1)=   [scale=0.4]at (1,0) g^;
at (1,1) g^;
at (1.5,0.5) g;
at (0.5,0.5) g;
    
[scale=0.4]at (1,0) 0;
at (1,1) 0;
at (1.5,0.5) 0;
at (0.5,0.5) 0;
     
[scale=0.4]at (1,0) a^;
at (1,1) 0;
at (1.5,0.5) 0;
at (0.5,0.5) 0;
     ⋯    
[scale=0.4]at (1,0) a^;
at (1,1) 0;
at (1.5,0.5) 0;
at (0.5,0.5) 0;
    
[scale=0.4]at (1,0) a^-1;
at (1,1) 1;
at (1.5,0.5) 0;
at (0.5,0.5) 0;





The following lemma follows immediately from considering this table. 
 Suppose Σ_W,H is a K3 orbifold with W=x_0^p +f(x_1,x_2,x_3). Then p-1 | 24.


Let X̃ be a crepant resolution of X=Σ(W,G), and X̃^∨ a crepant resolution of the mirror. The fixed locus of s is a disjoint union of curves an isolated fixed points. As before, let f_1 be the number of isolated fixed points, N_1 the number of curves, and g_1 the sum of the genera of the curves. 
 Suppose p>2. Then
N_1=g_1^∨+1 and

    f_1+f_1^∨+4=(p-2)/(p-1) 24.



Using the table, it is easy to see

    N_1=g^∨+1, g_1=g.

Additionally,

    f_1=(p-2)a^∨-2.

Combining this with (p-1)a+(p-1)a^∨=24, we obtain the statement in the theorem. 


This corollary implies that Berglund–Hübsch mirror symmetry agrees with mirror symmetry for lattice polarised K3 surfaces.  We briefly recall the latter. 

Given a smooth K3 surface Σ, Λ=H^2(Σ,)  is 
equipped with a lattice structure via the cup product taking values in H^4(Σ;)=. Let S_Σ:=Λ∩ H^1,1(Σ; ) be the Picard lattice of Σ. 

Let M be a hyperbolic lattice with signature (1,t-1). A K3 surface Σ is called M-polarized if there exists a primitive embedding M ↪ S_Σ. Given a non-symplectic automorphism s of Σ, the invariant sublattice S(s):=Λ^s is in fact a primitive sublattice of the Picard lattice.  

 Given M a primitive hyperbolic sublattice of Λ=H^2(Σ,) of rank at most 19 such that 

    M^⊥=U ⊕ M^∨,

M^∨ is defined to be the mirror lattice to M. 

  

Recall that we have restricted to the case where s has prime order p>2 (we have discussed p=2 in <cit.>). We now show that if two K3 surfaces with prime order automorphisms arise as crepant resolutions of a mirror pair of Berglund-Hübsch orbifolds, they have mirror lattices. In this case, M:=S(s) is p-elementary. That is, M^*/M =(/p )^⊕ a, and it is completely classified by its rank r and a. Then, by <cit.>, the fixed locus of s is either just isolated points or a disjoint union of N curves, of which N-1 are rational and the remaining one has genus g, and  f isolated points. Set m=22-r/p-1. Moreover, <cit.> states (in a slightly different form) that for p=3,5,7,13, if the fixed locus contains a curve,


  * m=2g+a, -g+N=r-11+p/p-1.

Notice that these are the only prime orders we need to consider, as we have shown that p-1 | 24. 

Lattice mirror symmetry exchanges (r,a) with (20-r,a).
 Let Σ_W,H and Σ_W^∨,H^∨ be mirror K3 orbifolds with prime order p>2 automorphisms s, s^∨, and let Σ and Σ^∨ be crepant resolutions with automorphisms also denoted s, s^∨. Then  Σ and Σ^∨ are mirror as lattice polarized K3 surfaces. 


Corollary <ref> relates the invariants (g,N,f) and (g^∨,N^∨,f^∨). It is enough to show that these relations give the mirror relations on (r,a), namely that 

    (r^∨,a^∨)=(20-r,a).
 Notice that there is always a fixed curve when the K3 is a hypersurface in weighted projective space of this form. Therefore, we see that

    r^∨=(-g^∨+N^∨)(p-1)+(11-p)=(-N+g+2)(p-1)+(11-p)
    
    =(p-1)(2-r-11+p/p-1)+11-p=20-r.

Finally, this implies

    a^∨=2g^∨-22-r^∨/p-1=2(N-1)-2+r/p-1.

Using that N=r-11+p/p-1+g, we obtain that

    a^∨=2g-22-r/p-1=a.






amsplain




A. Chiodo

Institut de Mathématiques de Jussieu – Paris Rive Gauche 

Sorbonne Université, UMR 7586 CNRS,

alessandro.chiodo@sorbonne-universite.fr 



E. Kalashnikov 

Department of Mathematics, FAS, Harvard University 

One Oxford Street, Cambridge, MA 02138

kalashnikov@math.harvard.edu


