Heavy subsets from microsupports
Tomohiro Asano

Casimir energy on the sphere and 6D CFT trace anomaly
    R Aros^1, F Bugini^2, DE Díaz^3 and C Núñez-Barra^4
    
=======================================================




    We construct partial symplectic quasi-states on a cotangent bundle with the use of microlocal sheaf theory. 
    We also give criteria and characterization for heaviness/superheaviness with respect to the partial symplectic quasi-state.






§ INTRODUCTION


Let X be a symplectic manifold. 
For closed subsets A,B⊂ X, A is said to be non-displaceable from B if there exists no Hamiltonian diffeomorphism φ X→ X with φ(A)∩ B=∅. 
The topic of non-displaceability has studied over the years from various viewpoints. 

Entov–Polterovich <cit.> introduced the notion of partial symplectic quasi-states and heavy/superheavy subsets. 
A partial symplectic quasi-state is a special functional defined on the function space of a symplectic manifold. 
For a partial symplectic quasi-state the notions of heavy/superheavy subsets are assigned. 
These notions provide a powerful tool to prove non-displaceability.


Tamarkin <cit.> introduced triangulated categories h (T^*M), h_∞ (T^*M) and proved non-displaceability results. 
The category h (T^*M) is a subcategory of the derived category of the sheaves on M× and h_∞(T^*M) is a quotient category of h (T^*M). 
For an object F of h (T^*M), a closed subset (F)⊂ T^*M called the reduced microsupport of F is defined. 
Tamarkin proved that the non-tiriviality of the morphism space between F and G in h_∞(T^*M) implies non-displaceability of (F) and (G) if they are compact.
See <ref>. 

In this paper, we construct partial symplectic quasi-states via sheaves. 
We also give criteria for heaviness/superheaviness with respect to these partial symplectic quasi-states. 



 §.§ Main results



We first define partial symplectic quasi-states from sheaves.


    One can construct a partial symplectic quasi-state ζ_F from a non-trivial object F∈ h_∞ (T^*M). 
    Moreover, if (F) is compact, it is ζ_F-superheavy.


We can extend this construction to more general idempotents on F. 
In this introduction, we restrict ourselves to the cases that the idempotent is the identity morphism 𝕀_F for simplicity. 
For F=_M× [0,∞), ζ_F coincides with the partial symplectic quasi-state ζ_MVZ defined via the Lagrangian intersection Floer theory by <cit.>. 

We then give a sufficient condition for ζ_F-heaviness. 
For objects F, G of h (T^*M), the subset (F,G) of  is defined and reflects some non-tiriviality of the morphism space between F and G in h_∞(T^*M). 

    Let G ∈ h(T^*M).
    Assume that (G) is compact and the set (F, G) of the spectral invariants is non-empty.
    Then (G) is ζ_F-heavy.


We also give sufficient conditions that the reduced microsupport is ζ_MVZ-superheavy (see <ref> and <ref>). 


For F's satisfying some conditions,
we also give a characterization for ζ_F-heaviness.


    For a special F, the following conditions for a compact subset A⊂ T^*M are equivalent:  
    
    
  * A is ζ_F-heavy,
    
  * there exist an object G∈ h(T^*M) with (G)⊂ A and a non-zero morphism F→ G in h_∞(T^*M), 
    
  * the unit morphism η_A,F F→ι_A_*ι_A^*F of the adjunction ι_A^*⊣ι_A_* satisfies η_A,F≠ 0 as a morphism of h_∞(T^*M). 


For the proof, we use the explicit description of the functor ι_A^* by Kuo <cit.>.

With these results and known results about Viterbo's spectral bound conjecture, we give several examples for ζ_MVZ-superheavy subsets, which include generalization of some of the results in Kawasaki–Orita <cit.>.
Furthermore, in conjunction with recent studies on γ-support <cit.>, we see that ζ_MVZ-superheavy subsets can become indecomposable continua.




We also construct partial symplectic quasi-states with a direct use of sheaf quantization of Hamiltonian diffeomorphisms and prove characterization of heaviness in <Ref>. 



 §.§ Related work


There are two kinds of well-studied partial symplectic quasi-states on the cotangent bundles of compact manifolds. 
Lanzat <cit.> defined partial symplectic quasi-states for Liouville domains using spectral invariants of Hamiltonian Floer homologies.
Monzner–Vichery–Zapolsky <cit.> defined a partial symplectic quasi-state on a cotangent bundle using spectral invariants of Lagrangian Floer homologies for the zero-section.

Chracterizations of heaviness already appeared in <cit.>. 
Ono–Sugimoto <cit.> constructed partial symplectic quasi-states for symplectic manifolds with boundaries of contact type from idempotents of symplectic cohomology and gave a characterization of heaviness.
Mak–Sun–Varolgunes <cit.> also proved a similar charaterization of heviness with respect to  partial symplectic quasi-states defined via quantum cohomologies of closed symplectic manifolds. 
Both studies use relative symplectic cohomologies to determine the heaviness. 
The main result of <cit.> restricted for cotangent bundles corresponds to <ref> of this paper.  
See <ref> for a difference. 


Since the relation between sheaf kernels and symplectic cohomologies of domains in cotangent bundles are described in <cit.> depending on a result of <cit.>, the author expects that more concrete relation between our work and Ono–Sugimoto's would be revealed. 










 §.§ Acknowledgements

The author would like to thank Yuichi Ike for his numerous helpful comments and suggestions regarding the structure of this paper.
The author is also thankful to Yoshihiro Sugimoto for his proposal, which inspired the content of <ref>.
The author would also like to thank Morimichi Kawasaki for enlightening him about the connection with Viterbo's conjecture.  
Furthermore, the author also thank Christopher Kuo, Tatsuki Kuwagaki, Wenyuan Li, and Ryuma Orita for helpful comments. 

This work was supported by JSPS KAKENHI Grant Number JP24K16920. 





§ PRELIMINARIES



In this paper, we work in the area of microlocal sheaf theory developed by Kashiwara–Schapira <cit.> with the framework of infinity categories. 
For the basics of infinity categories, refer to <cit.>. 
Regarding the implementation of microlocal sheaf theory within the setting of infinity categories, 
we will follow Kuo's setting <cit.> in this paper.
See <cit.> and its references. 




Throughout this paper, we fix a coefficient field .
For a manifold X, 
let (X) be the -linear stable derived category of sheaves of -vector spaces on X. 
For each object F∈(X), we write (F)⊂ T^*X for the microsupport of F, which is a closed conic subset.
For a closed subset A⊂ T^*X, _A(X) denotes the full subcategory of (X) consisting of objects whose microsupports are contained in A. 






 §.§ Tamarkin category


Until the end of this paper, let M be a connected manifold without boundary. 
We write (t;τ) for the homogeneous coordinate system on the cotangent bundle T^*.
The Tamarkin category (T^*M) is defined to be (M×)/_{τ≤ 0 }(M×). 
The homotopy category h(T^*M) is isomorphic to the quotient category h (M×)/h_{τ≤ 0 }(M×) of the homotopy categories by <cit.>. 
For an object F∈ (T^*M), (F)∩{τ>0 } is invariant under isomorphisms in (T^*M). 
See <cit.> for details. 
For an object F ∈(T^*M), we define 

    (F) ρ((F) ∩{τ>0 }),

where ρ T^*_τ>0(M ×_t) → T^*M; (x,t;ξ,τ) ↦ (x;ξ/τ).
The closed subset (F) is called the reduced microsupport of F ∈(T^*M).
For a closed subset A⊂ T^*M, let _A(T^*M) denote the full subcategory of (T^*M) consisting of objects F with (F)⊂ A. 



We introduce notation for basic operations in the Tamarkin categories.
We set 

    q_1 M ××→ M ×; 
        (x,t_1,t_2) ↦ (x,t_1) 
        q_2 M ××→ M ×; 
        (x,t_1,t_2) ↦ (x,t_2) 
        m  M ××→ M ×; 
        (x,t_1,t_2) ↦ (x,t_1+t_2)

and define 

    F ⋆ G m_!(q_1^-1 F ⊗ q_2^-1 G) ∈(_M ×)

for F,G ∈(_M ×).
This induces a functor ⋆(T^*M)×(T^*M)→(T^*M), which is called the convolution functor and provides (T^*M) with a monoidal structure. 
The monoidal structure is closed, and the monoidal unit is given by _M× [0,∞). 
The internal homomorphism is given by ^⋆, which is defined as 

    ^⋆(G,H) q_1_* (q_2^-1G,m^!H).


Set q_ M×→ be the projection. 


    Let F, G ∈(_M ×).
    Assume F is cohomologically constructible and the projection q_|_(F)(F) → is proper and has a bounded image.
    Then, one has an isomorphism
    
    ^⋆(F,G) ≃ D(i^-1F) ⋆ G,

    where D denotes the Verdier dual and i  M ×→ M ×; (x,t) ↦ (x,-t).
    In particular, the functor ^⋆(F,`-) preserves colimits.    


    First note that we have an isomorphism 
    
    ^⋆(F,G) ≃ m_* (q_2^-1i^-1F,q_1^!G).

    Since i^-1F is cohomologically constructible, by <cit.>, 
    
    (q_2^-1i^-1F,q_1^!G) ≃ G ⊠ D(i^-1F).

    By the assumption, m is proper on the support of G ⊠ D(i^-1F), which proves the first isomorphism.
    The second assertion follows from the fact that the functor D(i^-1F) ⋆ (`-) is a left adjoint.



    In the proof, the assumption about the boundedness of the image of q_|_(F)(F) → is only used for the properness of m on (G ⊠ D(i^-1F)). 
    This properness is satisfied also in other situations:
    
        
  * q_|_(G)(G) → has a bounded image, 
        
  * The image of q_|_(F) is bounded below and that of q_|_(G) is bounded above,
        
  * The image of q_|_(F) is bounded above and that of q_|_(G) is bounded below.
    
    Hence one can replace the boundedness of the image of q_|_(F) by either of them. 
    We do not use the other versions in this paper. 



We will use a variant of the convolution functor.
Let M_i  (i=1,2,3) be a manifold. 
We set 

    q_12 M_1 × M_2 × M_3 ××   → M_1 × M_2 ×; 
     
        (x_1,x_2,x_3,t_1,t_2)    ↦ (x_1,x_2,t_1) 
    
        q_23 M_1 × M_2 × M_3 ××   → M_2 × M_3 ×; 
    
        (x_1,x_2,x_3,t_1,t_2)    ↦ (x_2,x_3,t_2) 
    
        m_13 M_1 × M_2 × M_3 ××   → M_1 × M_3 ×; 
    
        (x_1,x_2,x_3,t_1,t_2)    ↦ (x_1,x_3,t_1+t_2).

With these maps, for F_12∈(_M_1 × M_2 ×) and F_23∈(_M_2 × M_3 ×), we define 

    F_12 F_23m_13_! (q_12^-1 F_12⊗ q_12^-1 F_23) ∈(_M_1 × M_3 ×),

which induces a functor (T^*M_1× T^*M_2)× (T^*M_2× T^*M_3)→ (T^*M_1× T^*M_3).





For c∈, T_c M×→ M× (x,t)↦ (x,t+c).
We shall abbreviate the functor T_c_* as T_c. 
For c≥ 0, there exists a natural transformation τ_c𝕀_(T^*M)⇒ T_c.
Asano–Ike <cit.> defined a pseudo-distance on the class of the objects of (T^*M) with the use of T_c and τ_c, following the sheaf-theoretic interleaving distance by Kashiwara–Schapira <cit.>.
In this paper, we use the following pseudo-distance (cf. <cit.>).


    Let F,G ∈(T^*M) and a,b ≥ 0.
    
        
  * The pair (F,G) is said to be (a,b)-isomorphic if there exist morphisms α F → T_a G and β G → T_b F in h(T^*M) such that 
        
    F  T_a G  T_a+b F  = τ_a+b(F), 
     G  T_b F  T_a+b G  = τ_a+b(G).

        
  * We define 
        
    d_(T^*M)(F,G) inf a+b (F,G) is (a,b)-isomorphic.

        
  * One defines Tor to be the full triangulated subcategory of h(T^*M) consisting of the objects F with d_(T^*M)(F,0)<+∞. 
        The quotient category h(T^*M)/Tor is denoted by h_∞(T^*M).
    


The Hom space in h_∞(T^*M) is described as follows. 


    For F, G ∈(T^*M), one has
    
    _h_∞ (T^*M)(F,G)≃_c _h(T^*M) (F,T_c G).




    The notation h_∞ (T^*M) suggests that this category is the homotopy category of a certain stable category _∞ (T^*M). 
    Indeed, the author expects that such _∞ (T^*M) is obtained as a quotient of (T^*M) by the full subcategory of torsion objects. 
    This fact seems to follow from, for example, the appendix of <cit.> or <cit.>, where the morphisms of the quotient of a small stable categories are described via colimits, and the fact that these colimits can be replaced by the smaller colimits as discussed in <cit.>.
    Precisely speaking, we need to work in a larger universe so that (T^*M) become a small category.
    Regardless, we only need the triangulated category h_∞ (T^*M), so we do not go into this aspect in this paper.




 §.§ Sheaf quantization of Hamiltonian diffeomorphisms and homeomorphisms


In this subsection, we recall sheaf quantization of Hamiltonian isotopies due to Guillermou–Kashiwara–Schapira <cit.>. 


First we introduce several classes of functions following Lanzat <cit.> and Ono–Sugimoto <cit.>. 
We let C^∞_c(T^*M) denote the set of compactly supported C^∞-functions on T^*M. 
Furthermore, we define 

    C^∞_cc(T^*M){ H T^*M→|there exists  C_H∈ such that  H-C_H∈ C^∞_c(T^*M) }.

We also write C_c^∞(T^*M× [0,1]) for the set of compactly supported C^∞-functions on T^*M × [0,1], and set

    C_cc^∞(T^*M × [0,1]) { H ∈ C^∞(M × [0,1]) |H|_T^*M×{s}∈ C_cc^∞ (T^*M) for each s∈ [0,1]}.

If M is 0-dimensional, we define C^∞_c(T^*M){ 0} and C^∞_c(T^*M× [0,1]){ 0} exceptionally.
We say that H∈ C_cc^∞(T^*M) or H∈ C_cc^∞(T^*M× [0,1]) is normalized if it is in H∈ C_c^∞(T^*M) or C_c^∞(T^*M× [0,1]). 



Each H∈ C_cc^∞(T^*M× [0,1]) generates an isotopy ϕ^H=(ϕ^H_s T^*M→ T^*M)_s∈ [0,1] called a Hamiltonian isotopy. 
Let _c(T^*M) be the set {ϕ^H_1 | H∈ C_cc^∞(T^*M× [0,1])} of the time 1-maps of the Hamiltonian isotopies, which forms a subgroup of the diffeomorphism group of T^*M. 
For H,H'∈ C_cc^∞(T^*M× [0,1]), put H♯ H'(p,s) H(p,s)+H'((ϕ^H_s)^-1(p),s) and H(p,s)-H(ϕ^H_s(p),s). 
Note that ϕ^H♯ H'_s=ϕ^H_s∘ϕ^H'_s and ϕ^H_s=(ϕ^H_s)^-1. 

For H∈ C_cc^∞(T^*M × [0,1]), one can uniquely associate an object _H∈ (T^*M^2× T^*[0,1]), which is called the sheaf quantization or the Guillermou–Kashiwara–Schapira (GKS) kernel <cit.>.
See also <cit.> and <cit.>. 
We set _H_H|_M^2×{1}×∈(T^*M× T^*M).



For φ∈_c(T^*M), there exists a normalized Hamiltonian function H with ϕ^H_1=φ. 
We define _φ_H. 
The following lemma asserts that this assignment is well-defined. 


    Let H and H' be normalized time-dependent Hamiltonian functions with ϕ^H_1=ϕ^H'_1. 
    Then _H≃_H'. 


The following lemmas are basic properties of the GKS kernel. 

    For H,H'∈ C_cc^∞ (T^*M × [0,1]), _H♯ H'≃_H _H'. 



    Let h [0,1]→ be a smooth function and s T^*M× [0,1]→ [0,1] be the projection. 
    For the timewise constant function H= h∘ s ∈ C_cc^∞ (T^*M × [0,1]), _H≃_Δ_M × [c,∞) where c=∫_0^1 h(s)  ds. 


    For H≤ H' ∈ C_cc^∞ (T^*M × [0,1]), there exists a natural morphism _H→_H' which corresponds to 1∈ H^0(M)≃_h_∞ (T^*M^2)(_H, _H'). 


For the pseudo-distance d_(T^*M), <cit.> proved the following Hamiltonian stability theorem.


    For H∈ C_c^∞ (T^*M× [0,1]), one has 
    
    d_(T^*M^2)(_0, _H)≤ H_osc,

    where 
    
    H_osc∫_0^1 max_p ∈ T^*M H_s(p) - min_p ∈ T^*M H_s(p)  ds.




We define C_c(T^*M), C_cc(T^*M), C_c(T^*M× [0,1]) and C_cc(T^*M × [0,1]) by replacing C^∞-functions by continuous functions. 
Moreover we regard C_c(T^*M)⊂  C_c(T^*M× [0,1]) and C_cc(T^*M)⊂ C_cc(T^*M × [0,1]). 
We have the direct sum decompositions

    C_cc(T^*M )     = C_c(T^*M ) ⊕, 
    
        C_cc(T^*M × [0,1])     =   C_c(T^*M × [0,1])⊕ C([0,1]).

Let H∈ C_cc(T^*M× [0,1]). 
We set H_norm∈ C_c(T^*M × [0,1]) to be the normalized part through the above decomposition and write (H)⊂ T^*M for the projection of the support of H_norm under the projection T^*M× [0,1]→ T^*M. 
Note that (H)⊂ T^*M is compact. 

It is proved in <cit.> that the pseudo-distance d_(T^*X) is complete, i.e., any Cauchy sequence converges. 
By combining the completeness and the Hamiltonian stability (<ref>), we can associate a GKS kernel for continuous Hamiltonian functions as follows.
Let H ∈ C_cc(T^*M × [0,1]).
Then there exists a sequence (H_n)_n ⊂ C^∞_cc(T^*M × [0,1]) satisfying H_n - H _C^0→ 0 as n →∞.  
Since (H_n)_n is a Cauchy sequence with respect to ·_osc, the sequence (_H_n)_n is also a Cauchy sequence with respect to d_(T^*M^2) by <ref>.
By the completeness of d_(T^*M^2), this sequence converges to an object of (T^*M^2). 
Moreover, by <cit.>, the object is unique. 
We write _H ∈(T^*M× T^*M) for the object. 
Note that (_H_n)_n also converges and 
we write _H^-1∈(T^*M× T^*M) for the object. 


In the later applications, we let _H^ k denote _H …_H_k-times for k ∈_≥ 1. 
We extend the notation as _H^ 0_Δ× [0,∞) and _H^ -k (^-1_H)^ k. 
Furthermore, for K∈ (T^*M× T^*M), the functor K  (`-) is also denoted as K^(`-), and (^ k_H)^(`-) is abbreviated as ^ k_H(`-).


Let _cc (T^*M× [0,1]) (resp. _cc^∞ (T^*M× [0,1])) be the nerve of the poset C_cc(T^*M× [0,1]) (resp. C_cc^∞ (T^*M× [0,1])).


    The assignment of the GKS kernels is refined to an infinity functor _cc (T^*M× [0,1])→(T^*M^2). 


    The functor _cc^∞ (T^*M× [0,1])→(T^*M^2) is defined by a similar argument to <cit.>. 
    By the left Kan extension along the inclusion _cc^∞ (T^*M× [0,1])→_cc(T^*M× [0,1]), we obtain a functor _cc (T^*M× [0,1])→(T^*M^2).
    
    This construction is compatible with the construction above since homotopy colimits <cit.> used in the previous papers <cit.> corresponds to colimits in the infinity categorical sense, which appear in the explicit description of the left Kan extension. 
    Precisely speaking, we need to be careful to see the correspondence of the colimits. 
    Note that for H∈ C_cc (T^*M× [0,1]), the poset {H'∈ C_cc^∞ (T^*M× [0,1]) | H'≤ H} may not be filtered. 
    For H, H'∈ C_cc (T^*M× [0,1]), we write H'≺ H if there exists a real number a<0 satisfying H'≤ H+a. 
    For H∈ C_cc (T^*M× [0,1]), the poset {H'∈ C_cc^∞ (T^*M× [0,1]) | H'≺ H} is filtered and the colimit indexed by this poset corresponds to the homotopy colimit in <cit.>. 
    The colimits indexed by 2 posets {H'∈ C_cc^∞ (T^*M× [0,1]) | H'≺ H} and {H'∈ C_cc^∞ (T^*M× [0,1]) | H'≤ H} are isomorphic since _a<0_H'+a≃_H' holds for any H'∈ C_cc^∞ (T^*M× [0,1]) by <ref>. 




 §.§ An explicit description of projectors



Let A⊂ T^*M be a compact subset.
The inclusion functor ι_A*_A(T^*M)→(T^*M) admits a left adjoint ι_A^* and a right adjoint ι_A^! by the presentavility of _A(T^*M) and (T^*M).
In this section, we give explicit descriptions for ι_A^* and ι_A^!. 

Kuo's explicit descriptions <cit.> of the adjoint functors of the inclusion functor _ρ^-1(A)∪ 0_M×(M×)→ (M×) descend to (T^*M) →_A(T^*M), where 0_M× is the image of the zero section of T^*(M×). 
In this paper, we use a variant of his description, which is also used in <cit.>. 

For a compact subset A⊂ T^*M, 
define 

    C_cc(T^*M× [0,1],A)    {H ∈ C_cc(T^*M× [0,1])|H ≡ 0 on a neighborhood of A}, 
    
        C_cc(T^*M,A)    {H ∈ C_cc(T^*M)|H ≡ 0 on a neighborhood of A}.

Define C_cc^∞ (T^*M× [0,1],A) and C_cc^∞(T^*M,A) similarly. 

Let _cc(T^*M,A) (resp. _cc(T^*M× [0,1],A), _cc^∞(T^*M,A), _cc^∞(T^*M× [0,1],A)) be the nerve of the poset C_cc(T^*M,A) (resp. C_cc(T^*M× [0,1],A), C_cc^∞(T^*M,A), C_cc^∞(T^*M× [0,1],A)). 

Kuo <cit.> considers compactly supported Hamiltonians on the sphere cotangent bundle, whereas we and Kuo–Shende–Zhang <cit.> consider Hamiltonians on J^1(M)=T^*M× that are translation invariant, and become compactly supported after taking the quotient of the domain by the -action. 
This is the only difference, and the proofs proceed parallelly.
Hence we obtain the following descripitions of the adjoint functors of ι_A_*

    
    ι^!_A F =lim_H∈_cc(T^*M,A)_H F,   ι^*_A F =_H∈_cc(T^*M,A)_H F.






    Every inclusion functor between any two of _cc(T^*M,A), _cc(T^*M× [0,1],A), _cc^∞(T^*M,A), _cc^∞(T^*M× [0,1],A) is initial and final if it exists. 
    So we can replace the index category of the limit or the colimit by other ones. 


Define

    _A_H∈_cc(T^*M,A)_H ∈ (T^*M^2).

Since  commutes with colimits, ι_A^* F is equivalent to _A  F. 
Since (ι_A_*ι_A^*)^2≃ι_A_*ι_A^*,  

    _A_A ≃_A

by <cit.>. 





§ PARTIAL SYMPLECTIC QUASI-STATES FROM SHEAVES



In this section, we construct a partial symplectic quasi-state from an idempotent of a sheaf category.
From now on, we assume the compactness of M. 
Hence M is a connected closed manifold. 



 §.§ Spectral invariants in Tamarkin category


In this subsection, we introduce spectral invariants for objects in the Tamarkin category.
See also Vichery <cit.>, Asano–Ike <cit.>, and  Guillermou–Viterbo <cit.> for example.


    Let F, G ∈(T^*M).
    One defines 
    
    h_c^d(F,G)    _h(T^*M) (F,T_cG[d]), 
    
            h_c^*(F, G)    ⊕_d∈ h_c^d(F, G), 
    
            h_∞^d(F,G)    _c h_c^d(F,G)≃_h_∞(T^*M)(F, G[d]), 
    
            h_∞^*(F,G)    _c h_c^*(F,G) ≃⊕_d∈_h_∞(T^*M)(F, G[d]).




    Let F, G ∈(T^*M). 
    For α∈ h^*_∞(F,G), one sets 
    
    c(α;F,G) 
             
            -inf{ c ∈|α∈(h^*_c(F,G) → h^*_∞(F,G)) }∈∪{ +∞}

    and 
    
    (F,G) { c(α;F,G)∈|α∈ h^*_∞(F,G) }⊂.



Note that our (F, G) never contains +∞ by definition. 



    The sign convention of spectral invariants differs among papers. In this paper, we follow that of Monzner–Vichery–Zapolsky <cit.>. This is consistent with the convention of Guillermou–Viterbo <cit.>, but it is opposite to that of Oh <cit.>, Frauenfelder–Schlenk <cit.> and the previous work <cit.>.
    See also <cit.>. 




For α∈ h^d_∞(F,G ), if we can take a lift α̃ F → T_c G[d] of α, then we have

    -c ≤ c(α;F, G).



 
     In our definition, c(0;F,G)=+∞. Generally, c(α; F,G) can be +∞ for non-zero α∈ h_∞^*(F,G). 
     See <cit.> for an example. 


For F,G∈(T^*M), we define 

    Σ(F,G):=-q_π(((G)⋆(F)^α ) ∩Γ_dt).

See <cit.> for the definitions of ⋆ and (`-)^α. 
However, we do not need to refer to these definitions directly; for the necessary properties of Σ(F,G), see the following.
By <cit.>, 

    (^⋆(F,G))⊂(G)⋆(F)^α

and its immediate consequence is the following. 

    For any α∈ h^*_∞(F,G), 
    
    c(α; F, G)∈Σ (F,G)∪{+∞}.



    The calculation of Σ(F, G) is not easy in general. 
    However we will use this result only for either case of the following:
    
   
        
  * (F) or (G) is compact,
        
  * both of F and G are of the form _H for some H∈ C_cc^∞(T^*M× [0,1]). 
    
    For both cases, it is easy to see
    
    Σ (F,G)={c∈|(T_cF)∩(G)∩{τ>0}≠∅}.




In favorable situations, phenomena like those mentioned in <ref> do not occur.


    Let F, G ∈(T^*M) and assume Σ(F,G) is bounded above or below.
    If α∈ h_∞(F,G) satisfies c(α;F,G)=∞, then α=0.


    Let us consider a morphism α F→ G in (T^*M) which is a lift of α with c(α ;F,G)=+∞. 
    For any s∈_≥ 0, there exists a lift α_s T_sF→ G of α. 
    There exists a_s≤ 0 such that α_s and α induces the same morphism T_a_sF→ G. 
    This implies the natural morphism F⋆_M× [a_s,s)→ G induced by α is 0.    
    
    Assume Σ(F,G)⊂ (c,∞) for some c∈, then we can take any c'∈ (-∞, c) as a_s above. 
    Hence F⋆_M× [c',s)→ G induced by α is zero. 
    Since _s F⋆_M× [c',s)≃ T_c'F, the morphism α T_c'F→ G is also zero.  
    This shows α=0. 
    
    Asuume Σ(F,G) ⊂ (-∞ ,c) for some c∈, then the canonical morphism h_-c'^*(F,G)→ h_-c^*(F,G) is isomophic for every c'≥ c. 
    Hence any morphism α_c T_cF→ G[d] uniquely factors through a morphism α_c' T_c' F→ G[d]. 
    By the uniqueness, we obtain a morphism α_∞_c' T_c'F→ G[d]. 
    For every object F∈(T^*M), _c' T_c'F≃ (_c'_[c',∞) )⋆ F≃ 0 holds. 
    Then we obtain α_∞ =0 and hence α_c=0 by (_c' T_c'F, G[d])≃lim_c' (T_c'F, G[d])≃ (T_cF, G[d]). 




We will use the following basic property of the spectral invariant. 


    One has
    
    c(βα; F,F” ) ≥ c(α; F,F')+c(β; F',F”).






 §.§ Sheaf-theoretic spectral norm


For ϕ, ψ∈_c(T^*M), we set

    γ (ϕ, ψ) -c(1;_ϕ, _ψ )-c(1;_ψ , _ϕ ).

We write γ (ϕ)γ (ϕ, 𝕀 ) for short. 
We call γ (ϕ) the sheaf-theoretic spectral norm of ϕ. 




    With the use of a version of Hamiltonian Floer theory, Frauenfelder–Schlenk <cit.> defined a spectral norm γ^Floer(ϕ) for each compactly supported Hamiltonian diffeomorphism ϕ. 
    Guillermou–Viterbo <cit.> combined with Viterbo's earlier result <cit.> proves the filtered isomorphism between Hamiltonian Floer cohomology and some sheaf cohomology defined from GKS kernels if the characteristic of the coefficient field is 2 or if the base manifold M is spin. 
    Hence γ^Floer and γ in this paper coincide for these cases.



We give some basic properties on γ, which is similar to those for γ^Floer in <cit.>. 


    Let H,H'∈ C_c^∞ (T^*M× [0,1]) and assume ϕ_H' displaces (H) then
    
    γ(ϕ_H) ≤ 2γ (ϕ_H').



To give a sheaf-theoretic proof of this lemma, we prepare the spectrality of the spectral invariants for GKS kernels and 
geometric description of Σ (_H,_H')

By <ref>, we obtain the following lemma. 
 
    One has
    
    c(α; _H, _H')∈Σ (_H,_H')={c∈|(T_c _H)∩(_H')∩{τ>0}≠∅}

    for any non-zero element α∈ h^*_∞ (_H,_H')≃ H^*(M).


For H∈ C_cc^∞ (T^*M× [0,1]), define 

    Σ(H){∫_0^1 (H_s-α (X_H_s))( γ(s) ) ds  γ [0,1]→ T^*M, γ̇(s)=X_H_s(γ (s)), γ(0)=γ(1) }.


By a direct calculation of (_H), we obtain the following lemma.  

    One has
    
    Σ(_0, _H)=Σ(H).



    The set Σ(_ϕ, _ψ) is nowhere dense in  for every ϕ, ψ∈_c(T^*M).


    It is enough to prove Σ(H) is nowhere dense for each H∈  C_c^∞ (T^*M). 
    We may assume H descends to a smooth function on T^*M×/ by a reparameterization. 
    By embedding M into a Euclidean space ^m, we can construct a compactly supported smooth function H T^*^m×/→ with Σ(H) ⊂Σ(H). 
    By <cit.>, Σ(H) is nowhere dense and hence, so is Σ(H). 



    We will mimic the proof by <cit.>. 
    Define c_+(H”) -c(1; _H”, _0) and γ (H)γ (ϕ_H”) for each H”∈ C_cc^∞ (T^*M× [0,1]). 
    
    By reparameterizing, we may assume that H_s=0 for s ∈ [0,1/2] and H'_s=0 for s ∈ [1/2,1]. 
    We can check Σ(_0, _ε H ♯ H') = Σ(_0, _H') for ε∈ [0,1].
    Since Σ(_0, _H') = Σ(_0, _ε H ♯ H') is nowhere dense, by the continuity of c_+ we find that the map 
    
    [0,1] →Σ(_0, _H');   ε↦ c_+(ε H ♯ H')

    is constant. 
    In particular, c_+(H ♯ H')=c_+(H'). 
    Since ϕ^H'_1 displaces H, its inverse ϕ^H'_1 displaces H= H. 
    An argument similar to the above shows that
    
    c_+(H ♯ H')
            =
            c_+(H'♯H)
            = 
            c_+(H').

    Hence, we have γ(H ♯ H') = γ(H'). 
    We thus conclude that 
    
    γ(H) = γ(H ♯ H' ♯H')
            ≤γ(H ♯ H') + γ(H') = 2γ(H').






    The inequality c(1;ϕ,ψ) ≤ 0 holds for each ϕ, ψ. 


The following proof is obtained through communication with Yuichi Ike. 


    Since c(1;ϕ,ψ)=c(1;𝕀,ϕ^-1ψ), it is enough to show that c(1;𝕀,ϕ^H_1) ≤ 0 for any H ∈ C_c^∞(T^*M× [0,1]).
    First let us study the case H of the form H=f(|ξ|), where f is a C^∞-function on [0,∞) such that, for some 0< C< C' and b>0,      
    
    
        
  * f ≡ b>0 on { z < C },
        
  * f ≡ 0 on { z > C' },
        
  * 0 ≤ f ≤ b, 
        
  * f is strictly decreasing and df/dz is non-zero on C < z < C',
        
  * df/dz is sufficiently small so that φ^H has no non-trivial fixed points.
    
    By microsupport estimate, we can see that _h(T^*M)(^0, T_c ^H) only changes at c=0 or -b.
    In each case, the change can be estimated by the microlocal stalk 
    
    μ_0 (q_* ^⋆(_0, T_c _H)),

    where q  M^2 ×→ is the projection.
    Although the intersection of  (_0) and (_H) is not clean, we can apply an argument similar to <cit.> to see that it is isomorphic to
    
    ({τ >0 } ; (_0,T_c _H)|_{τ >0 }).

    See for example <cit.> for the definition and treatment of . 
    Calculations similar to <cit.> show
    
    (_0, _H)|_{τ >0 }≃_{ |ξ/τ| ≥ C' }

    and 
    
    (_0,T_b_H)|_{τ >0 }≃_{ |ξ/τ|<C }.

    The contribution at c=0 is H^*(S^*M) and that at c=-b is H^*(M;or_M)[-n]. 
    We obtain a long exact sequence 
    
    H^*(M;_M)[-n] → H^*(M) → H^*(S^*M) .
 
    Since H^d(M)[-n] is zero for d<n, H^0(M)≃ in the middle term injects into H^*(S^*M). 
    This means that c(1;𝕀,ϕ^H_1) = 0.

    For a general H ∈ C_c^∞(T^*M× [0,1]), we can take H'=f(|ξ|) with f satisfying the above conditions such that H ≤ H'.
    By the monotonicity of c, we have c(1;𝕀,ϕ^H_1) ≤ c(1;𝕀,ϕ^H'_1).
    The right-hand side is 0 by the above argument.


    In this proof, we did not identified the morphisms in the exact sequence (<ref>) since it is not needed for the proof. 
    The author conjectures that it coincides with the Gysin exact sequence. 




    One has
    
    γ(ϕ,ψ)=d_(T^*M^2)(_ϕ, _ψ).



     With <ref> in hand, the rest of the discussion is parallel to those of <cit.> and <cit.>. 







 §.§ Partial symplectic quasi-states from sheaves


In this subsection, we construct a partial symplectic quasi-state from an idempotent of h_∞(T^*M).

First, let us recall the definition of partial symplectic quasi-states on T^*M.
The notion of partial symplectic quasi-states was first introduced by Entov–Polterovich <cit.> on closed symplectic manifolds. 
Later, Lanzat <cit.> and Monzner–Vichery–Zapolsky <cit.> defined  partial symplectic quasi-states for some open symplectic manifolds, especially for the cotangent bundles of closed manifolds. 



    A partial symplectic quasi-state ζ is a map ζ C_cc(T^*M)→ satisfying the following conditions:
    
    
        
  * (Lipschitz continuity) For all H_1, H_2 ∈ C_cc(T^*M), |ζ (H_1)-ζ (H_2)|≤H_1-H_2_C^0.
        
  * (Semi-homogeneity) For all λ≥ 0 and H ∈ C_cc(T^*M), ζ (λ H)=λζ (H).
        
  * (Monotonicity) If H_1, H_2 ∈ C_cc(T^*M) and H_1≤ H_2, then ζ (H_1)≤ζ (H_2).
        
  * (Additivity with respect to constants and Normalization) For all H ∈ C_cc(T^*M) and a ∈, ζ (H+a)=ζ (H)+a.
        
  * (Partial additivity) If H_1, H_2 ∈ C_cc^∞ (T^*M) and {H_1,H_2}=0, then ζ (H_1+H_2)≤ζ (H_1)+ζ (H_2). 
        
  * (Vanishing) If H ∈ C_c (T^*M) and (H) is displaceable from itself, then ζ (H)= 0. 
        
  * (Invariance) For each φ∈_c(T^*M) and H ∈ C_cc(T^*M), ζ (H∘φ)=ζ (H).
    




    Let ζ C_cc(T^*M) → be a partial symplectic quasi-state. 
    A compact subset A of T^*M is said to be ζ-heavy (resp. ζ-superheavy) if for any H ∈ C_cc(T^*M) one has ζ(H) ≥min_A H (resp. ζ(H) ≤max_A H). 


We will use the following equivalent definition. 


     A compact subset A of T^*M is ζ-heavy (resp. ζ-superheavy) if and only if for any H ∈ C_cc(T^*M) and c∈ satisfying H≡ c on a neighborhood of A one has ζ(H) ≥ c (resp. ζ(H) ≤ c). 


    Since the “only if" part is obvious, let us prove the “if" part. 
    For arbitrary H∈ C_cc(T^*M), take H'∈ C_cc(T^*M) with H'|_A≡ c=min_A H (resp. max_A H) and H≥ H' (resp. H≤ H'). 
    For example, min{c, H} (resp. max{c, H}) satisfies the condition for H'. 
    Then by the monotonicity, we get ζ(H) ≥ζ(H') (resp. ζ(H) ≤ζ(H')). 
    There exists a sequence (H'_n)_n C^0-converging to H' such that H'_n|_U_n≡ c for a neighborhood U_n of A for each n. 
    By the Lipschitz continuity, we obtain ζ(H')=lim_n ζ(H'_n). 
    By assumption, ζ(H'_n)≥ c (resp. ζ(H'_n)≤ c) and hence ζ (H)≥ c (resp. ζ (H)≤ c). 


The following are basic and important properties of heavy/superheavy subsets. 


    Let ζ be a partial symplectic quasi-state.
    
        
  * Every ζ-superheavy subset is ζ-heavy. 
        
  * Every ζ-heavy subset is non-displaceable from itself.
        
  * Every ζ-heavy subset is non-displaceable from every ζ-superheavy subset.
    


 
We can define a partial symplectic quasi-state from sheaves as follows.
Let F ∈(T^*M) and e F→ F be an idempotent in h _∞(T^*M) satisfying c(e;F,F)≠ +∞. 
We define ζ_e C_cc(M)→ by

    ζ_e(H)  -lim_kc(e;F, _H^ -k (F))/k.

We also define ζ_F as ζ_𝕀_F for F∈(T^*M) with c(𝕀_F;F,F)≠∞. 


    Let F, F' ∈ (T^*M) be objects which are isomorphic in h_∞(T^*M). 
    Then d_(T^*M) (F,F') <+∞ and an isomorphism F≃ F' in h_∞(T^*M) induces an isomorphism h_∞^*(F, F) ≃ h_∞^*(F', F'). 
    An idempotent e F→ F in h_∞(T^*M) corresponds to an idempotent e' F'→ F' through the isomorphism.
    The condition c(e;F,F)<+∞ implies c(e';F',F')<+∞ since d (F,F') <+∞. 
    The difference |c(e;F, _H^ -k (F))-c(e';F', _H^ -k (F'))| is bounded above by 2d_(T^*M) (F,F'), and hence ζ_e=ζ_e'. 



    The map ζ_e is a partial symplectic quasi-state. 


To prove the theorem, we shall prepare a lemma.
Define 

    ℓ_e (ϕ) -c(e;_ϕ^ (F), F).



The following hold:

    
  * ℓ_e (ϕψ)≤ℓ_e (ϕ)+ℓ_e(ψ),
    
  * |ℓ_e(ϕ)-ℓ_e(ψ)|≤γ (ϕ,ψ).



    (i) is a direct consequence of <ref> and e^2=e. 

    (ii) |ℓ_e(ϕ)-ℓ_e(ψ)|≤ d_(T^*M^2)(_ϕ, _ψ) is deduced from a basic property of the spectral invariant. Hence the assertion follows from <ref>




    We have prepared <ref>. 
    The rest of the proof is parallel to that of <cit.>. 


If (F) is compact, the partial symplectic quasi-state ζ_e satisfies the following important property.  

    If (F) is compact, then (F) is ζ_e-superheavy.


This is a straightforward consequence of the following standard lemma, to which similar statements and proofs can be found in several articles. 

 
    Let F∈ (T^*M) and H∈ C_cc(T^*M× [0,1]). Assume H-c ∈  C_cc(T^*M× [0,1], (F) ) for some constant c∈. Then _H^(F)≃ T_c F.


    It is enough to prove for H∈ C_cc^∞(T^*M× [0,1]) since the kernel _H for continuous H is defined as a colimit of _H' for smooth H' and the colimit commute with .  
    <cit.> is the statement for c=0. 
    The desired statement is obtained from it since _H≃ T_c_H-c by <ref>. 



As a special case, we can take F=_M × [0,∞) and e=𝕀_F. 
In this case, if =/2, ζ_e coincides with the partial symplectic quasi-state ζ_MVZ defined by Monzner–Vichery–Zapolsky <cit.>. 
Hence, we still denote the partial symplectic quasi-state by ζ_MVZ for a general coefficient.

For the later application, we describe ζ_MVZ in a different form. 



    One defines
    
    ℓ_-(H) c(1;_M× [0,∞), _H^ (_M× [0,∞))),  ℓ_+(H) -c(1;_M× [0,∞), _H^ -1 (_M× [0,∞))).




    We have the following:
    
        
  * ℓ_+(H)= -ℓ_-(H) for H∈ C_cc^∞(T^*M× [0,1]),
        
  * ℓ_+(H)= -ℓ_-(-H) for H∈ C_cc(T^*M),
        
  * ℓ_+(H)=c(μ_M; _M× [0,∞), _H^ (_M× [0,∞))⊗or_M×) for H∈ C_cc(T^*M× [0,1]). 
    
    The last equation follows from the work <cit.> by Viterbo. 
    Although an orientation for M is assumed in that work, parallel arguments can be applied in the absence of orientation once one use the orientation sheaf.





    For H ∈ C_cc(T^*M), one has 
    
    ζ_MVZ(H) = lim_k →∞ℓ_+(kH)/k=lim_k →∞c(μ_M; _M× [0,∞), _H^ k (_M× [0,∞))⊗or_M×)/k.







§ CRITERIA FOR HEVINESS/SUPERHEAVINESS


In this section, we give sufficient conditions for ζ_e-heaviness and ζ_MVZ-superheaviness. 



 §.§ Criteria for ζ_e-heaviness


Let F ∈(T^*M) and e F→ F be an idempotent in h_∞(T^*M) with c(e;F,F)≠ +∞. 
Let ζ_e be the partial symplectic quasi-state associated with e. 
In this section, we first prove the following theorem.


    Let G ∈(T^*M). Assume that that (G) is compact.  
    
        
  * If there exists an element β∈ hT^* (F,G) satisfying c(β e ;F,G)≠ +∞, then (G) is ζ_e-heavy. 
        
  * If there exists an element β∈  hT^* (G,F) satisfying c(e β ;F,G)≠ +∞, then (G) is ζ_e-heavy. 
    


    We only give a proof for (i) since the other is parallel. 
    By <ref>, it is enough to prove ζ_e(H)≥ c for H ∈ C_cc(T^*M) with H|_U≡ c where U is a neighborhood of (G). 
    
    Take k∈ and set ℓℓ_e(kH) -c(e;F, _-kH^(F))=-c(e;F, _H^ -k(F)).
    For any ε>0, we can take a lift 
    
    e F → T_ℓ+ε_H^ -k(F)

    of e ∈ Q_∞^*(F,F).
    We may assume there exist d∈ such that β∈ Q^d_∞(F,G). Put b c(β e; F,G)∈.
    For any ε'>0, we can take a lift 
    
    α F → T_-b+ε' G[d]

    of β e.
    By the assumption of H and <ref>, we find that 
    
    _H^ -k(G) ≃ T_-kcG    (k ∈).

    Hence, by composing the above morphisms, we get a morphism
    
    F  T_ℓ+ε_H^ -kF  T_ℓ-b+ε+ε' - kc G[d],

    which is also a lift of β e ∈ Q^*_∞(F,G).
    This means that 
    
    -ℓ+b-ε-ε' + kc ≤ c(β e;F, G)=b.

    Since ε and ε' are arbitrary, we obtain 
    
    ℓ_e(kH) = ℓ≥ kc.

    By the definition of ζ_e and taking the limit k→ +∞,
    we get ζ_e(H)≥ c. 



    Let us assume (F) and (G) are compact. 
    Under the assumption of (i) or (ii) in <ref>, _h_∞(T^*M)(F,G[d]) or  _h_∞(T^*M)(G,F[d]) for some d ∈ is non-zero. 
    Hence, by Tamarkin's non-displaceability theorem (see <cit.> and <cit.>), we find that (G) is non-displaceable from (F). 
    Since any ζ_e-heavy subset is non-displaceable from any ζ_e-superheavy subset (<ref>), one can view <ref> as a refinement of Tamarkin's non-displaceability theorem. 








    Let G ∈(T^*M). Assume that (G) is compact.  
    
        
  * If (_M× [0,∞), G) ≠∅, 
    then (G) is ζ_MVZ-heavy.
        
  * If (G,_M×[0,∞)) ≠∅,
    then (G) is ζ_MVZ-heavy.
    





    We can state our results in a different way by introducing a 1-category h_∞^*(T^*M) as follows.
    
    We set h_∞, -∞^d(F,G) {α∈ h_∞^d(F,G) | c(α;F,G)=+∞}. 
    This is closed under precomposition and postcomposition of morphisms in h_∞(T^*M).
    Let h_∞(T^*M) be the category that has the same object as h_∞(T^*M) and the morphism spaces is defined by 
    
    _h_∞(T^*M)(F,G) h_∞^0(F,G) /h_∞, -∞^0(F,G).

    The triangulated structure on h_∞(T^*M) is revoked by this quotient.  
    Define 
    
    h_∞^d(F,G)    h_∞^d(F,G) /h_∞, -∞^d(F,G) =_h_∞(M)(F,G[d]), 
    h_∞^*(F,G)   ⊕_d∈h_∞^d(F,G).

    Notice that the assignment c( `-; F,G) h_∞^*(F,G) →∪{ +∞} descends to an assignment h_∞^*(F,G) →∪{ +∞}. We also write this as c(`-; F,G). 
    By definition, α∈h_∞^*(F,G) is zero if and only if c(α ;F,G)=+∞. 
    
    In some reasonable situations, h_∞^*(F,G) is isomorphic to h_∞^*(F,G) (see <ref>).

    In fact, e can be taken as a non-zero idempotent in h_∞(T^*M). 
    However the author do not know an example of idempotents in h_∞(T^*M) which does not lift to that of h_∞(T^*M).




 §.§ Criteria for ζ_MVZ-superheaviness


We also give a sufficient condition for ζ_MVZ-superheaviness.



    Let G ∈(T^*M).
    Assume that (G) is compact.
    
        
  * We assume that there exists β∈ h_∞^*(_M× [0,∞), G) such that c(βμ_M;_M× [0,∞),  G⊗or_M×)≠ +∞. Then (G) is ζ_MVZ-superheavy.
        
  * We assume that there exists β∈ h_∞^*(G,_M× [0,∞)) such that c(μ_Mβ; G, _M× [0,∞)⊗or_M×)≠ +∞. Then (G) is ζ_MVZ-superheavy.
    



    We only give a proof for (i) since the other is parallel. 

    By <ref>, it is enough to prove ζ_e(H)≤ c for H ∈ C_cc(T^*M) with H|_U≡ c where U is a neighborhood of (G). 

    Take k∈ and set ℓℓ_+(kH)=c(μ_M;_M × [0,∞),_H^ k(_M × [0,∞))⊗or_M×) (see <ref>).
    For ε>0, we take a lift
    
    μ_M _M × [0,∞)→ T_-ℓ+ε_H^ k(_M × [0,∞))⊗or_M×[n]

    of μ_M.
    For any a < c(β; _M × [0,∞),G), we can also take a lift 
    
    β_M × [0,∞)→ T_-a G[d]

    of β.
    By composing the above morphisms and noticing that _H^ k (G) ≃ T_kc G by <ref>, we get a morphism
    
    _M × [0,∞) T_-ℓ+ε_H^ k(_M × [0,∞))⊗or_M×[n]  T_-ℓ-a+ε + kc G⊗or_M×[n+d],

    which is a lift of βμ_M ∈ Q^*_∞(_M × [0,∞), G).
    This means that 
    
    ℓ+a-ε - kc ≤ c(βμ_M;_M × [0,∞), G⊗or_M×).

    Since ε >0 and a < c(β;_M × [0,∞), G) are arbitrary, we obtain 
    
    ℓ_+(kH) = ℓ≤ kc + c(βμ_M;_M × [0,∞), G⊗or_M×)-c(β;_M × [0,∞), G).

    Taking the limit k→ +∞ completes the proof.








 §.§ ζ_MVZ-heavy/superheaviness subsets and Viterbo's conjecture


In this subsection, we briefly review the relationship between heavy/superheavy subsets and Viterbo's spectral bound conjecture.

We define DT^*M{(x,ξ)∈ T^*M |ξ_g ≤ 1} for a Riemannian manifold (M,g). 
The following was first conjectured by Viterbo in an earlier verion of <cit.>, as the special case M=T^n.
In this paper, we call this conjecture the Viterbo conjecture. 
Let 0_M be the image of the zero section of T^*M. 

Let (M,g) be a closed Riemannian manifold. There exists a constant R>0 such that, if ϕ∈_c(T^*M) satisfies ϕ (0_M)⊂ DT^*M, then γ (ϕ(0_M))<R holds.


Note that, by <cit.>, we have 

    γ (ϕ(0_M))= c(μ_M; _M×[0,∞), _ϕ^( _M× [0,∞))⊗or_M×)-c(1; _M×[0,∞), _ϕ^( _M× [0,∞)))

where μ_M∈ H^n(M;or_M) is the fundamental class. 
The validity of this conjecture does not depend on the choice of the metric g on M. It may depend on the coefficient field .

This conjecture is related to our discussion by the following theorem:


If M satisfies the Viterbo conjecture, then any ζ_MVZ-heavy subset A⊂ T^*M is ζ_MVZ-superheavy. 

This statement is well-known to experts. 
See, for example, <cit.>, <cit.>,  <cit.>, <cit.> for related statements. 
In this paper, we will see a slightly stronger statement <ref>. 

The following is an obvious corollary of <ref>.

    Let G ∈(T^*M). Assume that (G) is compact and that the Viterbo conjecture for M over  is true. 
    
        
  * If (_M× [0,∞), G) ≠∅,
    then (G) is ζ_MVZ-superheavy.
        
  * If (G,_M×[0,∞)) ≠∅,
    then (G) is ζ_MVZ-superheavy.
    







To the best of the author's knowledge, no counterexamples have been found, and the conjecture has been proven for some manifolds, which we will explain below.
We say that a -orientation is an isomorphism _M≃or_M.
We call a manifold M equipped with a -orientation a -oriented manifold. 
Note that the element μ_M can be regarded as an element of H^n(M)=H^n(M;_M) if M is -oriented. 





First, Shelukhin <cit.> proved the conjecture for =/2 and M=S^n,  P^n,  P^n,  P^n.
Next, Shelukhin <cit.> provided a proof for the case where M is -oriented and string point-invertible. The condition of string point-invertibility is a condition on the Gerstenhaber algebra structure on the homology of the free loop space of M.
Also, with a minor assumption on the coefficient field, Guillermou–Vichery <cit.> proved the conjecture for homogeneous spaces of compact Lie groups. 
At around the same time, Viterbo <cit.> proved the conjecture for a mysterious class that includes the compact Lie groups, and provided an alternative proof for homogeneous spaces without the assumption of the coefficient field.
To be precise, the following holds:




  * For a compact Lie group 𝖦 and its closed subgroup 𝖧, the homogeneous space 𝖦/𝖧 satisfies the Viterbo conjecture.

  * If M is -oriented and string point invertible, then it satisfies the Viterbo conjecture.

  * If there exist compact manifolds V, P and morphisms f P→ M, g V→Diff (P) such that (f ∘ev_y∘ g )^*(μ_M)≠ 0 where ev_y is the evaluation at some point y∈ P, then M satisfies the Viterbo conjecture.

  * If there exists a morphism p N→ M such that p^*(μ_M)≠ 0, and N satisfies the Viterbo conjecture, then M also satisfies it.









§ A CHARACTERIZATION OF HEAVINESS

 

Let e F→ F be an idempotent in h_∞(T^*M) with c(e;F,F)≠ +∞ and denote the associated partial symplectic quasi-state by ζ_e. 
To prove a given compact subset A⊂ T^*M is ζ_e-heavy using <ref>, one need to look for G∈_A(T^*M) satisfying a condition in the theorem. 

In fact, there is a most sensitive pair of an object G with (G)⊂ A and a morphism β F→ G (resp. β G→ F) with respect to the condition in (i) (resp. (ii)) of <ref>. 
These object and morphism are given by the left (resp. right) adjoint ι_A^*(resp. ι_A^!) of the inclusion functor ι_A_* _A(T^*M)→(T^*M) and unit (resp. counit) morphism of the adjunction. An explicit description of these adjoint functors are given by Kuo <cit.>. 
Since ι_A_* is the natural inclusion, it will be omitted in many cases below.


Let us consider the following conditions for a compact subset A⊂ T^*M and an idempotent e F→ F in h_∞(T^*M) with c(e;F,F)≠ +∞:

    
  * A is ζ_e-heavy,
    
  * there exist an object G∈_A(T^*M) and an element β∈ h_∞^* (F,G) satisfying c(β e;F,G)≠ +∞,
    
  * the morphism η_A,F F→ι_A^*F in h_∞(T^*M) induced by the unit morphism of the adjunction satisfies  c(η_A,F e; F, ι_A^*F)≠ +∞,
    
  * there exist an object G∈_A(T^*M) and an element β∈ h_∞^*(G,F) satisfying c( e β ; G,F)≠ +∞, 
    
  * the morphism ϵ_A,Fι_A^!F→ F in h_∞(T^*M) induced by the counit morphism of the adjunction satisfies  c( e ϵ_A,F ; ι_A^!F, F)≠ +∞. 



The implications (iii) ⇒ (ii) and (v) ⇒ (iv) are trivial. 
The implications (ii) ⇒ (i) and (iv) ⇒ (i) are <ref>. 
The implications (ii) ⇒ (iii) and (iv) ⇒ (v) follow from formal properties of the adjunctions and the spectral invariants. 

Under some assumption on F, we will show that the conditions (i), (ii), (iii) are equivalent. 
See <ref> for the dual statements (iv), (v). 


    Assume that F is cohomologically constructible and q_π ((F)∩{τ >0}) is bounded in .
    Then the following conditions for a compact subset A⊂ T^*M and an idempotent e F→ F are equivalent:  
    
    
  * A is ζ_e-heavy,
    
  * there exist an object G∈_A(T^*M) and an element β∈ h_∞^* (F,G) satisfying c(β e; F,G)≠ +∞,
    
   there exist an object G∈_A(T^*M) and an element β∈ h_∞^* (F,G) satisfying β e≠ 0 ∈ h_∞^* (F,G),
    
  * the morphism η_A,F F→ι_A^*F in h_∞(T^*M) induced by the unit morphism of the adjunction satisfies c(η_A,F e;F, ι_A^*F)≠∞. 
    
   the morphism η_A,F F→ι_A^*F in h_∞(T^*M) induced by the unit morphism of the adjunction satisfies η_A,F e≠ 0∈ h_∞^*(F, ι_A^*F), 







There are many F satisfying the assumption of the proposition. 
However, this assumption imposes a non-trivial constraint on the partial symplectic quasi state ζ_e. See <ref> below. 

The main part of the proof is to prove (i) ⇒ (iii).  
The following proposition is a key observation for this part. 


    Let F be an object of (T^*M) and G_∙→(T^*M) be a functor.
    Assume the following:
    
    
        
  * F is cohomologically constructible and q_π ((F)∩{τ >0}) is bounded in .
        
  * q_π ((G_i) ∩{τ >0}) are uniformly bounded below. 
    
    Let (β_i F→ G_i)_i∈ be morphisms which are compatible with structure morphisms of G_∙ and β_∞ F→ G_∙ be the induced morphism. 
    Then 
    
    lim_i c(β_i;F,G_i)=c(β_∞;F, G_∙).




    lim_i c(β_i;F,G_i)≤ c(β_∞;F,_i G_i) is obvious. 
    If the conclusion is not true, we can take a,b∈ satisfying lim_i c(β_i;F,G_i)<a<b<c(β_∞;F,_i G_i). 
    

    For any R'≥ 0, i∈ and ε∈ [0,b-a], 
    the morphism F⋆_M× [-R',∞)→ G_i induced by β_i can not be lifted to a morphism F⋆_M× [a+ε,∞)→ G_i.
    This means that the induced morphism F⋆_M× [-R',a+ε)→ G_i is non-zero. 
    On the other hand, there exists a real number R>0 such that
    F⋆_M× [-R+ε , a+ε)→_i G_i is zero for any ε∈ [0,b-a].  

    By the adjunctions and <ref>, for any G∈(T^*M), 
    
    _(T^*M)(F⋆_M× [-R+ε, a+ε),G)≃_ (pt)(_[ε,∞), q_ *(D^T(F⋆_M× [-R,a))⋆ G))

    holds. 

    Since q_ *≃ q_ !, 
    
    _i q_ *(D^T(F⋆_M× [-R,a))⋆ G_i) ≃ q_ *(D^T(F⋆_M× [-R,a))⋆_i G_i ).


    Put H_∙→ (pt) be the functor q_ *(D^T(F⋆_M× [-R,a))⋆ G_∙)[-1] and H_∞ H_∙. 
    Since _[ε, ∞)≃_(-∞, ε)[1] in (pt), 
    for any ε∈ [0,b-a] and i∈, the induced morphism _(-∞, ε)→  H_i is non-zero. 
    On the other hand, the induced morphism 
    _(-∞, ε)→ H_∞ is zero for each ε∈ [0,b-a].

    For each H∈_{τ≥ 0 }() and c∈,  
    the stalk H_c is isomorphic to _ε'→ +0Γ((-∞ , c+ε');H). 
    Let us identify () with the subcategory of _{τ≥ 0} (). 
    Hence the morphisms (_(-∞, ε)→ H_i)_ε∈ [0,b-a] above induce a nonzero element in the cohomology H^0((H_i)_ε) of the stalk for each ε∈ [0,b-a) and i∈. 
    
    Since stalk and cohomology commute with colimits, 
    _i H^0((H_i)_ε)≃ H^0((H_∞)_ε) and hence a non-zero element in H^0((H_∞)_ε) is induced for each ε∈ [0,b-a). 
    This contradicts to that 
    _(-∞, ε')→ H_∞ is zero for each ε' ∈ (ε,b-a]. 



    Assume that F is cohomologically constructible and q_π ((F)∩{τ >0}) is bounded in .
    Then the following are equivalent:
    
        
  * c(β e;F,_A  F)=+∞,
        
  * c(β e;F,_A  F)>0,
        
  * for any s∈ there exists H^s∈ C_cc(T^*M, A) with c(e;F,_H^s F)>s,
        
  * there exists H∈ C_cc(T^*M, A) with c(e;F,_H  F)>0.
    


    (iii) ⇒ (iv) is trivial. 
    
    (iv) ⇒ (ii) is obvious by the inequality c(e;F,_H  F)≤ c(β e;F,_A  F) for any H∈ C_cc(T^*M, A). 

    (ii) ⇒ (i) is deduced by <ref> _A _A≃_A . 
    If c(β e;F,_A  F)>0, then for each s∈ (0, c(β e;F,_A  F)) there exists a lift α F→ T_-s_A  F of β e. 
    Then the composition 
    
    T_-s_A^(α)∘α F→ T_-s_A  F→ T_-2s_A _A  F≃  T_-2s_A  F

    is also a lift of β e and hence 2s≤ c(β e;F,_A  F). This means c(β e;F,_A  F)=+∞.

    (i) ⇒ (iii) follows from <ref> as follows. 
    Let us prove the contraposition (iii) ⇒ (i) and assume that there exists s∈ such that c(e;F,_H F)≤ s for any H∈ C_cc(T^*M, A). 
    By fixing a final functor H_∙→_cc(T^*M, A), we can write _A F as the sequential colimit _n _H_n F. 
    By the assumption, c(e;F,_H_n F)≤ s holds for each n∈. By <ref>, c(β e;F,_A  F)=lim_n c(e;F,_H_n F)≤ s <+∞ is deduced. Note that the assumption (2) in the <ref> is guaranteed by the assumption on F. 




    “(iii)⇒ (ii)” and “(ii) ⇒ (ii')” are trivial implications. 
    By <ref>, the condition (iii) is equivalent to (iii'). 
    By the universal property of ι_A^*F, “(ii') ⇒ (iii')” holds.
    Hence all the four conditions (ii), (ii'), (iii) and (iii') are equivalent. 
    
    <ref> asserts “(ii) ⇒ (i)”. 
    
    We prove the contraposition “(iii') ⇒ (i)”. 
    Let us assume c(β e;F, ι_A^*F)=+∞.
    Then there exists H∈ C_cc(T^*M, A) with c(e;F,_H F)>0 by <ref>. 
    Hence ζ_e(-H)≤ℓ_e (-H)=-c(e;F,_H F)<0, which means that A is not ζ_e-heavy.


For the case e=𝕀_F, we obtain a slightly refined version of <ref> by the universal property of ι_A^*F.

    Assume that F is cohomologically constructible and q_π ((F)∩{τ >0}) is bounded in .
    For any compact subset A⊂ T^*M,
    A is ζ_F-heavy if and only if ι_A^*F≄0 in h_∞ (T^*M). 


    If A is ζ_F-heavy, ι_A^*F≄0 in h_∞ (T^*M) by (i) ⇒ (iii') of <ref>. 

    If ι_A^*F≄0 in h_∞ (T^*M), the natural morphism τ_cι_A^*F→ T_c ι_A^*F is non-zero for any c≥ 0.
    By the adjunction isomorphism _(T^*M) (F,T_c ι_A^*F)≃__A(T^*M) ( ι_A^*F,T_c ι_A^*F), the composite τ_c∘η_A,F is also non-zero for any c≥ 0. 
    This means η_A,F is non-zero as a morphism of _∞(T^*M). 
    The assertion follows from (iii') ⇒ (i) of <ref>. 



    The author do not know whether the dual statement (i) ⇒ (v) holds or not. 
    At least, the dual statement of the key proposition <ref> does not hold. 
    
    Let V_0 be the direct sum ^. For each n∈, define V_n⊂ V_0 as the subspace consisting of the elements (a_i)_i∈ with a_i=0 for each 0≤ i <n.
    Consider the linear map V_0→ as taking the sum of its components.
    Let F_n (V_n)_[0,∞) and G_[0,∞).
    Then we obtain a projective system F_∙^op→ (pt) and consistent morphisms β_i F_i→ G. In this case, c(β_i;F_i, G)=0 for each i∈, however c(β_∞; lim F_∙, G)=+∞ since lim F_∙=0.



    A compact subset A⊂ T^*M is cohomologically superheavy if the  morphism βμ_M_M× [0,∞)→ι_A^*_M× [0,∞)⊗or_M×[n] 
    is non-zero in h_∞ (T^*M). 


By <ref>, cohomological superheaviness implies ζ_MVZ-superheaviness.
The author do not know whether the inverse holds in general. 
Heaviness/superheviness is preserved by Hausdorff limits, that is clear by <ref>. 
On the other hand, the corresponding property for cohomological superheaviness is not obvious. 
However, the Viterbo conjecture implies the inverse statement. 


    If the Viterbo conjecture for M is true over , every ζ_MVZ-heavy subset in T^*M is cohomologically superheavy.


    Take A⊂ T^*M is a ζ_MVZ-heavy subset.
    We may assume A is contained in the interior of DT^*M by choosing the Riemannian metric on M sufficiently small. 
    By (i) ⇔ (iii) of <ref>, c(η_A,_M×[0,∞); _M×[0,∞), _A^ (_M× [0,∞)))≠∞ holds. 
    By (i) ⇔ (iv) of <ref>, for any H∈ C_cc(T^*M,A), c(1; _M×[0,∞), _H^ (_M× [0,∞)))≤ 0. 
    Take a final functor H_∙→_cc(T^*M,A) satisfying 0≤ H_n and H_n is constant outside DT^*M for each n.  
    By the monotonicity of c, c(1; _M×[0,∞), _H_n^(_M× [0,∞)))=0 for each n. 
    If the Viterbo conjecture for M is true over , since ϕ_1^H_n(0_M)⊂ DT^*M, there exists a real number R>0 such that γ(ϕ_1^H_n(0_M))<R for each n. 
    Hence by <ref>, we obtain
    
    c(μ_M; _M×[0,∞), _H_n^ (_M× [0,∞))⊗or_M×)<R

    for each n. 
    By <ref>, we also get 
    
    c(η_A,_M×[0,∞)μ_M; _M×[0,∞), _A^ (_M× [0,∞))⊗or_M×)≤ R.

    This implies A is cohomologically superheavy. 



Let us conclude this section by observing that the assumption on F imposes a constraint on ζ_e. 

    Assume that F is cohomologically constructible and q_π ((F)∩{τ >0}) is bounded in .
    Let e F→ F be an idempotent in _∞(T^*M) with c(e;F,F)<+∞.
    There exists G∈(M) such that
    ζ_e=ζ_G⊠_[0,∞). 
    Moreover if (F) is compact, then G is cohomologically locally constant.  


    Consider the morphism
    
    r M××_>0→ M×; (x,t,s)↦(x,t/s)

    and define F_n r^-1(F)|_M××{1/n} for each n∈_≥ 1.
    By <cit.>, d(F_n, F_m)< +∞ for any m,n ∈_≥ 1 and the sequence (F_n)_n is Cauchy. 
    Let F_∞ be a convergence point of (F_n)_n, which exists by <cit.> or <cit.>. 
    Let e' F_∞→ F_∞ be an idempotent corresponding to the idempotent e F→ F through some isomorphism F≃ F_∞ in h_∞(T^*M) by <ref>.

    By a microsupport estimation, we can see that F_∞ is of the form G'⊠_[0,∞) with G'∈(M). 
    If (F) is compact, the microsupport estimate also says G' is cohomologically locally constant. 
    The idempotent in e' corresponds to an idempotent e” G→ G in h(M). 
    Since h(M) is idempotent complete by <cit.>, e” corresponds to some object G which is a retract of G'. 
    Note that if G' is cohomologically locally constant, so is the retract G.  





§ EXAMPLES


In this section, we give some examples that we can check the ζ_MVZsuperheaviness by our theorems and known results about the Viterbo conjecture.


    For a continuous function f M→, we define a subset Z_f⊂ M× as {(x,t) ∈ M×| t≥ -f(t) } and define F_f_Z_f.
    If f is of class C^1, the derivative df M→ T^*M is defined as a continuous section. In this case, we denote by Γ_df⊂ T^*M the image of df. We then have that (F_f)=Γ_df.
    For a general continuous function f, 
    (F_f) is called the Vichery subdifferential <cit.>. 
    Note that (F_f) is not necessarily compact for a general continuous function. However, if f is Lipschitz continuous, then (F_f) is compact.

    If (F_f) is compact, it is ζ_MVZ-superheavy by <ref>.




    Any compact exact Lagrangian submanifold in a cotangent bundle is ζ_MVZ-superheavy. This is deduced from <ref> and the existence result <cit.> of the sheaf quatizations of compact exact Lagrangian submanifolds.


The completion of the space of Lagrangians with respect to the spectral metric gives interesting examples. 
As a common generalization of <ref>, we have the following. 


    Viterbo <cit.> introduced the notion of γ-supports for elements of the completion of the space of some Lagrangian submanifolds with respect to the spectral distance.
    The γ-support of an element of the completion coincides with the reduced microsupport of an associated sheaf quantization <cit.>. 
    Especially, a γ-support is realized as a reduced microsupport of a sheaf F with d(_M× [0,∞), F)<∞, 
    Hence, compact γ-supports are ζ_MVZ-superheavy by <ref>.



    A Birkhoff attractor is a closed subset of T^*S^1 associated to a map called conformally exact symplectic map on T^*S^1. 
    It can be an indecomposable continuum. 
    See <cit.> and its references.
    The Birkhoff attractors are realized as γ-supports by <cit.>. 
    Hence we obtain an examples of a ζ_MVZ-superheavy subset which is an indecomposable continuum in T^*S^1.


With a morphism of manifolds f  X → Y, we associate the following commutative diagram of morphisms of manifolds:

    T^*X [d]_-π    X ×_Y T^*Y [d] [l]_-f_d[r]^-f_π    T^*Y [d]^-π
    
    	X @=[r]     X [r]_-f     Y,

where f_π is the projection and f_d is induced by the transpose
of the tangent map f'  TX → X ×_Y TY.

The following lemma is a consequence of <ref>. 

    Let p  N → M be a submersion.
    Let G ∈(T^*N) and assume that (_M× [0,∞), G) ≠∅ and p̃=p ×𝕀_ is proper on (G) and moreover (G)∩ (p_d) is compact.
    Then (p̃_*G) is ζ_MVZ-heavy.


Note that under the situation of the above lemma, we have (p̃_*G) ⊂ p_π p_d^-1(G).




    Take S^n={(x_0,…, x_n+1)∈^n+1|∑_i=0^n+1x_i^2=1 }, P^n S^n /(/2) and let p S^n→ P^n be the quotient map. 
    Let us take f S^n→; (x_0,… ,x_n+1)↦ x_0. 
    Then ι p_π∘(p_d)^-1∘ df S^n→ T^* P^n is an exact Lagrangian immersion. 

    Applying <ref> to p S^n→ P^n and F_f=_Z_f∈(T^*S^n), we obtain ι (S^n)⊂ T^* P^n is ζ_MVZ-heavy. 
    Since the Viterbo conjecture for P^n is true over any coefficient field, ι (S^n)⊂ T^* P^n is ζ_MVZ-superheavy by <ref>. 
    If the characteristic of  is not 2, the superheaviness can be deduced directly from <ref>. 

    
    
    For n=1, ι (S^1)⊂ T^*S^1 is the singular fiber of the pendulum. 



    Let us take S^2n+1={(z_0,…, z_n+1)∈^n+1|∑_i=0^n+1|z_i|^2=1 } and consider the quotient map p S^2n+1→ P^n by the U(1)-action. 
    
    Let us take f S^2n+1→; (z_0,… ,z_n+1)↦ z_0 and then the fiberwise singular points of f along p is the sphere S^2n={(z_0,…, z_n+1)∈ S^2n+1| z_0∈}. 
    Then the differential of f on S^2n along the base direction gives an exact Lagrangian immersion ι S^2n→ T^* P^n.  

    Applying <ref> to p S^2n+1→ P^n and F_f=_Z_f∈(T^*S^2n+1), we obtain ι (S^2n)⊂ T^* P^n is ζ_MVZ-heavy. 
    Since the Viterbo conjecture for P^n is true over any coefficient field, ι (S^2n)⊂ T^* P^n is ζ_MVZ-superheavy. 


    For n=1, ι (Σ_f)⊂ T^* S^2 is the superheavy singular fiber of the spherical pendulum, whose superheaviness is proved by <cit.>. 
    See <cit.> for detailed description for the fibers of the spherical pendulum. 
    


     <ref> can be extended to the quaternionic version.[Lagrangian immersions in <ref> were treated in an intensive lecture series by Kenji Fukaya at Kyoto University in January 2023.]



[The Lagrange top] 
    Let be qSO(3)→ S^2 the map picking up the first column of each matrix. 
    In n=1 case of <ref>, the superheaviness of the singular fiber of the spherical pendulum is proved by the existence of the object p_*F_f∈(T^*S^2). 
    The reduced microsupport of q^-1p_*F_f∈(T^*SO(3)) is a singular fiber of the Lagrange top. 
    By <ref>, the singular fiber is ζ_MVZ-superheavy. 
    This is also proved by <cit.> and
    see <cit.> also for detailed description for the fibers of the Lagrange top. 








§ PARTIAL SYMPLECTIC QUASI-STATES FROM KERNELS


In this appendix, we discuss partial symplectic quasi-states that are directly constructed from GKS kernels. 
Many of the proofs are parallel to those handled in the main text.

We define ζ_Ham C_cc(T^*M)→ by

    ζ_Ham(H)  -lim_kc(1;_Δ_M× [0,∞), _H^ -k)/k.

This functional will be a partial symplectic quasi-states and correspond to one defined in <cit.>.
More generally, for K ∈(T^*M× T^*M) an idempotent e K → K in _∞ (T^*M× T^*M) with c(e;K,K)=+∞,
we can define

    ζ_Ham, e(H)  -lim_kc(e;K, _H^ -k(K))/k.


    For each K ∈(T^*M× T^*M) and each idempotent e K → K in _∞ (T^*M× T^*M) with c(e;K,K)=+∞, 
     ζ_Ham, e is a partial symplectic quasi-state. 


    The proof is parallel to that of <ref> by replacing ℓ_e (ϕ) -c(e;_ϕ^ (F), F) by -c(e;_ϕ^ (K), K).


    
        
  * For each F ∈(T^*M), each idempotent e F → F in _∞ (T^*M) with c(e;F,F)=+∞, and each H∈ C_cc(T^*M), ζ_e (H)≤ζ_Ham(H).
        
  * For each K ∈(T^*M× T^*M), each idempotent e K → K in _∞ (T^*M× T^*M) with c(e;K,K)=+∞, and each H∈ C_cc(T^*M),
        ζ_Ham, e (H)≤ζ_Ham(H).
    


    (i) For each k,  
    
    c(e;F,_H^ -k(F))≥ c(1;_Δ_M× [0,∞), _H^ -k) +c(e;F,F)

    holds. Hence by taking the limit k→∞, we obtain the assertion. 

    (ii) is parallel to (i). 


The following hold:

    
  * Every ζ_e-heavy subset is ζ_Ham-heavy for each F ∈(T^*M) and each idempotent e F → F in _∞ (T^*M) with c(e;F,F)=+∞.
    
  * Every ζ_Ham, e-heavy subset is ζ_Ham-heavy  for each K ∈(T^*M× T^*M) and each idempotent e K → K in _∞ (T^*M× T^*M) with c(e;K,K)=+∞.



The following proposition is a direct consequence of <ref>.

    ζ_Ham(H)≥ 0 for each H∈ C_c(T^*M).



    There is no ζ_Ham-superheavy subset. 


For a compact subset A of T^*M, we defined

    _A_H∈_cc(T^*M,A)_H.



    For a compact subset A of T^*M, the following are equivalent:
    
        
  * A is ζ_Ham-heavy,
        
  * the canonical morphism _Δ× [0,∞)→_A is non-zero as a morphism of h_∞(T^*M× T^*M),  
        
  * _A ≄0 in h_∞(T^*M× T^*M). 
    


    A proof of (i) ⇒ (ii) is parallel to that of (i) ⇒ (iii') of <ref>.

    A proof of (ii) ⇒ (i) is parallel to that of (i) of <ref>, once one replaced <ref> by <ref> below.

    The implication (ii) ⇒ (iii) is trivial. 

    Applying _A^ to the canonical morphism _Δ× [0,∞)→_A, we obtain the identity morphism 𝕀__A of _A. 
    Hence (iii) implies (ii). 


    Let H∈ C_cc(T^*M).
    If H≡ c on some open neighborhood of A then 
    _H_A≃ T_c_A. 


    We may assume c=0 since _H≃ T_c_H-c. 
    Moreover we may assume H is smooth since the kernel _H for continuous H is defined as a colimit of _H' for smooth H' and the colimit commute with . 

    Since H♯(`-)  C_cc^∞(T^*M× [0,1],A)→  C_cc^∞(T^*M× [0,1],A) is an order preserving bijection, 
    the induced functor _cc^∞(T^*M× [0,1],A)→_cc^∞(T^*M× [0,1],A) is an equivalence. 
    In particular, it is a final functor. 
    By the commutativity of  and , we obtain
    
    _H_A
               ≃_H _H'∈_cc^∞(T^*M× [0,1],A)_H'
       ≃_H'∈_cc^∞(T^*M× [0,1],A)_H♯ H'
       ≃_H”∈_cc^∞(T^*M× [0,1],A)_H”≃_A.




    The main result of <cit.> restricted for cotangent bundles is very similar to <ref>. 
    Their relative symplectic cohomologies are defined as some colimits of Hamiltonian Floer cohomologies. 
    In other words, the colimit is taken at the Hom-space level. 
    On the other hand, our colimit is taken in the sheaf category (T^*M× T^*M). 
    Due to the fact that Hom and colim do not commute in general, 
    the direct relationship between our results and theirs is not clear. 
    The author expect that colimit in this paper, being taken in the sheaf category, would be easier to study.
    
    
    Since the relation between sheaf kernels and symplectic cohomologies of domains in cotangent bundles are described in <cit.>,
    the author expects a concrete relation between the kernel _A and relative symplectic cohomology of A ⊂ T^*M. 


For ζ_Ham-heavy subset A⊂ T^*M, we define ζ_Ham, A C_cc(M)→ by

    ζ_Ham, A(H)  -lim_kc(𝕀__A;_A, _H^ -k( _A))/k.

This is a special case of above construction of ζ_Ham,e. 
At the end of this appendix, we introduce the basic properties of ζ_Ham,A. 

    For ζ_Ham-heavy subset A⊂ T^*M, we have the following:
    
        
  * ζ_Ham, A is a partial symplectic quasi-state. 
        
  * For every H∈ C_cc(T^*M), ζ_Ham, A(H)≤ζ_Ham(H). 
        
  * A is ζ_Ham, A-superheavy. 
    


    (i) and (ii) are already stated for a more general setting. 

    (iii) is a direct consequence of <ref>. 










Tomohiro Asano: 
Research Institute for Mathematical Sciences, Kyoto University, Kitashirakawa-Oiwake-Cho, Sakyo-ku, 606-8502, Kyoto, Japan.

E-mail address: , 


