Specialization of cycles and the K-theory elevator
Pedro Luis del Angel, Charles Doran, Jaya Iyer, Matt Kerr, James D. Lewis, Stefan Müller-Stach, Deepam Patel
SSPECIALIZATION OF CYCLES AND THE K -THEORYELEVATOR P. LUIS DEL ÁNGEL R., C. DORAN, M. KERR, J. LEWIS, J. IYER,S. MÜLLER-STACH, AND D. PATEL
Abstract.
A general specialization map is constructed for higherChow groups and used to prove a “going-up” theorem for algebraiccycles and their regulators. The results are applied to study thedegeneration of the modified diagonal cycle of Gross and Schoen,and of the coordinate symbol on a genus-2 curve.
They have ladders that will reach further, but no one will climb them. – A. Sexton, “Riding the Elevator into the Sky”
Contents Introduction
A first view of going up: semi-nodal degenerations
Motivic picture: Specialization and going-up
Degeneration of a modified diagonal cycle
Limits of higher normal functions
Application to a conjecture from topological stringtheory
Introduction
The aim of this paper is to describe limiting invariants for general-ized normal functions of geometric origin at a singularity of the un-derlying period mapping. To describe the underlying geometry, let¯ π : X → S be a proper, dominant morphism of smooth quasi-projectivevarieties over C , with dim S = 1 and smooth restriction π : X ∗ →S ∗ = S\{ s } . Write X s = ¯ π − ( s ), and set V := R p − r − π ∗ Q ( p ),with monodromy operator T about s . Consider a higher Chow cy-cle Z ∗ ∈ CH p ( X ∗ , r ) Q ∼ = H p − r M ( X ∗ , Q ( r )), and if r = 0 assume that Mathematics Subject Classification.
Primary: 14C25, 19E15; Secondary:14C30. a r X i v : . [ m a t h . AG ] S e p DEL ÁNGEL, DORAN, KERR, LEWIS, IYER, MÜLLER-STACH, AND PATEL the restrictions Z s = ı ∗ s Z ∗ are homologous to zero. Then there is an as-sociated (“higher”, if r >
0) admissible normal function ν ∈ ANF r S ∗ ( V ),given by AJ p,rX s ( Z s ) ∈ Ext ( Q , H p − r − ( X s , Q ( p ))) on fibers of π .General formulas for the regulator maps AJ p,r , first constructed byBloch [B5], were given in [KLM]. They can often be difficult to com-pute directly; even for showing that the normal function is nonzero,one often makes do with the associated infinitesimal invariant, inho-mogeneous Picard-Fuchs equation, or (if r >
0) the presence of a non-torsion singularity at s . In the absence of a singularity, one can alsoconsider the limit of the normal function at s : indeed, if the cycleclass cl p,r X ∗ ( Z ∗ ) ∈ Hom ( Q , H p − r ( X ∗ , Q ( p ))) has vanishing residue on X s , then ν extends to S , with ν ( s ) in the generalized Jacobian ofker( T − I ) ⊆ H p − r − ( X s , Q ( p )).A useful technique for computing this limiting value is given by spe-cialization : if Z ∗ lifts to Z ∈ CH p ( X , r ) Q , then we obtain a class ı ∗ s Z in the motivic cohomology H p − r M ( X s , Q ( p )). This formalism, and itsrelation to the “naive” specialization to CH p ( X s , r ) Q , is discussed indetail in §
3. As a simple example, one can think of a difference of sec-tions of a family of elliptic curves that degenerate to a nodal rationalcurve: the class of the naive specialization is always zero, whereas thespecialization into motivic cohomology takes values in C ∗ .Given the specialized cycle ı ∗ s Z , then, we can use of a semi-simplicialhyperresolution of X s to compute its Abel-Jacobi class in absoluteHodge cohomology H p − r H ( X s , Q ( p )) ∼ = Ext ( Q , H p − r − ( X s , Q ( p )) . The main general result of this paper (Theorem 5.2) is that the imageof this class under the Clemens retraction computes ν ( s ). Note thatthe case of a semistable degeneration has been treated carefully for r = 0 [GGK], so we concentrate in § higher normal functionsetting, which behaves a bit differently.The even-numbered sections are devoted to worked examples andspecial cases, all of which exhibit the phenomenon referred to in thetitle: this is a 7-author paper, and some of us prefer “ K -theory eleva-tor”, others “going up”. Whatever one wishes to call it, we all felt itmerited a systematic exposition, given the many contexts in which itarises (e.g. [JW], [dS], [DK], [Ke], [GGK], [Co]). In the event that X s is a normal crossing variety, and ı ∗ s Z “comes from” its c th coskeleton(with desingularization Y [ c ] ), the basic point is that we can interpretpart of ν ( s ) as the regulator of a class in CH p ( Y [ c ] , r + c ) Q . So in effectone goes up from K alg r ( X s ) to K alg r + c ( Y [ c ] ).The special case we study in § X s the product of a nodal rational curve Q by a -THEORY ELEVATOR 3 smooth variety. We briefly recall results from [KLM, KL], and thenuse them to directly compute the limit of the fiberwise regulator maps(Theorem 2.2). This is applied in § K classes on elliptic curves. A relatedexample comes much later, in §
6, where we specialize a K class on afamily of genus two curves. The resulting number-theoretic identities,(6.13) and (6.14), had been proposed by M. Mariño in recent privatecorrespondence with two of the authors, on the basis of the t’ Hooftlimit of a far-reaching conjectural relationship between the spectrum ofa quantum curve and the enumerative geometry of its mirror [CGM].But the motivation for this paper goes back much further, to theseminal work of Collino [Co], based on a fascinating idea which heattributes to Bloch. Let C/ C be a general genus 3 curve, with Ja-cobian J ( C ). Then the Ceresa cycle ξ := C − C − ∈ CH ( J ( C ))defines a non-torsion element of the Griffiths group Griff ( J ( C )) [Ce].Collino considers a one-parameter deformation of J ( C ), degeneratingto a singular variety “isogenous to” J ( D ) × Q , where D is a generalgenus 2 curve. In the sense described above, ξ “goes up” to a K class ξ ∈ CH ( J ( D ) , D varies) to be regulator indecomposable . This gives analternative proof of the nontriviality of ξ .A further degeneration to E × Q × Q (up to isogeny), for somegeneral elliptic curve E , leads (by iteration of the “going up” procedure)to a non-torsion class ξ ∈ CH ( E, Q leaves us with a class ξ ∈ CH (Spec( C ) ,
2) (in fact defined over Q ( i )). Alternatively, onemay degenerate C directly to a rational curve with three nodes and godirectly to ξ as in [GGK, § IV.D], where the regulator of this class iscomputed (and shown to be nontorsion) directly.In §
4, the first step ( K (cid:32) K ) of this procedure is made muchmore precise, and applied to study “going up” for the modified (small)diagonal cycle ∆ ∈ CH ( C × C × C ) [GS], which is closely relatedto Ceresa’s cycle. In particular, we obtain a regulator indecomposablecycle in CH ( D × D, §§ Q -coefficients, denoted by a subscript Q . (This is a basic requirement forHanamura’s construction [Ha].) When describing the construction of DEL ÁNGEL, DORAN, KERR, LEWIS, IYER, MÜLLER-STACH, AND PATEL motivic cohomology, we also require intersection conditions on cycles(and higher cycles) which permit them to be pulled back. In particular,if Y I = Y i ∩ · · · ∩ Y i ‘ is a substratum of a normal crossing variety, and Z ∈ Z p ( Y I , r ) is a higher Chow precycle, we might impose the conditionthat Z properly intersect the products of all Y J ( J ⊃ I ) and all facesof (cid:3) r . Such conditions will be denoted throughout by a subscript “ Acknowledgments.
The authors acknowledge partial support underNSF FRG grant DMS-1361147 (Kerr), NSF grant DMS-1502296 (Pa-tel), grants from the Natural Sciences and Engineering Research Coun-cil of Canada (Doran, Lewis), and DFG grant SFB/TRR 45 (Müller-Stach). We thank M. Mariño for bringing [CGM] to our attention,and D. Ramakrishnan for suggesting to look at the degeneration of themodified diagonal cycle.2.
A first view of going up: semi-nodal degenerations
We begin by providing a concrete view of “going up” in the verysimplest setting: that of a semi-stable degeneration with singular fiberthe product of a smooth variety and a nodal rational curve. In additionto setting the stage for §§ § Bloch’s higher Chow groups.
The higher Chow groups are analgebraic version of ordinary simplicial Borel-Moore homology. Given W/ C quasi-projective, let Z p ( W ) denote the free abelian group gener-ated by subvarieties of codimension p in W . Consider the “algebraic r -simplex” ∆ r = Spec ( C [ t , . . . , t r ] (cid:16) − P rj =0 t j (cid:17) ) ’ C r , and put Z p ∆ ( W, r ) = ( ξ ∈ Z p ( W × ∆ r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ meets all faces { t i = · · · = t i ‘ = 0 , ‘ ≥ } properly ) . Denoting by ∂ j : Z p ∆ ( W, r ) → Z p ∆ ( W, r −
1) the restriction to j -th facet t j = 0, we note that ∂ = P rj =0 ( − j ∂ j : Z p ∆ ( W, r ) → Z p ∆ ( W, r − ∂ = 0. Definition 2.1.1. CH r ( W, m ) := homology of ( Z r ∆ ( W, • ) , ∂ ) at • = m . -THEORY ELEVATOR 5 Alternate take: Cubical version.
Let (cid:3) r := ( P \{ } ) r , withcoordinates z i , and ∂ i , ∂ ∞ i the restriction maps to the facets z i =0 , z i = ∞ respectively. The rest of the definition is completely analo-gous (with c p ( W, r ) denoting cycles meeting all faces properly) exceptthat one has to divide out degenerate cycles. More specifically, letPr j : (cid:3) r → (cid:3) r − be the projection forgetting the j th factor. Then thedegenerate cycles are the subgroup d p ( W, r ) := r X j =0 Pr ∗ j (cid:16) c p ( W, r − (cid:17) ⊂ c p ( W, r ) , and we take Z p ( W, r ) := c p ( W, r ) /d p ( W, r ) with differential ∂ = r X j =1 ( − j − (cid:16) ∂ j − ∂ ∞ j (cid:17) : Z p ( W, r ) → Z p ( W, r − . By [L2, Thm. 4.7], the simplicial and cubical complexes are quasi-isomorphic (with Z -coefficients), so that H r ( Z p ( W, • )) ∼ = CH p ( W, r ) . Remark 2.2.1.
In [Ha], Hanamura defines Chow cohomology groups CH p ( W, r ) for quasi-projective varieties through a hypercovering, as-suming resolution of singularities for varieties over the ground field. Inthe case of smooth varieties this coincides with Bloch’s higher Chowgroups. See the discussion below Remark 3.1.7 for details.2.3.
The currents.
If ( z , ..., z r ) ∈ (cid:3) r are affine coordinates, set T r := (2 π i ) r T r := (2 π i ) r δ [ −∞ , r , Ω r := Z (cid:3) r r ^ j =1 d log z j , and R r := Z (cid:3) r log z r ^ j =1 d log z j − (2 π i ) Z [ −∞ , × (cid:3) r − log z r ^ j =3 d log z j + · · · +( − π i ) r Z [ −∞ , r − × (cid:3) d log z r . For ξ ∈ Z p ( X, r ), let π : | ξ | ⊂ X × (cid:3) r → X , π : | ξ | ⊂ X × (cid:3) r → (cid:3) r .We put(2.1) R ξ = ( π , ∗ ◦ π ∗ ) R r , Ω ξ = ( π , ∗ ◦ π ∗ )Ω r , T ξ = ( π , ∗ ◦ π ∗ ) T r , and T ξ = (2 π i ) r T ξ . Recall that in the Deligne cohomology complex, M •D = Cone n C p + • X ( X, Z ( p )) ⊕ F p D p + • X ( X ) → D p + •− X ( X ) o [ − , DEL ÁNGEL, DORAN, KERR, LEWIS, IYER, MÜLLER-STACH, AND PATEL the differential D is given by D (cid:16) (2 π i ) p − r ( T ξ , Ω ξ , R ξ ) (cid:17) = (2 π i ) p − r ( dT ξ , d Ω ξ , T ξ − Ω ξ − dR ξ ) . = (2 π i ) p − r +1 ( T ∂ξ , Ω ∂ξ , R ∂ξ ) ;the resulting cohomology at • = − r is H p − r D ( X, Z ( p )). To guaranteethat the currents in (2.1) are defined, we have to restrict to a subcom-plex Z p R ( X, • ) of cycles meeting real faces of [ −∞ , m properly. Themain results we shall need are summarized in: Theorem 2.1. (i) [KLM]
The formula ξ (2 π i ) p − r ( T ξ , Ω ξ , R ξ ) in-duces a morphism of (cohomological) complexes Z p R ( X, −• ) → M •D . (ii) [KL] The inclusion Z p R ( X, • ) , → Z p ( X, • ) is a rational quasi-isomorphism. In view of (ii), we shall work with higher Chow groups with Q -coefficients CH p ( X, r ) Q for the remainder of this section.2.4. A key prototypical situation.
Let ∆ ⊂ C be a disk centeredat 0 ∈ ∆, with ∆ ∗ = ∆ \{ } , and consider the diagram(2.2) X , → Xf y y f ∆ ∗ , → ∆ , where f is a proper family of complex projective varieties of (relative)dimension d , and further, f is smooth. This should be seen as a re-striction of a global setting(2.3) X , → X y y B , → B, where all varieties are smooth and quasi-projective, B is a smoothscheme of dimension 1, and X → B is smooth and proper, with ∆ ⊂ B and ∆ ∗ = B ∩ ∆. Put X t = f − ( t ), for t ∈ ∆. Obviously X t is smoothprojective for t ∈ ∆ ∗ , and we can consider the monodromy operator T ∈ Aut ( H p − r − ( X t )( p )). Let us assume that X is reduced and ofthe form Y × Q , where Y is smooth, projective, and Q is a rationalcurve with a single node as singular set. In particular, T is unipotent.Now a cycle ξ ∈ CH p ( X , r ) Q can be assumed to meet all fibers { X t } t ∈ ∆ properly; and setting ξ t := X t · ξ , we will assume that ξ t A similar story holds if Q is replaced by a rational curve with multiple nodes. -THEORY ELEVATOR 7 belongs to CH p hom ( X t , r ) Q for t ∈ ∆. For t = 0, additional conditionswill be imposed in § ξ furnishes an element ofChow cohomology of X .Recall that for t ∈ ∆ ∗ we have the Abel-Jacobi invariantAJ( ξ t ) ∈ J p,r (cid:16) X t (cid:17) ’ h F d − p +1 H d − p + r +1 ( X t , C ) i ∨ H d − p + r +1 ( X t , Q )( p ) , given by the functional(2.4) ω t (2 π i ) r − m R m ( ξ t ) + (2 π i ) m Z ∂ − ( T m ( ξ t )) ! ( ω t ) . modulo periods, on test forms ω t ∈ F d − p +1 A d − p + r +1 d -closed ( X t ). Here T r ( ξ t )is T ξ t = P r X t ( ξ t ∩ { X t × [ −∞ , r } ), and R r ( ξ t ) = R ξ t ; writing themthis way will clarify the computation below.Consider the (co)homological situation on X . First of all, if p ∈ Q is the node, then Q \{ p } = C ∗ ; write S for the unit circle. Workingwith Q -coefficients, we have Q (1) ∼ = H ( Q )(1) ∼ = ←− H c ( C ∗ )(1) ∼ = H ( C ∗ ) = Q h S i with duals Q ( − ∼ = H ( Q )( − ∼ = −→ H BM1 ( C ∗ )( − ∼ = H ( C ∗ ) = Q h dlog( z )2 π i i . (One may also view ( −∞ ,
0) as the generator of the untwisted Borel-Moore homology group H BM1 ( C ∗ ).) The perfect pairing(2.5) { H p − r − ( Y )( p ) ⊗ H ( Q ) } × { H d − p + r ( Y )( d − p ) ⊗ H ( C ∗ ) } → Q may thus be interpreted via intersection or integration (on X ), withthe second factor identified with a summand of homology (of X ). Theplan is to view the limiting cycle ξ as defining an element in Chow cohomology , with Abel-Jacobi invariant in the generalized Jacobian ofthe first factor of (2.5).2.5. The limiting regulator.
We seek a formula for(2.6) AJ( ξ ) := lim t → AJ( ξ t ) ∈ J p,r ( X ) , where J p,r ( X ) := Ext (cid:16) Q , H p − r − ( X )( p ) (cid:17) ∼ = Ext ( Q , ker( T − I )( p )) DEL ÁNGEL, DORAN, KERR, LEWIS, IYER, MÜLLER-STACH, AND PATEL is the “limiting generalized Jacobian”. (The precise sense in which thelimit (2.6) is to be interpreted is discussed in § ξ ) ∈ (cid:16)h F d − p H d − p + r ( Y , C ) i ⊗ H ( C ∗ , C ) (cid:17) ∨ H d − p + r ( Y , Q )( − d + p ) ⊗ Q h S i corresponding to H ( C ∗ ) (rather than H ( C ∗ )).We shall use as “test form”(2.8) ω = π i η ∧ Ω , where η ∈ F d − p A d − p + r ( Y , C ) is closed and Ω = dlog z . Note that ω is a limit of classes ω t ∈ F d − p +1 H d − p + r +1 ( X t , C ) as t
0. This isa classical result stemming from an explicit description of the canonicalextension of the bundle with fibers H d − p + r +1 ( X t , C ) for t = 0 ∈ ∆(cf. [Zu, p. 190] or [GGK, III.B.7]).Next we impose several requirements on ξ at t = 0: first, that ξ meet properly X × (cid:3) r , sing( X ) × (cid:3) r , and all their subfaces. We canthen “naively” define ξ by using the canonical desingularization ˜ X := Y × P → Y × Q ⊂ X (sending { , ∞} to the node P ∈ Q ) to pull ξ back to ˜ ξ followed by push-forward under ˜ X (cid:16) X to CH p ( X , r ).But this process factors through the Chow cohomology group CH p ( X , r ) := H − m (cid:26) Cone (cid:18) Z p ( ˜ X , • ) ı ∗ − ı ∗∞ −→ Z p ( Y , • ) (cid:19) [ − (cid:27) and the image by CH p ( X , r ) → CH p ( X , r ) has no invariant in (2.7).So it is appropriate to consider ξ as an element of CH p ( X , r ) (andthereby view T ξ = Pr X ( ξ ∩ { X × [ −∞ , m } ) in F r H r − m ( X , Q ) = { } ). The general perspective will be covered in §§ ξ defines a class in Z p ( X , r ) ∂ − closed , as well as aclass in Z p ( Y , r + 1) ∂ − closed , the latter via this schema:(2.9) ξ ∈ Z p ( Y × Q × (cid:3) r ) Z p ( Y × P × (cid:3) r ) Z p ( Y × (cid:3) r +1 ) Z p ( Y , r + 1) . To give a brief glimpse of the idea: the generalized Jacobian bundle ∪ t ∈S ∗ J p,r ( X t )admits a canonical extension across the origin (cf. § ξ t ) extends holomorphically. The value in the fiber over the originis what we call lim t → A( ξ t ). This may be computed by taking limits of pairingswith families of test forms representing sections of the dual canonically extendedcohomology bundle (cf. Cor. 5.3). -THEORY ELEVATOR 9 In order to easily compute the regulator, we will also assume that ξ andits pullbacks (to ˜ X , sing( X )) meet the real sub-cube faces properly(resp. those of Y × (cid:3) r +1 ). Then (in view of (2.8)) we have the limitingformula(2.10) AJ( ξ t )( ω t ) t → (2 π i ) p − r − (cid:16) R r ( ξ ) + (2 π i ) r δ ζ (cid:17) ( η ∧ Ω ) , where ζ is a (2 d − p + r + 1)-chain on X = Y × Q with ∂ζ = T ξ ,properly meeting sing( X )( ∼ = Y ). The nodal point p ∈ Q corre-sponds to | ∂ [ −∞ , | in the schema (2.9) above. Let ζ be a lift of ζ in Y × (cid:3) . Then(2.11) ∂ n ζ ∩ { Y × [ −∞ , } o = ∂ζ ∩ n Y × [ −∞ , o ± ζ ∩ n Y × ∂ [ −∞ , o , and P r Y (cid:16) ζ ∩ n Y × ∂ [ −∞ , o(cid:17) = 0, since the lift arises from the samecopies of a membrane over a given nodal singularity. Therefore(2.12) ∂ (cid:16) P r Y (cid:16)n ζ ∩ { Y × [ −∞ , } o(cid:17)(cid:17) = ∂ζ ∩ n Y × [ −∞ , o . Again, via the schema (2.9) above, ξ has a lift (which we still denoteby ξ ) with support in Y × (cid:3) r +1 . With the aid of (2.12), intersectingthis lift with Y × [ −∞ , r +1 , followed by a projection to Y , is precisely ∂ζ Y , where ζ Y = P r Y (cid:16)n ζ ∩ { Y × [ −∞ , } o(cid:17) .To compute the limiting AJ invariant, we shall utilize the relation ofcurrents (cf. [KLM, (5.2)]) on (cid:3) n dR n = Ω n − (2 π i ) n T n − π i R ∂ (cid:3) n in the case n = 1, where it reads(2.13) Ω = dR + (2 π i ) T . In (2.10), we first consider the term δ ζ ( η ∧ Ω ) , which by (2.13) decomposes into two pieces:(2.14) (2 π i ) δ ζ ( η ∧ T ) = by (2.12) (2 π i ) δ ζ Y ( η );and δ ζ ( η ∧ d [ R ]) = ( − r δ ζ ( d [ η ∧ R ]) , This is possible (even if m = 0) since we assumed ξ ≡ hom
0, and 0 = [ T ξ ] ∈ H d − p + r ( X ) = ⇒ T ξ ] ∈ H p − r ( X ) due to the specific form of X . which by Stokes’s theorem (2.15) = ( − r T ξ ( η ∧ R ) = ( − r (( T r ∧ R )( ξ )) ( η )Recalling the relation ( − π i ) r T r ∧ R + R r ∧ Ω = R r +1 from [KLM],the remaining part of (2.10)(2.16) ( R m ( ξ ))( η ∧ Ω ) = (( R m ∧ Ω )( ξ )) ( η )now combines with (2 π i ) r (2.15) to yield simply R r +1 ( ξ )( η ) , so that altogether (2.10) becomes(2 π i ) p − r − ( R r +1 ( ξ )( η ) + ( − π i ) r +1 Z ζ Y η ) ≡ pds. AJ( ξ )( η ) . Summarizing, we have
Theorem 2.2.
Given the above setting of subsection 2.4 of a normalfunction induced by
AJ( ξ t ) ∈ J p,r (cid:16) X t (cid:17) , where t ∈ ∆ ∗ , ξ t ∈ CH p hom ( X t , r ) Q , and where X = Y × Q , then lim t → AJ( ξ t )( ω t ) = AJ( ξ )( η ) , where ξ is interpreted as defining a class in CH p ( Y , r + 1) Q . Remark 2.5.1. (i) The situation X = Y × Q can be replaced by Y × Q ‘ ( Y smooth) for ‘ ≥
1, and a parallel analysis expresses the limitingregulator as the regulator of a class in CH p ( Y , r + ‘ ) Q . But there is acaveat in order here: the total space X over ∆ cannot be both smoothand semistable if ‘ >
1. It all boils down to the situation V ( x y − t, ..., x N y N − t ) ⊂ C N × ∆, a variety which is singular at (0 , ..., N >
1. This can be remedied in a number of ways: by blowingup (along the lines of § § V ( x y − t , ..., x N y N − t N ) ⊂ C N × ∆ N ; not pursued here).(ii) Many natural moduli spaces do not contain singular fibers of theform X = Y × Q . For instance, let Z ⊂ P be a very general hyper-surface of high degree. Then Z does not contain any rational curves,and hence neither does any hyperplane section X of Z . Furthermore,there are Hodge-theoretic obstructions to having such a degeneration.This is another reason to develop the more general perspectives in §§ we are also using the general fact that R n vanishes along ( P ) n \ (cid:3) n = S nj =1 P ×· · · × { } × · · · × P ⊂ [ P ] × n , which here is just the vanishing of R = log z at 1. -THEORY ELEVATOR 11 A toy model.
Let π : X → P be the elliptic surface defined by y = x + x + t =: h ( x ) , and let Σ = { , ∞ , } ⊂ P denote the singular set of π . (Note that X and X − are nodal curves, while X ∞ is a simply-connected treeof P ’s. We wish to verify, as a first application of Theorem 2.2, thatCH ( X t , Q = { } for very general t ∈ P . Of course, this is a knownfact in view of Theorem 2.3. [Le2, As]
Let U = X \ n X , X − , X ∞ o . Then Γ (cid:16) H ( U, Q (2)) (cid:17) ’ Q ; moreover it is generated by [Ω ξ ] , [Ω ξ ] , where ξ = ( ( y − x ) , ( y + x ) )( y + xy − x , t ) ,ξ = ( ( i y + x + ) , ( i y − x − ) )( i y − x − i y + x + , − t − ) , are classes in CH ( U, Q ) . Indeed, given any class ξ ∈ CH ( U,
2) such that [Ω ξ ] is nonzeroin Γ( H ( U, Q (2))), standard arguments (injectivity of the topologicalinvariant) imply that AJ( ξ t ) (hence CH ( X t , t .For the approach based on limits, take a small disk ∆ centered at t = 0. For t ∈ ∆ ∗ , ξ t belongs to CH ( X t , Q , and for t = 0, weshall interpret ξ as an element of CH (Spec( C ) , Q . We attend toseveral details. First, X = V ( y = x + x ) is a nodal rational curveparameterized by P / { , ∞} via z z ( z − , z ( z + 1)( z − ! = ( x ( z ) , y ( z )) . The restriction of ξ to X may be written ξ = 9 z, (cid:16) − i y ( z ) + x ( z ) + (cid:17) , − i y ( z ) + x ( z ) + i y ( z ) + x ( z ) + ! , as a cycle in (cid:3) , and we set w ( z ) := 32 (cid:16) − i y ( z ) + x ( z ) + (cid:17) . Write γ for the closed path T z = [ −∞ ,
0] on X ; and note that, on γ , w ( z ) winds once clockwise about 0. Moreover one easily sees that(2.17) 2 − ≤ | w || γ ≤ − and − i y + x + i y + x + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ = w ¯ w (cid:12)(cid:12)(cid:12)(cid:12) γ . So along γ , ξ looks like ( z, w, w ¯ w ), and log( w ¯ w ) is zero at γ ∩ T w = { w = − } .For the regulator, then, R := AJ( ξ )(1) = π i Z ξ R = 9 Z γ log( w )dlog( w ¯ w )= 18 i Z γ log( w )darg( w )= ⇒ Im( R ) = 18 Z γ log | w | darg( w ) . Using the bounds (2.17) and reversing the path (for a positive measure),we conclude that(2.18) 36 π · log(2) ≤ Im( R ) ≤ π · log(2) . Consequently we have
Theorem 2.4.
AJ( ξ ) = 0 ∈ H D (Spec( C ) , Q (2)) ’ CQ · π . Remark . From a different point of view, limiting calculations wereperformed in [DK, § E T : XY = T (cid:16) X + Y (cid:17) , which is birational to a base change of the Tate curve viaΘ : ( X , Y , T ) (cid:16) − (2 T ) Y , (2 T ) X − (2 T ) Y , − (2 T ) (cid:17) = ( x, y, t ) . The symbol studied in [op. cit.] is { X , Y } = Θ ∗ ξ ; and there is abirational automorphism α : ( x, y, t ) (cid:16) − x − , i y, − t − (cid:17) of theTate curve with α ∗ ξ = ξ . Overall, α − ◦ Θ sends the fiber E − − =: E T isomorphically to X , and pulls ξ back to { X , Y } . Modulo a -THEORY ELEVATOR 13 conjectural relation in the Bloch group, it was shown in [op. cit.] that π i AJ (cid:16) { X , Y } E T (cid:17) = π G , where(2.19) G := X n ≥ ( − n (2 n + 1) − = L ( χ , R ) = 120 · G , whichagrees with (2.18) above.2.7. Speculation.
As another application of the semi-nodal instanceof the going-up principle, we briefly address a relationship between theGriffiths group of a threefold and the group of indecomposables on agiven surface.Begin with a diagram X , → X f y y fB , → B where X is a smooth projective fourfold, B is a smooth projective curveand f is smooth and proper. Put X t := f − ( t ), a smooth threefold. Acycle ξ ∈ CH ( X ) which is relatively homologous to zero determinesa normal function ν ξ : B → a t ∈ B ( C ) J , (cid:16) X t (cid:17) , with topological invariant [ ν ξ ] ∈ Hom
MHS ( Q , (cid:16) H ( B, R f ∗ Q (2)) (cid:17) . Whenthis is nonzero, then under suitable monodromy conditions, Griff ( X t ) Q = { } for very general t ∈ B ( C ).Now consider the situation where for some 0 ∈ B \ B , X = Y × Q .Viewing ξ as a class in CH ( Y , Q , we may ask whether it is inde-composable , i.e. nonzero in CH ( Y , Q / (CH ( Y ) ⊗ C ∗ ). A strongercondition is regulator indecomposability , which is to say that AJ( ξ ) isnonzero in J , ( Y ) / (NS( Y ) ⊗ C ∗ ).The point is that the limiting Abel-Jacobi calculation (Theorem 2.2)gives a connection between these conditions on ν ξ and ξ . First notethat for very general t ∈ ∆ ∗ , N H ( X t , Q (2)) has constant rank. Onehas a map Griff ( X t ) → J H ( X t , Q (2)) N H ( X t , Q (2)) ! . There are natural isomorphisms N H ( X t ) ’ N H ( X t ) ∨ , [ N H ( X t )] ⊥ ’ (cid:16) [ N H ( X t )] ⊥ (cid:17) ∨ , and so J (cid:16) [ N H ( X t , Q (2))] ⊥ (cid:17) ’ (cid:16) [ N F H ( X t , C )] ⊥ (cid:17) ∨ [ N H ( X t , Q (2))] ⊥ . At t = 0, a similar calculation holds, leading to a specialized analogueof Theorem 2.2, where the limiting calculation is of the formAJ( ξ t ) ∈ J (cid:16) [ N H ( X t , Q (2))] ⊥ (cid:17) AJ( ξ ) ∈ J (cid:16) H ( Y , Q (2)) (cid:17) . A well-known conjecture (see [dJL]) states thatAJ : CH ( Y ; 1; Q ) → J (cid:16) H ( Y , Q (2)) (cid:17) , is injective. Assuming this, we have a diagram(2.20) { ξ t } ∈ Griff ( X t ; Q ) (?) (cid:15) (cid:15) AJ( ξ t ) (cid:47) (cid:47) J (cid:0) [ N H ( X t , Q (2))] ⊥ (cid:1) ’ J (cid:18) H ( X t , Q (2)) N H ( X t , Q (2)) (cid:19) lim t → (cid:15) (cid:15) { ξ } ∈ CH ( Y , Q ) (cid:31) (cid:127) AJ( ξ ) (cid:47) (cid:47) J (cid:0) H ( Y , Q (2)) (cid:1) ’ J (cid:18) H ( Y , Q (2)) N H ( Y , Q (2)) (cid:19) where the limiting map (?) is defined by making the diagram com-mutative. In particular, then, we expect that indecomposability of ξ implies nontriviality of ξ t in the Griffiths group. This line of inquiry,as well as various generalizations, will be pursued in a later work.On the other hand, there is nothing at all conjectural about regula-tor indecomposability of ξ implying nontriviality of ξ t in the Griffithsgroup (for t general). This will be spelled out in the worked example of § Motivic picture: Specialization and going-up
In this section, we recall the construction of specialization maps inthe settings of higher Chow groups and motivic cohomology, and provesome elementary properties. These results are then applied to articu-late a more general perspective on “going up” in K -theory. both to higher degrees of K -theory and to higher AJ maps and the Bloch-Beilinsonfiltration [Le1]. Note that we do not see a way to define the dotted arrow withoutassuming injectivity of the bottom Abel-Jacobi map. -THEORY ELEVATOR 15 Specialization for Higher Chow groups.
In the following, f : X → B will denote a flat morphism of regular noetherian (equi-dimensional) schemes where B = Spec ( R ) is the spectrum of a discretevaluation ring. In this setting, Levine ([L1]) has defined a theory ofhigher Chow groups CH d + r − p ( X, r ) ∼ = CH p ( X, r ) ( d = relative dimen-sion of f ). The CH q ( X, r ) are defined as the homology groups of acertain complex Z q ( X, • ). These satisfy the following properties:(1) If X and B are essentially of finite type over a field k , thenthese are the usual higher Chow groups defined by Bloch.(2) If Z ⊂ X is a closed (pure codimension) subscheme (of finitetype over B ) of codimension c , then there is a long exact local-ization sequence → CH p − c ( Z, r ) → CH p ( X, r ) → CH p ( X \ Z, r ) ∂ −→ CH p − c ( Z, r − → . Remark 3.1.1.
In our applications, we work in the setting of a de-generating family over a one-dimensional base B of equi-characteristiczero.Let π be a fixed uniformizer in R , s denote the closed point of B , and η denote the generic point. Furthermore, let X s (resp. X η ) denotethe corresponding special (resp. generic) fiber; note that by virtue ofregularity of X , X η is smooth. Let f s (resp. f η ) denote the restrictionof f to the special fiber (resp. generic fiber). Finally, let i : X s , → X and j : X η , → X denote the natural inclusions. Then ψ := f ∗ η ( π ) ∈ CH ( X η ,
1) and one can define a specialization map(3.1) Sp π : CH p ( X η , r ) → CH p ( X s , r ) . by setting Sp π ( y ) := ∂ ( ψ · y ), where ∂ : CH p +1 ( X η , r + 1) → CH p ( X s , r )is the boundary map coming from the localization sequence. Notethat pullback morphisms induce a CH ∗ ( X, ∗ )-module structure on bothCH ∗ ( X η , ∗ ) and CH ∗ ( X s , ∗ ). Moreover, since the localization sequencerespects the module structure, the boundary map ∂ is a morphism ofCH ∗ ( X, ∗ )-modules. It follows that Sp π is also compatible with thismodule structure. Remark 3.1.2. (1) If n = 0, these specialization maps are already con-sidered in Fulton ([Fu]). In this case, the morphisms are independentof the choice of uniformizer, and preserve ring structures. In particular, Sp π : CH ∗ ( X η ) → CH ∗ ( X s ) is a ring homomorphism.(2) If X = B , then the specialization morphisms above were consid-ered by Bloch [B4, § B contains its residue field, the specialization map isan algebra map. It is likely that the construction of the specialization map and thefollowing properties are known to the experts. However, we give thedetails here due to the lack of a reference.
Proposition 3.1.3. (1) With notation as above, the following di-agram commutes:CH p ( X, r ) i ∗ (cid:15) (cid:15) j ∗ (cid:47) (cid:47) CH p ( X η , r ) Sp π (cid:119) (cid:119) CH p ( X s , r ) . (2) Let g : X → X denote a proper morphism of regular schemessmooth over B . Then the following diagram commutes:CH q ( X η , r ) Sp π (cid:47) (cid:47) g η ∗ (cid:15) (cid:15) CH q ( X s , r ) g s ∗ (cid:15) (cid:15) CH q ( X η , r ) Sp π (cid:47) (cid:47) CH q ( X s , r ) . (3) Let g : X → X denote a flat morphism of regular schemessmooth over B which is equi-dimensional of relative dimension d . Then the following diagram commutes:CH q ( X η , r ) Sp π (cid:47) (cid:47) g ∗ η (cid:15) (cid:15) CH q ( X s , r ) g ∗ s (cid:15) (cid:15) CH q + d ( X η , r ) Sp π (cid:47) (cid:47) CH q + d ( X s , r ) . (4) Let i : Z ⊂ X denote a regular (codimension c ) immersionwith smooth generic fiber over B . Then the following diagramcommutes: CH q ( X η , r ) Sp π (cid:47) (cid:47) i ∗ η (cid:15) (cid:15) CH q ( X s , r ) i ∗ s (cid:15) (cid:15) CH q − c ( Z η , r ) Sp π (cid:47) (cid:47) CH q − c ( Z s , r ) . (5) Let ζ ∈ CH p ( X η , ζ is decomposable, then Sp π ( ζ ) is de-composable. Proof.
1: Given y ∈ CH p ( X, n ), one has Sp π ( j ∗ ( y )) = ∂ ( j ∗ ( y ) · ψ ) = i ∗ ( y ) ∂ ( ψ ) = i ∗ ( y ) . -THEORY ELEVATOR 17
2: This follows from an application of the projection formula combinedwith the fact that ∂ commutes with push-forward. Namely, let f : X → B denote the structure map and ψ := f ∗ η ( π ). Note that g ∗ η ( ψ ) = f ∗ η ( π ) = ψ . One has: g s ∗ ( Sp π ( z )) == g s ∗ ( ∂ ( z · ψ )) = ∂ ( g η ∗ ( z · g ∗ η ( ψ ))) = ∂ ( g η ∗ ( z ) · ψ ) = Sp π ( g η ∗ ( z )) .
3: This follows from the fact that pull-back is a ring homomorphism.Namely, g ∗ s ( Sp π ( z )) = g ∗ s ( ∂ ( z · ψ )) = ∂ ( g ∗ η ( z · ψ )) = ∂ ( g ∗ η ( z ) · ψ ) = Sp π ( g ∗ η ( z )) .
4: The proof is the same as in Part (3).5: Recall, by defintion:CH pdec ( X,
1) = Im (CH ( X, ⊗ CH p − ( X ) → CH p ( X, . Let ζ ∈ CH p ( X η ,
1) be a decomposable element. Since specializationis additive, it suffices to prove the result for z which is the image ofa tensor ζ ⊗ ζ for ζ ∈ CH ( X η ,
1) and ζ ∈ CH p − ( X η ). Note that ζ can be lifted to an element ˜ ζ ∈ CH p − ( X ). Since specialization iscompatible with CH ∗ ( X, ∗ )-module structure, one has Sp π ( ζ ) = Sp π ( ζ · ζ ) = ˜ ζ Sp π ( ζ ) = Sp π ( ζ ) · Sp π ( ζ ) . It follows that Sp π ( ζ ) is decomposable. (cid:3) Remark 3.1.4.
Note that proof of Part (2) above does not require thesmoothness of f or f , only that the generic fibers are smooth. Theanalogous remark also applies to Part (3). Remark 3.1.5.
The last part of Proposition 3.1.3 was proved byCollino and Fakhruddin ([CF], Theorem 2.1) under the assumptionthat the cycle ζ lifts to X . The proof here also partially appliesto CH p ( X η , r ). Namely, the same proof shows that if an element ofCH p ( X η , r ) lies in the image of CH r ( X, r ) ⊗ CH p − r ( X η ) (whenever thismakes sense), then the same can be said of its specialization.Note that Sp π depends on the choice of uniformizer in the settingof higher Chow groups. However, one has the following comparisonresult. Lemma 3.1.6.
With notation as above, let π = uπ be another choiceof uniformizer where u is a unit in R . Then Sp π ( a ) = Sp π ( a ) +( − r ( u∂ ( a )) for any a ∈ CH p ( X η , r ). Proof.
This follows directly from the fact that the boundary maps ∂ inthe localization sequence are CH ∗ ( X, ∗ )-module maps. (cid:3) Remark 3.1.7.
We note that on ker ( ∂ : CH p ( X η , r ) → CH p − ( X s , r − Q -coefficients) we simply recall some of the properties.(1) Given any quasi-projective variety S over a field k of charac-teristic zero (or more, generally characteristic p , assuming resolutionof singularities) one can associate to it the Chow cohomology groups CH p ( S, r ). Briefly, these are defined by choosing a semi-simplicialhyper-resolution X • → S , and then taking the total complex of thedouble complex formed by the Bloch higher cycle complex associatedto the corresponding semi-simplicial scheme. It can be shown that theconstruction is independent of the chosen hyper-resolution. We referto ([Ha]) for the details.(2) The Chow cohomology groups come equipped with a contravariantfunctoriality (for arbitrary maps) and a ring structure.(3) These are covariantly functorial under proper maps with smoothtarget, and under flat maps of projective varieties.(4) They agree with the usual higher Chow groups in the smooth case.Suppose now we have a f : X → B as before, where X is regular,and f is proper and generically smooth. Suppose further that we arein the equi-characteristic zero case. In this case, X η and X are smooth.The previously stated properties of motivic cohomology allow one tospecialize cycles on X η which are liftable to X . For usual cycles, onehas a diagram CH p ( X s ) i ∗ ←− CH p ( X ) (cid:16) CH p ( X η ) . We may lift a cycle ζ ∈ CH p ( X η ), and then pull-back to the motiviccohomology group. In general, this ‘specialization’ depends on thelift. However, in the following we shall work with examples that comeequipped with canonical extensions to X . Similarly, for higher cyclesone has a diagram: CH p ( X s , r ) i ∗ ←− CH p ( X, r ) (cid:16) ker ( ∂ : CH p ( X η , r ) → CH p − ( X s , r − . One should be aware that even (or perhaps especially) in this situation, propertiessuch as cohomological or algebraic equivalence to zero on nearby fibers need notspecialize. -THEORY ELEVATOR 19
In particular, if we are given natural extensions of classes ζ in the right-most term to all of X , then we can specialize them to the motivic coho-mology of X . These constructions are functorial in families. Namely,suppose we are given two families f : X → S and f : X → S , as above.Suppose, moreover that we have a proper S -morphism F : X → X ofrelative dimension c . Then we have a natural commutative diagram: CH p ( X s ) (cid:15) (cid:15) CH p ( X ) (cid:111) (cid:111) (cid:47) (cid:47) (cid:15) (cid:15) CH p ( X η ) (cid:15) (cid:15) CH p − c ( X s ) CH p − c ( X ) (cid:111) (cid:111) (cid:47) (cid:47) CH p − c ( X η )Here the vertical maps are given by push-forward. Remark 3.1.8. (1) In the following subsection, our cycles will be natu-rally liftable to X , and the previous method combined with the descentspectral sequence will allow one to construct higher Chow cycles on sin-gular strata of the special fiber.(2) One could also work with the motivic cohomology of Suslin andVoevodsky; indeed, it is known that CH p ( X, n ) ∼ = H p − n M ( X, Q ( p )).However, in the following we shall use convenient hyper-resolutions(in the spirit of Hanamura and Levine) to explicitly compute motiviccohomology.3.2. Examples of going-up for algebraic cycles.
We now demon-strate how to use the specialization map to produce a “going-up” cal-culus for higher Chow cycles, which will be elaborated in §
5. Namely,we show that in certain types of degenerations, the specialization mor-phisms combined with edge morphisms in a certain descent spectralsequence allows one to construct higher weight Chow cycles from lowerweight cycles.Working over a field of characteristic zero, we continue to assumethat X is regular, and f generically smooth; write dim( X ) = d + 1. Inthis setting, we have constructed specialization morphisms: Sp π : CH p ( X η , r ) → CH p ( X s , r ) ,ı ∗ : CH p ( X, r ) → CH p ( X s , r ) . Of course, we can compose Sp π with the restriction to obtain a map sp : CH p ( X, r ) → CH p ( X s , r )that is independent of π .Suppose we are given a smooth proper semi-simplicial hypercover X • → X s . In this setting, one has a (1 st quadrant) descent spectral sequence:(3.2) E ‘,k ( q ) := CH q ( X ‘ , k ) ⇒ CH q ( X s , ‘ + k ) . (See for example [Ge, Thm. 1.4]; this also follows from the doublecomplex for Chow homology in [Ha, Def. 2.10], by taking the associ-ated spectral sequence [We, § th quadrant cohomological spectral sequence:(3.3) E ‘,k ( p ) := CH p ( X ‘ , − k ) ⇒ CH p ( X s , − ( ‘ + k )) . Rewriting (3.2) as a 3 rd quadrant cohomological spectral sequence E ‘,k ( p ) := E − ‘, − k ( d − p ), there is an obvious map E • , • ( p ) → E • , • ( p )given by the identity on the (0 , k )-entries and by zero elsewhere. Thisinduces a homomorphism CH p ( X s , r ) θ → CH p ( X s , r ) factoring sp = θ ◦ ı ∗ . However, θ tends to lose much of the information we want tounderstand in the limit (via ı ∗ ). Example 3.2.1.
We now apply this to the simple situation of a semi-nodal degeneration, to give the abstract perspective on §
2. Write X s = Y × Q , with Q a nodal rational curve. In this case, a smooth hypercovercan be constructed by taking the usual normalization. Then X = Y × P → Y × Q is given by identity on the first component and is justthe normalization on the second component. Moreover, X = Y andthe semi-simplicial scheme X • → X s is a proper smooth hypercover.In this setting, the 4th-quadrant descent spectral sequence for motiviccohomology has two columns. Moreover, the differentials on the E -terms are given by the difference of pullbacks via i , i ∞ : Y → Y × P .Since this difference is zero, the descent spectral sequence degenerates .In particular, one has a natural map CH p ( X s , r ) → CH p ( Y, r + 1) , which does not factor through θ . One can now compose this with thepull-back map, to get a going-up map:CH p ( X, r ) → CH p ( Y, r + 1) . In particular, given an extension of a cycle on the generic fiber to allof X , one can specialize it to a higher Chow cycle on Y .Again we emphasize that im(sp) ⊆ im( θ ), where θ is a motivic ana-logue of taking the “image of cohomology in homology”. Often thissimply has the effect of killing everything. For example, if p = r = 2and Y = Spec( F ) is a point over a number field, then CH ( X s , ∼ =CH ( F, ∼ = K ind3 ( F ) Q while CH ( X s , ∼ = CH ( F, ∼ = K ( F ) Q . In -THEORY ELEVATOR 21 this scenario, we have image(sp) = { } = image( θ ). So only ı ∗ (andnot sp) captures the K ind3 information in the limit.Typically one cannot expect the descent spectral sequence to degen-erate at E . In order to formulate more general “going-up” statements,we introduce a filtration, writing W − b CH p ( X s , r ) ⊂ CH p ( X s , r )for the image of the cohomology of E ‘ ≥ b,k ( p ). Example 3.2.2.
One can apply a similar argument in the setting ofdegenerations of triple products of curves. Namely, suppose we are in asetting where F : C → B is a semistable family of genus 3 curves, andlet X := C × F C × F C denote the triple fiber-product. Suppose that thespecial fiber C s = f C s ∪ P where f C s is the normalization of an irreduciblecurve C s of arithmetic genus three with one node. Moreover, in thatcase, f C s is a smooth hyperelliptic curve of genus 2, and we assume thatthe inverse image of the node consists of the two Weierstrass points on f C s . Finally, suppose f C s ∩ P consists precisely of these two Weierstrasspoints. In this setting, Gross and Schoen [GS] have constructed a goodfamily f : X → B such that f is flat, proper, and the total space issmooth. Moreover, the generic fiber X η = X η , and the special fiber X s has eight components (cf. § § ( C η × C η × C η ), which has a natural extension to X . Theprevious constructions then allow one to specialize the modified diag-onal to a cycle in W − CH ( X s ) . Furthermore, the previous descriptionof the components of X s give rise to a natural smooth proper hyper-cover of X s . Considering the associated descent spectral sequence as inthe previous example gives rise to edge maps(3.4) W − CH ( X s ) → CH ( C × C , . It follows that the image of the specialization of the modified diago-nal under the image of this map gives rise to a higher Chow cycle inCH ( C × C , Degeneration of a modified diagonal cycle
In this section, we provide details on the Example sketched in §3.2.2.Furthermore, we show that the specialization is an indecomposablehigher Chow cycle.
Modified diagonal cycle on a triple product of a curve.
Given a smooth projective curve C of genus g (defined over C ), the modified diagonal cycle of Gross and Schoen [GS] on X := C × C × C can be described as follows. Fixing a closed point e ∈ C ( C ), considerthe codimension-2 subvarieties∆ := { x, x, x ) : x ∈ X } ∆ := { ( x, x, e ) : x ∈ X } ∆ := { ( x, e, x ) : x ∈ X } ∆ := { ( e, x, x ) : x ∈ X } ∆ := { ( x, e, e ) : x ∈ X } ∆ := { ( e, x, e ) : x ∈ X } ∆ := { ( e, e, x ) : x ∈ X } of X ; then the cycle(4.1) ∆ e := ∆ − ∆ − ∆ − ∆ + ∆ + ∆ + ∆ ∈ Z ( X )is homologous to zero [GS, Prop. 3.1]. Furthermore: • if g C = 0, then ∆ e ≡ rat
0; and • if C is hyperelliptic, then 6∆ e ≡ rat p ∈ C ( C ), we have Abel maps ϕ ± p : C → J ( C ) q
7→ ±
AJ( q − p )with image C ± p = ϕ ± p ( C ), and(4.2) f : X → Sym C → J ( C )( q , q , q ) Σ q i AJ(Σ q i − p )Recall that the Ceresa cycle is defined by Z C,p := C + p − C − p ∈ Z g − ( J ( C ));when we consider it in Griff g − ( J ( C )) = Z g − ( J ( C )) /Z g − ( J ( C )) , whereit is nontorsion for C general (in particular, non-hyperelliptic), we maydrop the “ p ”. The same goes, of course, for the subscripts on f p and∆ e . According to results of Colombo and van Geemen [CvG, Props.2.9 and 3.7], in Griff g − ( J ( C )) we have(4.3) f ∗ ∆ ≡ alg Z C whenever C is hyperelliptic or trigonal – in particular, if g C = 3. Fur-thermore, we have the following: -THEORY ELEVATOR 23 Lemma 4.1.1. If g C = 3, then f ∗ f ∗ ∆ ≡ alg
6∆ (in Griff ( C × )). Proof.
In fact, we claim that for p = e , f ∗ f ∗ ∆ = 6∆ in Z ( C × ).Indeed, this formula holds for the morphism f : C × → Sym C by[GS, (4.4)]. Now write f = h ◦ f , where h : Sym C → Pic C ∼ = J ( C ).Here Pic C is the degree-3 Picard scheme, with the isomorphism givenby e ; and h is a birational morphism, namely the blow-up of Pic C along the curve − C + ω C = { ω C ( − x ) | x ∈ C } ⊂ Pic C (cf. [BL, p.360, Ex. 2(b)]). As the support of f ∆ e does not lie in the exceptionallocus of the blow-up morphism, we have h ∗ h ∗ ( f (∆ e )) = f (∆ e ); andso f ∗ f ∗ (∆ e ) = f h ∗ ( h ∗ ( f (∆ e ))= f f (∆ e )= 6∆ e as desired. (cid:3) Together with (4.3), the Lemma implies that for C of genus 3, wehave (in Griff ( C × ))(4.4) f ∗ Z C ≡ alg . In what follows, we shall explain how to use the behavior of Z C underdegeneration to understand that of ∆. (We shall also take p = e .)4.2. Degeneration of C × and J ( C ) . Let
C →
Spec( R ) =: B bea (flat, proper) family of stable curves over a DVR, with regular to-tal space. The Jacobian J ( C η ) of the (smooth) generic fiber (over η = B \{ s } ) is extended over B by the Néron model N g ( C /B ), whose specialfiber is a finite disjoint union of semi-abelian varieties [BLR]. One com-pletion (to a proper B -scheme) is given by the moduli scheme ¯ P g ( C /B )of degree g semibalanced line bundles, which contains N g ( C /B ) as adense open subscheme [CE]. Write N g ( C s ) ⊂ ¯ P g ( C s ) for the specialfibers.On the other hand, if C is a semistable family and the componentsof C s are smooth, Gross and Schoen construct a “good model” X → B for C × B C × B C . In particular, X is flat and proper over B , with regulartotal space, such that X η = C × η .The particular case of interest for us is when C has genus g = 3,and C s is irreducible, with one node q . Then J := ¯ P ( C /B ) is smooth(over C ); and one may describe the special fiber J s = ¯ P ( C s ) as follows.First observe that its normalization f J s is a P -bundle over ˜ A := J ( f C s ).Then J s is formed by attaching the 0- and ∞ -sections of this bundle with a shift by ε := AJ e C s (˜ q − ˜ q ) ∈ ˜ A ( C ), where { ˜ q , ˜ q } ⊂ f C s lie over q . This shift records the Hodge-theoretic extension class of(4.5) 0 → H ( ˜ A ) → H ( J s ) → H ( ¯ G m ) → , where ¯ G m := P / { , ∞} is the nodal rational curve. The open smoothsubset J ∗ s = N ( C /B ) ⊂ J s is itself an extension of ˜ A by G m, C ; thecorresponding extension of Hodge structures(4.6) 0 → H ( G m ) ı → H ( J ∗ s ) ˜ ρ → H ( ˜ A ) → { ˜ q , ˜ q } to be Weierstrass points on f C s , so that (4.5) and (4.6) are 2-torsion extensions of MHS. In this case, there exists a homomorphism σ : J ∗ s → G m with ( σ ◦ ı )( z ) = z , so that ˜ ρ × σ : J ∗ s (cid:16) ˜ A × G m is a2:1 isogeny. Writing ρ for the composition of ˜ ρ with ˜ A (cid:16) A := ˜ A/ h ε i , ρ × σ extends to a map(4.7) ρ : J s (cid:16) A × ¯ G m =: A which is 4:1 on J ∗ s (and 2:1 on sing( J s ) ∼ = ˜ A ). Write J • s → A • for themap of semi-simplicial schemes, where J s = ˜ J s , J s = sing( J s ) = ˜ A (resp. A = A × P , A = A ).Now our chosen C doesn’t satisfy the hypotheses of [CE]: the solecomponent of C s is singular. To fix this, we take the base change of C under t t ( B → B ) and blow up the double point to get C → B semistable, with C s = f C s ∪ P ( f C s ∩ P = { ˜ q , ˜ q } = { , ∞} ). Thespecial fiber of the associated good model X is X s = ∪ i =1 Y i , where[GS, Ex. 6.15]: • Y (resp. Y , Y ) is the blow-up of P × f C s × f C s (resp. f C s × P × f C s , f C s × f C s × P ) along the { P × { ˜ q i } × { ˜ q j }} ; • Y (resp. Y , Y ) is the blow-up of f C s × P × P (resp. P × f C s × P , P × P × f C s ) along the nf C s × { ˜ q i } × { ˜ q j } o ; • Y (cid:16) f C s × (resp. { ˜ q i }× P × P ), ˜ P (= degree-6 del Pezzo)-fibersover the 8 points { ˜ q i } × { ˜ q j } × { ˜ q k } , and point fibers elsewhere.We will write X • s for the corresponding semi-simplicial scheme, where X ‘s := ‘ | I | = ‘ +1 Y I ( I ⊂ { , . . . , } , Y I := ∩ i ∈ I Y i ). (4.5) is obtained by identifying the end terms of 0 → H ( f J s ) → H ( J s ) → H ( ˜ A ) → H ( ˜ A ) and H ( ¯ G m ), respectively; the second identification seemslike a cheap trick (both are Q (0) as Hodge structures), but is natural once we makethe 2-torsion assumption below (which yields a projection from J s to ¯ G m ). -THEORY ELEVATOR 25 Extension of the Abel map.
Likewise, we can base-change theextended Jacobian J (via t t ) and blow up the preimage of ˜ A ; thisresults in a smoth total space J and singular fiber J s = J s, ∪ J s, ( J s,i ∼ = f J s ), where J s, is the “identity” component.Fix a section e : B → C such that e s is a Weierstrass point on f C s ⊂ C s , distinct from ˜ q and ˜ q . Together with (4.2), this yields amap X η F η → J η over η , which extends continuously to a well-definedmorphism F : X → J . On the smooth locus X sm s = (( C s \ { q } ) ∪ G m ) × of the singular fiber X s , this extension may be described Hodge-theoretically, or alterna-tively (at least on ( C s \ { q } ) × ) by pulling back the Abel-Néron map of[CE]. Explicitly, we send ( p , p , p ) P i =1 R p i e (cid:15)is ∈ ω ( C s ) ∨ /H ( C s ) ∼ = J ∗ s, | (cid:15) | , where e s := e s , e s := 1 ∈ G m , | (cid:15) | := P (cid:15) i (mod 2), and (cid:15) i = 0(resp. 1) if p i ∈ C s \ { q } (resp. G m ). In particular, Y , Y , Y , Y aremapped to J s, while Y , Y , Y , Y go to J s, .Below we shall only need the composition π : X → J of F with the finite morphism J (cid:16) J of degree 2. On the singularfiber, the composition ρ ◦ π s : X s → A (= A × ¯ G m ) is easy to describe: Y , Y , Y , Y are collapsed to sing( A ); Y , Y , Y have 2-dimensional im-age; Y ( (cid:16) f C s × ) → ( ˜ A (cid:16) ) A is the AJ map for the genus 2 (hyper-elliptic) curve f C s ; and Y ( (cid:16) C × s ) → ¯ G × m × → ¯ G m is the product ofthe hyperelliptic maps on factors. Our situation is summarized by thediagram X π (cid:40) (cid:40) (cid:47) (cid:47) J (cid:47) (cid:47) JX s (cid:63)(cid:31) ı X (cid:79) (cid:79) (cid:47) (cid:47) π s (cid:54) (cid:54) J s (cid:63)(cid:31) (cid:79) (cid:79) (cid:47) (cid:47) J s (cid:63)(cid:31) ı J (cid:79) (cid:79) ρ (cid:47) (cid:47) A . Extension and specialization of cycles.
The choice of e givesus a natural family of modified diagonal cycles on X η and Ceresacycles on J η ; the naive extensions (obtained by taking closures ofeach irreducible component ∆ i , ∆ ij , ∆ ijk , C + , C − ) will be denoted by∆ = ∆ e ∈ CH ( X ) and Z C = Z C ,e ∈ CH ( J ). We may consider thespecializations ı ∗X ∆ ∈ CH ( X s ) and ı ∗J Z C ∈ CH ( J s ) in motivic co-homology. The idea is then that if these are cohomologically trivialin H ( X s ) resp. H ( J s ), we expect they are rationally equivalent to zero (with Q -coefficients) on the normalizations X s resp. J s , whichwould allow us to “go up” into (subquotients of) CH ( X s ,
1) resp. CH ( J s , Z C by amodification of the form ˆ∆ := ∆ − ( ı X ) ∗ W ∆ ( W ∆ ∈ Z ( X s )) resp.ˆ Z C := Z C − ( ı J ) ∗ W Z ( W Z ∈ Z ( J s )). Since ı ∗X ∆ is nonzero on each component Y i ⊂ X s , the direct con-struction of W ∆ becomes a complicated exercise in intersection the-ory and combinatorics. Instead we shall proceed indirectly, using thefact that ı ∗J Z C is already cohomologically trivial. Here it is conve-nient to use ρ ; while ρ is not flat, we can construct an ad hoc push-forward map, CH ( J s ) ρ ∗ → CH ( A ) by the map of double complexes Z ( J • s , −• ) → Z ( A • , −• ) given by ρ ∗ on J s and 2 ρ ∗ on J s . Thenwe have ρ ∗ ρ ∗ = 4 · Id on CH ( J s ), and ρ ∗ Z C ,s =: Z + A − Z − A ∈ Z ( A ) = Z ( A × P )is evidently rationally equivalent to zero. Indeed, writing z A : f C s → P for the hyperelliptic map and φ ± A for the composition f C s → ϕ ± es ˜ A (cid:16) ρ A,Z ± A = (cid:16) φ − A × z ± A (cid:17) ( f C s ) = (cid:16) φ + A × z ± A (cid:17) ( f C s ) ⊂ A × P may be viewed as the graph of z ± A over the nodal curve φ + A ( f C s )( ∼ = C s ). (Moreover, the zero and pole of z A are located at the node.)The rational equivalence is given by the push-forward of z − z A z − z − A under f C s × P → φ + A × Id A × P , whose divisor is precisely Z + A − Z − A . Viewing thispushforward as an element of Z ( A ,
1) from ρ ∗ Z C ,s ∈ Z ( A ) yields(4.8) Z (1) C := (cid:16) φ + A ( f C s ) , z A (cid:17) ∈ ker( ∂ ) ⊂ Z ( A ,
1) = Z ( A, . By the projective bundle formula, CH ( A × P , → ı ∗ − ı ∗∞ CH ( A, ( J s , → ı ∗ − ı ∗∞ CH ( ˜ A,
1) are zero; we conclude: in view of the triviality ( ⊗ Q ) of Ceresa cycles and modified diagonal cycles forhyperelliptic curves (hence for the genus-2 curve e C s ). In fact, for codimension-2 cycles this can be accomplished integrally, after multi-plying the original cycle by the exponent of the (finite) singularity group G := im { H ( X s , Q ) → H ( X , Q ) } Z im { H ( X s , Z ) → H ( X , Z ) } . -THEORY ELEVATOR 27 Proposition 4.4.1.
The specialization ı ∗J Z C of the Ceresa cycle, be-longs to W − CH ( J s ) = ρ ∗ W − CH ( A )(= CH ( ˜ A,
1) = ρ ∗ CH ( A, Z (1) C .4.5. Indecomposability of the specialization.
Recall the higherAbel-Jacobi maps associated to this situation:CH ( A, AJ , (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) J ( H ( A, Q (2))) (cid:15) (cid:15) (cid:15) (cid:15) CH ( A, AJ , (cid:47) (cid:47) J ( H ( A, Q (2)))where J ( H ) := Ext ( Q , H ) = H C { F H C + H Q } , and AJ , ( Z ) ( Z = ( C, φ ))is given by the class of the current 2 π i R C (log φ )( · )+(2 π i ) R Γ ( · ) (where ∂ Γ = φ − ( R − )). We say that Z is regulator indecomposable if AJ , ( Z ) =0; by the diagram, this implies indecomposability. Proposition 4.5.1.
For f C s very general in the moduli space M ofgenus 2 curves, Z (1) C is regulator indecomposable. (Hence for f C s general, Z (1) C is indecomposable.) Proof. Z (1) C is a multiple of Collino’s cycle; apply the main result of[Co]. (cid:3) By (4.4), π ∗ Z C =: ˜∆ is algebraically equivalent to ∆ on the genericfiber. To describe the precise sense in which(4.9) ı ∗X ˜∆ = ı ∗X π ∗ Z C = π ∗ s ı ∗J Z C ∈ W − CH ( X s )remains regulator indecomposable, we look at the spectral sequence E a,b = ⊕ | I | = a +1 Z ( Y I , − b ) computing CH ( X s ) ( d = ∂ , d = δ ). Let (cid:16) Gr W − CH ( X s ) (cid:17) ind denote the quotient ofGr W − CH ( X s ) = (cid:26) ker( d ) ∩ ker( d ) ⊂ ⊕ CH ( Y ij , δ ( ⊕ CH ( Y i , ) (cid:27) by the subspace of (equivalence classes of) decomposable cycles; fur-ther, S acts on X s , and we let ( · · · ) S denote invariants. Lemma 4.5.2.
We have isomorphisms(a) (cid:16) Gr W − CH ( X s ) (cid:17) S ind ∼ = CH ( f C s × f C s , S and(b) (cid:16) Gr W H ( X s ) (cid:17) S tr ∼ = H ( f C s × f C s ) S ∼ = ← ( π (1) s ) ∗ H ( ˜ A ) . Proof.
First note that CH ( Y ij ,
1) is zero for all but Y , Y , Y , eachof which has two components (because of ˜ q , ˜ q ). Moreover, we canignore blowups, which only change the decomposable cycles (by theprojective bundle formula). Looking at f C s × k ( k = 2 or 3), there arehyperelliptic involutions σ i on the factors, with quotients P i permuta-tions of P × C × ( k − s and fixed points containing Q i = a permutationof { ˜ q , ˜ q } × C × ( k − s . We may of course decompose CH a ( f C s × k , b ) = P χ CH a ( f C s × k , b ) χ according to the character thorugh which Z × k acts.In fact, writing Z = X χ k X σ ∈ Z k χ ( σ ) σ ∗ Z = X χ Z χ , we can do this on the level of cycles . If χ ( σ i ) = −
1, then Z χ pullsback to zero on Q i ; while if χ ( σ i ) = +1, Z χ is pulled back from P i .From this, one deduces that the image of δ merely equates cycles oneach pair of components, leaving us with 3 copies of CH ( f C s × f C s ,
1) =CH ( f C s × f C s , χ . Here χ ( σ i ) = − i = 1 , f C s × P or P × P are decomposable. Since this χ -part restricts tozero on { ˜ q j } × f C s and f C s × { ˜ q j } , it already lies in ker( d ) ∩ ker( d ).Taking S -invariants gives (a). The same proof applies verbatim for(b). (cid:3) Proposition 4.5.3.
The regulator of ı ∗X ˜∆ is nonzero in the Jacobianof Lemma 4.5.2(b) (which implies it is nonzero also in (a)). Proof.
Follows at once from the commutative diagram (cid:16) Gr W − CH ( X s ) (cid:17) S indAJ (cid:15) (cid:15) CH ( ˜ A, AJ (cid:15) (cid:15) ( π (1) s ) ∗ (cid:111) (cid:111) J (cid:18)(cid:16) Gr W H ( X s ) (cid:17) S tr (2) (cid:19) J (cid:16) H ( ˜ A )(2) (cid:17) ( π (1) s ) ∗ (cid:111) (cid:111) and the fact that AJ( ı ∗J Z C ) = 0 on the right-hand side. (cid:3) The normal function.
We assume that C extends to a family C an over an analytic disk D (with s its central point); this is necessaryin order to consider the normal functions associated to Z an C and ∆ an ,which are sections of a family of nonalgebraic complex tori. We will That is, C an → D resp. C → B are analytic resp. algebraic localizations of afamily of genus 2 curves over a complex algebraic curve. -THEORY ELEVATOR 29 drop the “an” in what follows. Write t for the coordinate on D (with t ( s ) = 0).Let V denote the VHS over D ∗ = D \ { } associated to { H ( X t ) } t ∈ D ∗ , V alg the maximal level-one sub-VHS, and V tr the quotient. Write W ··· for the corresponding objects for H ( J ( C t )), so that W tr , → π ∗ V tr withimage the S -invariants. Denote the normal functions by ν Z C ∈ ANF( D ∗ , W (2)) and ν ∆ , ν ˜∆ ∈ ANF( D ∗ , V (2))where ν ˜∆ = π ∗ ν Z C . These are the sections of J ( W (2)) resp. J ( V (2))obtained via fiberwise AJ of the cycles. We write ¯ ν for the projectionsto ANF( D ∗ , W (2)tr ) resp. ANF( D ∗ , V (2)tr ); these record fiberwise AJ of theclass of the cycles in the Griffiths group Griff ( X t ) resp. Griff ( J ( C t )).But then since ∆ t ≡ alg ˜∆ t , we have ¯ ν ∆ = ¯ ν ˜∆ .Write ( · · · ) N to denote ker( N ) ⊂ ( · · · ). By [GGK, Thm. II.B.9], wehave a well-defined limit mapping(4.10) lim s : ANF( D ∗ , V (2)) → J ( V N lim )(2)) . Moreover, lim s ν ˜∆ is given by r ∗ AJ( ı ∗X ˜∆) in view of [GGK, Thm. III.B.5],where r ∗ : H ( X s ) → V lim is the pullback via the Clemens retrac-tion. We need an extension of (4.10) to V tr . Consider the preimage W lim2 ANF( D ∗ , V (2)) of J (cid:16) ( W V N lim )(2) (cid:17) under (4.10): its intersection W lim2 ANF( D ∗ , V alg (2)) has lim s in J (cid:16) ( W V N alg,lim )(2) (cid:17) , and W V N alg,lim isof pure type (1 , W V N lim ) tr . So (4.10) descends to awell-defined mapping(4.11) lim s : W lim2 ANF( D ∗ , V tr (2)) → J (cid:16) (Gr W V N lim ) tr (2) (cid:17) . From (4.9) it is clear that ν ˜∆ belongs to W lim2 ANF( D ∗ , V (2)) and so wemay apply lim s to ¯ ν ˜∆ (= ¯ ν ∆ ), to obtain r ∗ AJ( ı ∗X ˜∆) = r ∗ ( π (1) s ) ∗ AJ( ı ∗J Z C ) = r ∗ ( π (1) s ) ∗ AJ( Z (1) C ) . But AJ( Z (1) C ) = 0 in the left-hand side of(4.12) J (cid:16) H ( A )(2) (cid:17) , → ( π (1) s ) ∗ J (cid:16) { Gr W H ( X s ) } tr (cid:17) ∼ = → r ∗ J (cid:16) { Gr W V N lim } tr (2) (cid:17) and r ∗ is an isomorphism on W . We conclude: Theorem 4.1.
Let ¯ ν ∆ be the section of J ( H tr ( X t )) over D ∗ associatedto the Gross-Schoen cycle. Then:(i) lim s (¯ ν ∆ ) is nonzero, and given by AJ( Z (1) C ) via (4.12) , where Z (1) C ∈ CH ind ( A, is Collino’s cycle; this implies that (ii) ¯ ν ∆ is nonzero in ANF( D ∗ , V tr (2)) , and so(iii) ∆ is nontorsion in Griff ( X t ) for very general t . We have thus used regulator indecomposability of the specializationof the modified diagonal to check its generic algebraic inequivalence tozero.4.7.
Second and third specializations of Z C and ∆ . By addinga second parameter, we can allow C s to acquire an additional node q , with normalization an elliptic curve ˜ E . Suppose moreover that hepreimages { ˜ q , ˜ q } (of q ) and { ˜ q , ˜ q } (of q ) on ˜ E are such that we havethe equalities ˜ q − ˜ q ≡ ˜ q − ˜ q ≡ q − ˜ q ) =: ε of two-torsion classes.Then A semistably degenerates to E × ¯ G m , where E := ˜ E/ h ε i , and Z (1) C specializes (goes up) to a class Z (2) C ∈ CH ( E,
2) which may bedescribed as follows. Let f, g ∈ C ( ˜ E ) ∗ have divisors ( f ) = 2[˜ q ] − q ]and ( g ) = 2[˜ q ] − q ], and satisfy f (˜ q i ) = 1, g (˜ q i ) = 1 ( i = 1 , { f, g } belongs to CH ( ˜ E, Z (2) C is theprojection to E of { f, g } − { f − , g − } ≡ { f, g } . Its regulator can beshown to be nontorsion as in [Co, § { f, g } as anEisenstein symbol [DK, Example 10.1].Degenerating once more, in such a way that our four 4-torsion points“remain finite”, Z (2) C goes up to a cycle Z (3) C ∈ CH ( C ,
3) given para-metrically by ( z ) (cid:18) z, − (cid:16) − z z (cid:17) , − (cid:16) z − i z + i (cid:17) (cid:19) − (cid:18) z − , − (cid:16) − z z (cid:17) − , − (cid:16) z − i z + i (cid:17) − (cid:19) , with regulator 32 i G (cf. (2.19)). This can be directly computed (as in § or done using two different formulas in [DK] (cf. Example 10.1,and “ D ” in § Z C , Z (1) C ,and Z (2) C are all nontorsion.Here is an easy implication for the cycle ∆ and its associated normalfunction, if we consider instead a good model for the triple fiber-productof the trinodal degeneration of C . We get a specialization map fromANF( D ∗ , V tr (2)) to C / Q (2) (along the lines of [GGK, (IV.D.3)ff]), un-der which ¯ ν ∆ goes to 16 i G . This corresponds to specializing ˜∆ to thespecial fiber of the good model, which is a complicated configurationof rational threefolds, with Gr W H of rank one. This is done in [GGK, § IV.D], but with a small error as regards branches of log(which produces an extraneous term). -THEORY ELEVATOR 31 Limits of higher normal functions
In this section we extend Proposition 6.2 of [DK] to the non-semistablesetting, and provide a proof, which is omitted in [DK] for even the(semistable) case presented there. We have found it more natural towork with motivic cohomology notation here; the reader who findsChow cohomology notation more convenient may replace H a M ( X, Q ( b ))by CH b ( X, b − a ) Q . All cycle groups in this section are taken to have Q -coefficients.5.1. The Abel-Jacobi map for motivic cohomology of a normalcrossing divisor.
Let X ¯ π → S be a proper, dominant morphism ofsmooth varieties, with unique singular fiber ¯ π − (0) = X = ∪ Y i , anddim X = d , dim S = 1. Assume first that X is a SNCD, so as to be ableto make the descent spectral sequence for H M and H D explicit. To thisend, we shall write Y I := ∩ i ∈ I Y i , Y [ ‘ ] = q | I | = ‘ +1 Y I , I,j : Y I ∪{ j } , → Y I , Y I := ∪ j / ∈ I Y I ∪{ j } ⊂ Y I , and h i i I for the position of i in I .Recall (from [KL, GGK]) that there are double complexes Z ‘,kY ( p ) := ⊕ | I | = ‘ +1 Z p ( Y I , − k ) resp. K ‘,kY ( p ) := B ‘,kY ( p ) ⊕ F p D ‘,kY ( p ) ⊕ D ‘,k − Y ( p ):= ⊕ | I | = ‘ +1 n C p + k ( Y I ; Q ( p )) ⊕ F p D p + k ( Y I ) ⊕ D p + k − ( Y I ) o , with d = ∂ B (Bloch differential) resp. D (cone differential D ( α, β, γ ) :=( − dα, − dβ, dγ − β + δ α )) and d = ∂ I = P | I | = ‘ +1 P j / ∈ I ( − h j i I ∪{ j } ( I,j ) ∗ ,whose associated simple complexes compute motivic resp. Deligne co-homology: H p − r M ( X , Q ( p )) = H − r ( Z • Y ( p ) , ∂ B ) ,H p − r D ( X , Q ( p )) = H − r ( K • Y ( p ) , D ) . Briefly, D • ( Y I ) := N • { Y I } ( Y I ) denotes normal currents of intersectiontype, C • ( Y I ; Q ( p )) := I • { Y I } ( Y I ) ⊗ Z Q ( p ) the locally integral currentscontained therein, and Z p ( Y I , • ) := Z p R ( Y I , • ) Y I ⊂ Z p ( Y I , • ) the quasi-isomorphic subcomplex of higher Chow precycles in (real) good positionwith respect to Y I . (For background on currents of intersection type,the reader may consult the treatments in [GGK, § III.A] and [KL, § W 7→ ( − π i ) p + k (cid:16) (2 π i ) − k T W , Ω W , R W (cid:17) , provides a morphism of double complexes Z ‘,kY ( p ) → K ‘,kY ( p ) whichinduces the Abel-Jacobi map AJ p,rX : H p − r M ( X , Q ( p )) → H p − r D ( X , Q ( p )) ∼ = r> J p,r ( X ) . Given Z = { Z [ ‘ ] } ‘ ≥ = { Z [ ‘ ] I } ‘ ≥ , | I | = ‘ +1 ∈ ker( ∂ B ) ⊂ ⊕ ‘ ≥ Z ‘ − ‘ − rY ( p ) = Z − rY ( p ) , there exist Ξ [ ‘ ] ∈ F p D − r − Y ( p ), Γ [ ‘ ] ∈ B − r − Y ( p ) such that n ( − π i ) p − ‘ (cid:16) (2 π i ) ‘ T Z [ ‘ ] , Ω Z [ ‘ ] , R Z [ ‘ ] (cid:17)o ‘ ≥ − d n ( − π i ) p − ‘ (cid:16) (2 π i ) ‘ Γ [ ‘ ] , Ξ [ ‘ ] , (cid:17)o ‘ ≥ = n ( − π i ) p − ‘ (cid:16) , , R Z [ ‘ ] + Ξ [ ‘ ] − (2 π i ) ‘ δ Γ [ ‘ ] (cid:17)o ‘ ≥ ∈ ker( d ) ⊂ D − r − Y ( p )yields a class (cid:94) AJ( Z ) ∈ H p − r − ( X , C ) projecting toAJ( Z ) ∈ J p,r ( X ) = H p − r − ( X , C ) F p H p − r − ( X , C ) + H p − r − ( X , Q ( p ))(5.1) ∼ = n F − p +1 H p − r − ( X , C ) o ∨ (cid:30) H p − r − ( X , Q ( p )) . Now consider the double complex[ F q ] D Y‘,k ( − p ) := ⊕ | I | = ‘ +1 [ F d + q − ‘ − ] D d − p − ‘ ) − k − ( Y I )with d = d , d = Gy := 2 π i P | I | = ‘ ( − h i i I ( I \{ i } ,i ) ∗ , which computeshomology:(5.2) H − r (cid:16) F − p +1 D Y • ( − p ) (cid:17) = F − p +1 H p − r − ( X , C ) . By [GGK, Prop. III.A.13], (5.2) can be represented by elements of theform ω = { ω [ ‘ ] } ‘ ≥ ⊂ ⊕ ‘ ≥ F d − p − ‘ A d − p ) − ‘ + r − ( Y [ ‘ ] ) h log( ∪ | I | = ‘ +1 Y I ) i , and then ( − π i ) r − p h (cid:94) AJ( Z ) , ω i =(5.3) X ‘ ≥ (cid:18)Z Y [ ‘ ] R Z [ ‘ ] ∧ ω [ ‘ ] − (2 π i ) r + ‘ Z Γ [ ‘ ] ω [ ‘ ] (cid:19) . The integrals here converge by [GGK, Lemma III.A.6]. In the eventthat Z = { Z i } ∈ Z p ( X , r ) := ker( ∂ B ) ∩ Z , − rY ( p ) , We concentrate on the r > r = 0 has been treated in [GGK]. -THEORY ELEVATOR 33 we can arrange to have Γ [ ‘ ] = 0 ∀ ‘ > − π i ) r − p h (cid:94) AJ( Z ) , ω i = X i (cid:18)Z Y i R Z i ∧ ω i − (2 π i ) r Z Γ i ω i (cid:19) . Limits of higher normal functions in the semistable set-ting.
Turning to normal functions, we begin with the morphisms X ∗ := X \ X π → S\{ } =: S ∗ , → S , and write V = R p − r − π ∗ Q ( p ), V (resp. V e ) for the correspondingweight-( − r −
1) VHS (resp. its canonical extension). Below we willabuse notation by writing V (resp. V e ) also for its sheaf of sections V ⊗ O ∆ ∗ (resp. ˜ V ⊗ O ∆ ) for a disk ∆ ⊂ S about { } . Denote theLMHS at { } by (cid:16) ˜ V , F • lim , W • (cid:17) = V lim , with the monodromy logarithm N = log( T ) and ˜ V = e − log( s )2 π i N V . (A general reference for the canonicalextension and degenerations of Hodge structure may be found in [PS, § ν ∈ AN F r S ∗ ( V ), we shall mean an ad-missible VMHS of the form(5.5) 0 → V → E ν → Q S ∗ (0) → N extends to the underlying local system E ν (which yields˜ E ν , E limν , E ν,e ). Write ı : { } , → S and ı : X , → X . Applying thecomposition AVMHS( S ∗ ) R ∗ → D b MHM( S ) ı ∗ → D b MHSof exact functors to V yields (up to quasi-isomorphism) the complex K • := n V lim N → V lim ( − o . Therefore, applying it to (5.5) yields a diagram 0Hom MHS ( Q (0) , H K • ) (cid:79) (cid:79) ANF r S ∗ ( V ) sing (cid:53) (cid:53) ı ∗ R ∗ (cid:47) (cid:47) Ext D b MHS ( Q (0) , K • ) (cid:79) (cid:79) ker(sing ) (cid:63)(cid:31) (cid:79) (cid:79) lim (cid:47) (cid:47) Ext ( Q (0) , H K • ) . (cid:79) (cid:79) (cid:79) (cid:79) defining the invariants sing and lim . Of course, we may also view ν as a (horizontal, holomorphic) section of the Jacobian bundle J ( V ) = V / ( F ⊕ V ) over S ∗ , by taking the difference of lifts ν Q ( s ) ∈ E ν,s resp. ν F ( s ) ∈ F E ν,s of 1 ∈ Q (0) in V s . In this context, admissibility meansthat we also have (for some disk ∆ ⊂ S about 0):(a) a lift ν F of 1 ∈ Q S (0) to E ν,e with ν F | ∆ ∗ in F E ν ; and(b) a lift ν Q of 1 to ˜ E ν, satisfying N ν Q ∈ W − ˜ V .One then hassing ( ν ) = [ N ν Q ] ∈ Hom
MHS Q (0) , V lim N V lim ( − ! ∼ = H (∆ ∗ , V ) (0 , . If this vanishes, then ν Q may be chosen in ker( N ), so that ˜ ν := ν Q − ν F gives a well-defined section of V e (over ∆) with ∇ ( ν Q − ν F ) | ∆ ∗ ∈ Γ(∆ ∗ , F − V ) by horizontality. Using Res ( ∇ ) = − π i N , we find that (cid:93) lim ν := ˜ ν (0) = ν Q − ν F (0) ∈ ker ( V lim N → V lim F − ) = ker( N ) + F V lim which projects to compute lim ν ∈ Ext ( Q (0) , ker( N )).By [GGK, III.B.7], a holomorphic section ω ( s ) ∈ Γ ( S , ( F V ∨ ) e )of the canonical extension may be represented by a d rel -closed C ∞ relative log h X i (2( d − p ) + r − X ∆ , and we write lim ω forits restriction to ( F V ∨ ) e, . Referring to the (dual) portions → H p − r − ( X )( p ) r ∗ → H p − r − lim ( X t )( p ) N → H p − r − lim ( X t )( p − → , we will freely shrink this as needed without further comment -THEORY ELEVATOR 35 and → H d − p )+ r − lim ( X t )( d − p ) N → H d − p )+ r − lim ( X t )( d − p − r ∗ → H p − r − ( X )( − p ) → of the Clemens-Schmid sequence, the pullbacks ω i (and their iteratedresidues ω I on substrata) define a representative (as described after(5.2)) of r ∗ (lim ω ) =: ω (0) ∈ F − p +1 H p − r − ( X , C ) . Note that D (cid:93) lim ν, lim ω E = lim s → h ˜ ν ( s ) , ω ( s ) i ∈ C . To construct a normal function with sing ( ν ) = 0, let z ∈ ker( ∂ B ) ⊂ Z p ( X , r ) be a representative of a class Ξ ∈ CH p ( X , r ) meeting all Y I properly, and define z = { Z i } ∈ Z p ( X , r ) by Z i := z · Y i . This rep-resents ı ∗ Ξ ∈ H p − r M ( X , Q ( p )), where ı : X , → X . In a neighborhood X ∆ := ¯ π − (∆) of X , z (hence T z ) meets all fibers properly, and (since H p − r ( X ∆ ) ∼ = H p − r ( X )) we may choose an integral current ˜Γ on X ∆ with ∂ ˜Γ = T z meeting the Y i and all fibers properly. Clearly then˜ R z := R z − (2 π i ) r δ ˜Γ is a closed current on X ∆ , of intersection type withrespect to the Y I . Setting Γ i := ˜Γ · Y i , we have by (5.4) that the restric-tion of ˜ R z to the Y i computes a lift to H p − r − ( X , C ) of AJ X ( ı ∗ Ξ).Moreover, over ∆ ∗ the normal function ν ( s ) = AJ X s (Ξ s ) associated toΞ ∗ ∈ CH p ( X ∗ , r ) is computed by the fiberwise restrictions h ˜ R z | X s i ∈ H p − r − ( X s , C ) (cid:16) J p,r ( X s ) ∼ = n F d − p H d − p )+ r − ( X s , C ) o ∨ periods . Putting everything together, we have h (cid:93) lim ν, lim ω i = lim s → Z X s ˜ R z | X s ∧ ω ( s )= X i Z Y i ˜ R z | Y i ∧ ω i = h (cid:94) AJ X ( ı ∗ Ξ) , ω (0) i = h r ∗ (cid:94) AJ X ( ı ∗ Ξ) , lim ω i . The second equality is the crucial one; it comes about by noting that˜ R z ∧ ω is of X -intersection type, hence the 0-current (¯ π | X ∆ ) ∗ (cid:16) ˜ R z ∧ ω (cid:17) isof { } -intersection type. Since it is also holomorphic on ∆ ∗ , it followsthat it is holomorphic (hence continuous) on ∆. So we have provedthat(5.6) lim s → AJ X s (Ξ s ) = J ( r ∗ )AJ X ( ı ∗ Ξ) , for z as above and X ∆ → ∆ semistable. Remark . It is the SSD case which most clearly exhibits the phe-nomenon of “going up in K -theory in the limit”. Recall from § X gives rise to a “weight” filtra-tion W • on H p − r M ( X , Q ( p )), with W − b consisting of the classes whichadmit a representative in ⊕ ‘ ≥ b Z ‘, − ‘ − rY ( p ), and Gr W − b a subquotient ofCH p ( Y [ b ] , r + b ). So the degree of K -theory “goes up” if ı ∗ Ξ ∈ W − b for b > Limits in the general setting.
To state the more general re-sult, we now drop the SSD assumption on ¯ π , hence the assumption ofunipotency of V at { } (i.e. of T ). One still has pullback and AJ mapsCH p ( X , r ) ı ∗ → H p − r M ( X , Q ( p )) AJ p,rX → J p,r ( X ) , where J p,r ( X ) := Ext ( Q (0) , H p − r − ( X , Q ( p ))). Write T = T ss T un for the Jordan decomposition, κ for the order of T ss , s for the coordinateon ∆, and N := log T un . Note that ker( N ) (= ker( T κ − I ) ) (cid:41) ker( T − I ),unless κ = 1. The portions of Clemens-Schmid → H d − p + r +1 ( X )( − d ) ı ∗ ı ∗ → H p − r − ( X ) r ∗ → H p − r − lim ( X s )and H p − r − lim ( X s )( − r ∗ → H d − p + r − ( X )( − d ) ı ∗ ı ∗ → H p − r +1 ( X ) → remain exact sequences of MHS, with im( r ∗ ) = H p − r − lim ( X s ) := ker( T − I ) ⊆ H p − r − lim ( X s ). (This is a sub-MHS although T − I itself is not amorphism of MHS from H p − r − lim to H p − r − lim ( − V for the VHS and J ( V ) for the family of generalized intermediateJacobians.Let σ : ˆ S ∗ → S ∗ be a cyclic cover extending the map t t κ (= s ) from ∆ ∗ → ∆ ∗ , with µ ∈ Aut( ˆ S ∗ / S ∗ ) a generator, and ˆ V resp.ˆ V the (unipotent) pullback variation resp. local system. We havethe canonical extension J ( ˆ V e ) := ˆ V e / { ˆ F e + ∗ ˆ V } , with fiber over { }J p,rlim := ˆ V e, / { ( ∗ ˆ V ) + ˆ F e, } , and write J ( r ∗ ) : J p,r ( X ) → J p,rlim -THEORY ELEVATOR 37 for the map induced by r , with image J p,rinv := Ext ( Q (0) , ker( T − I )).Moreover, there is a diagram X (cid:127) (cid:95) ı (cid:15) (cid:15) ˆ X (cid:127) (cid:95) ˆ ı (cid:15) (cid:15) P (cid:111) (cid:111) (cid:111) (cid:111) Q (cid:47) (cid:47) (cid:47) (cid:47) ˆ X (cid:127) (cid:95) ˆ ı (cid:15) (cid:15) X π (cid:15) (cid:15) ˆ X ˆ π (cid:15) (cid:15) ¯ Q (cid:47) (cid:47) (cid:47) (cid:47) ¯ P (cid:111) (cid:111) (cid:111) (cid:111) ˆ X ˆ π (cid:15) (cid:15) S ˆ S ¯ σ (cid:111) (cid:111) (cid:111) (cid:111) ˆ S with ˆ X ∆ := ˆ π − (∆) → ∆ semistable, ˆ X , ˆ X smooth, and ˆ X \ ˆ X =ˆ X \ ˆ X = ˆ X ∗ := X ∗ × σ ˆ S . (That is, ¯ Q restricts to the identity on ˆ X ∗ ;write P for the restriction of ¯ P to ˆ X ∗ → X ∗ .) Note that we have H ∗ lim ( ˆ X t ) ∼ = H ∗ lim ( X s ). The natural map J (ˆ r ∗ ) : J p,r ( ˆ X ) → J p,rlim has image ˆ J p,rinv := Ext ( Q (0) , ker( T κ − I )).By definition of admissibility, we have a pullback map σ ∗ : ANF S ∗ ( V ) → ANF ˆ S ∗ ( ˆ V ) ν ˆ ν , and if sing ( ν ) := sing (ˆ ν ) = 0, we define lim ν := lim ˆ ν ∈ ˆ J p,rinv . Thefollowing result extends Proposition 6.2 of [DK]: Theorem 5.2.
Let Ξ ∗ ∈ CH p ( X ∗ , r ) ( r > be given, with cl p,r (Ξ ∗ ) ∈ Hom
MHS ( Q (0) , H p − r ( X ∗ , Q ( p ))) and ν Ξ ∗ ( s ) := AJ X s (Ξ s ) ∈ ANF S ∗ ( V ) , where Ξ s := ı ∗ X s (Ξ ∗ ) .(a) Suppose Res X ( cl p,r (Ξ ∗ )) = 0 ∈ Hom
MHS (cid:16) Q (0) , H d − p )+ r − ( X , Q ( − d )) (cid:17) . Then sing ( ν Ξ ∗ ) = 0 , and lim ( ν Ξ ∗ ) lies in J p,rinv .(b) If Ξ ∗ is the restriction of Ξ ∈ CH p ( X , r ) , then we have lim ( ν ) = J ( r ∗ ) (AJ X ( ı ∗ Ξ)) . Proof. ( a ) Set ˆΞ ∗ := P ∗ (Ξ ∗ ). The assumption implies that cl p,r (Ξ ∗ )lifts to ξ ∈ Hom
MHS ( Q (0) , H p − r ( X , Q ( p ))) , and then cl p,r (ˆΞ ∗ ) lifts to ¯ Q ∗ ¯ P ∗ ξ , hence has trivial Res ˆ X . It follows atonce that (sing ( ν Ξ ∗ ) =) sing ( ν ˆΞ ∗ ) = 0, in view of the diagramANF ˆ S ∗ ( ˆ V ) sing (cid:41) (cid:41) [ · ] (cid:47) (cid:47) H ( ˆ S ∗ , ˆ V ) | ∆ (cid:47) (cid:47) (cid:15) (cid:15) H (∆ ∗ , ˆ V ) (cid:20) (cid:116) (cid:39) (cid:39) (cid:15) (cid:15) CH p ( ˆ X ∗ , r ) (cid:79) (cid:79) cl p,r (cid:47) (cid:47) H p − r ( ˆ X ∗ ) | ∆ (cid:47) (cid:47) H p − r ( ˆ X ∗ ∆ ) Res (cid:47) (cid:47) H p − r +1ˆ X ( ˆ X ) . Using admissibility, ν ˆΞ ∗ lifts to a section of J ( ˆ V e ) with value lim ( ν ˆΞ ∗ ) ∈ ˆ J p,rinv at 0.Now µ lifts to M ∈
Aut( ˆ X ∗ / X ∗ ), which evidently acts on ( ∗ ˆ V ) asan automorphism of MHS. That is, the restriction of T to ker( T κ − I ) ⊂ H p − r − lim is MHS-compatible, and so T acts on ˆ J p,rinv with fixed locus J p,rinv .Since ν ˆΞ ∗ = σ ∗ ν Ξ ∗ , we have ν ˆΞ ∗ = µ ∗ ν ˆΞ ∗ and taking lim on both fibersgives lim ( ν ˆΞ ∗ ) = T lim ( ν ˆΞ ∗ ).( b ) Write ˆΞ := ¯ P ∗ Ξ, ˆΞ := ¯ Q ∗ ˆΞ , ˆΞ := ¯ Q ∗ ˆΞ, Ξ := ı ∗ Ξ, ˆΞ :=(ˆ ı ) ∗ ˆΞ , etc.; note that P ∗ Ξ = ˆΞ , Q ∗ ˆΞ = ˆΞ , and ˆΞ − ˆΞ = ˆ ı ξ for some ξ ∈ CH p − ( ˆ X , r ). We have the motivic homology AJ mapAJ ˆ X : CH p − ( ˆ X , r ) → Hom
MHS ( Q (0) , H d − p )+ r +1 ( X , Q ( − d ))), andusing functoriality of AJ J ( P ∗ ) (AJ X (Ξ )) = AJ ˆ X (ˆΞ )= AJ ˆ X (ˆΞ ) + AJ ˆ X ((ˆ ı ) ∗ ˆ ı ξ )= J ( Q ∗ ) (cid:16) AJ ˆ X (ˆΞ ) (cid:17) + J ((ˆ ı ) ∗ ˆ ı ) AJ ˆ X ( ξ ) . Since (ˆ r ) ∗ ◦ (ˆ ı ) ∗ ˆ ı = 0, P ◦ ˆ r = r , and Q ◦ ˆ r = ˆ r , applying J ((ˆ r ) ∗ )and using (5.6) gives J ( r ∗ ) (AJ X (Ξ )) = J (ˆ r ∗ ) (cid:16) AJ ˆ X (ˆΞ ) (cid:17) = lim ( ν ˆΞ ∗ ) = lim ( ν Ξ ∗ ) . (cid:3) Remark 5.3.1.
A similar result holds for r = 0; details are left to thereader.5.4. Limits of truncated normal functions.
Continuing in the set-ting of §5.3, recall that the fiber over { } of the canonical extension( V ∨ ) e decomposes as a direct sum of generalized eigenspaces E λ for Res s =0 ( ∇ ), with eigenvalues in [0 , σ ∗ ( V ∨ ) e ρ → ˆ V ∨ e has kernel the skyscraper sheaf ⊕ λ ∈ (0 , E λ over { } . We may use -THEORY ELEVATOR 39 the compositionΓ (∆ , ( V ∨ ) e ) lim → ( V ∨ ) e, ρ | → ˆ V ∨ e, ∼ = H d − p )+ r − lim ( ˆ X t )( d − p − ˆ r → H p − r − ( ˆ X )( − p ) ( P ) ∗ → H p − r − ( X )( − p )to define ω (0) ∈ H p − r − ( X , C ) by ω ( s ) lim ω lim (¯ σ ∗ ω ) (¯ σ ∗ ω )(0) =: ω (0) . Note that a section of F ( V ∨ ) e lands in F − p +1 H p − r − ( X , C ). FromTheorem 5.2(b) we have at once the Corollary 5.3.
Given ω ( s ) ∈ Γ(∆ , F ( V ∨ ) e ) and Ξ ∈ CH p ( X , r ) , thereexist lifts ˜ ν of ν Ξ ∗ to V e that make F Ξ ,ω ( s ) := h ˜ ν ( s ) , ω ( s ) i holomorphicand single-valued on ∆ . For any such lift, we have (5.7) lim s → F Ξ ,ω ( s ) ≡ h AJ X ( ı ∗ X Ξ) , ω (0) i modulo periods of ω (0) over H p − r − ( X , Q ( p )) . Of course, this limiting value may lie in C modulo some horrible sub-group with lots of generators. This corollary is used most successfullywhen one has a splitting H p − r − ( X )( p ) η (cid:16) Q ( p ) [dually Q (0) η ∨ , → H p − r − ( X )]of the MHS on the singular fiber, with ω (0) = η ∨ (1): then (5.7) be-comes lim s → F ( s ) ≡ J ( η ) (cid:16) AJ X ( ı ∗ X Ξ) (cid:17) ∈ J ( Q ( p )) ∼ = C / Q ( p ) . The tempered Laurent polynomials of [DK] give one method of con-structing such splittings, for maximal unipotent degenerations of Calabi-Yau varieties. Example 5.4.1.
Consider the Fermat quintic family defined by f ( t, X ) := t X i =0 X i − Y i =0 X i = 0in P ( t in a small disk about 0). Let X ∆ be its semistable reduction.(See [GGK] for an explicit description; X is a union of 4 P ’s blownup along Fermat quintic curves.) Then the standard residue (3 , { ω t } t ∈ ∆ produces a splitting Q (0) , → H ( X ) over { } , essentiallybecause f ( t, X ) / Q i =0 X i is tempered [DK]. In [GGK], this was used tostudy limits of usual normal functions (paired with ω ) in C / Q (2). In the special case where V is a VHS of CY type, and ω is a section of the topHodge filtrand, F Ξ ,ω is called a truncated normal function . Of course, there are many cases where H p − r − ( X ) (or at least itsimage by r ∗ ) is Q (0), and here the Corollary applies automatically; forexamples, see [JW] and [dS].6. Application to a conjecture from topological string theory
In this section we apply Theorem 5.2(b) (or (5.6)) to an an algebraic K -class on a 2-parameter family X of genus-2 curves. The fibers X z ,z of our family are obtained by compactifying Y z ,z := { φ ( X , Y ) = 0 } ⊂ ( C ∗ ) in the toric Fano surface P ∆ associated to the Newton polytope ∆ =∆( φ ), where φ ( X , Y ) := x + x X + x Y + x X − Y − + x X − Y − and z = x x x x , z = x x x . For the total space X (resp. Y ), we take the union of the X z (resp. Y z )for z ∈ ( P z \{ z = 0 } ) × ( P z \{ z = 0 } ); note that the base containsthe “conifold point” z (0) := ( z (0)1 , z (0)2 ) := (cid:18) − , (cid:19) . (This is actually an ordinary double-point of the conifold curve .) Ineffect, we will be applying the Theorem to a 1-parameter slice throughthis point, which is a 1-parameter semistable degeneration.We shall begin by describing two vanishing cycles α , α ∈ H ( X z , Z ),corresponding respectively to z = 0 and z = 0. Fix a small (cid:15) > α vanishing at z = 0, we reason that z → z constant corresponds to x → x → α be the cycle pinchedto the node at x = 0. If we make the coordinate change u = X − Y , v = Y − , then φ = x + φ := x + n x u − v − + x v − + x uv + x u v o and (for very small | x | , | x | ) the image of α under T ube : H ( X ) → H ( P ∆ \ X ) (dual to 2 π i Res ) is given by τ = {| u | = | v | = (cid:15) } . Similarly, z → z constant corresponds to x →
0. Taking α to be thecycle pinched to the node there, the coordinate change ˜ u = X Y ,˜ v = X − Y − makes XY φ = ˜ u ˜ vφ = x + φ := x + n x ˜ u ˜ v + x ˜ v − + x ˜ u ˜ v + x ˜ u − ˜ v − o ; -THEORY ELEVATOR 41 and (for very small | x | , | x | ) T ube ( α ) = τ := {| ˜ u | = | ˜ v | = (cid:15) } . Itshould be emphasized that both cycles vanish at z = 0, but (as wedescribe below) neither cycle vanishes at z = z (0) .By rescaling φ, X , Y , (cid:15) , etc., we may both retain the descriptions T ube ( α i ) = τ i and have x = x = x = 1, so that φ is tempered(and z = x /x , z = x /x ). This implies that the symbol { X , Y } ∈ CH ( Y ,
2) lifts to a class Ξ ∈ CH ( X , z := ı ∗ X z Ξunder the Abel-Jacobi mapsAJ : CH ( X z , → H ( X z , C / Q (2))for z ∈ U := n ( z , z ) | < | z | < , < | z | < o may be computed as in [DK, §4.2] for elliptic curves, suitably modi-fied for genus 2 and two vanishing cycles. We now briefly sketch theprocedure, using the regulator current notation of [DK, § Referring to the toric coordinate changes above, note the equality ofsymbols { u, v } = { X , Y } = { ˜ u, ˜ v } in K ( G m ), hence in CH ( Y z ,
2) (for Y z smooth). By temperedness, for sufficiently small nonzero | z | , | z | we have1(2 π i ) Z τ R { φ, u, v } ≡ Q (1) π i Z α R { u, v } ≡ Q (1) π i Z α R { X , Y } = 12 π i AJ(Ξ z )( α ) ;and similarly1(2 π i ) Z τ R { ˜ u ˜ vφ, ˜ u, ˜ v } ≡ Q (1) π i Z α R { ˜ u, ˜ v } ≡ Q (1) π i AJ(Ξ z )( α ) . Moreover, for small arg( z ) and arg( z ) we have T φ ∩ τ = ∅ , and so R { φ, u, v } = log φ duu ∧ dvv . Noting that z z = x − , this yields(6.1) π i AJ(Ξ z )( α ) ≡ Q (1) 1(2 π i ) Z τ log( x + φ ) duu ∧ dvv (6.2) ≡ Q (1) log( x ) − X n> ( − n x − n n h(cid:16) x u − v − + x v − + x uv + x u v (cid:17) n i In brief, we have R { f, g } = log( f ) dgg − π i log( g ) δ T f and R { f, g, h } = log( f ) dgg ∧ dhh + 2 π i log( g ) dhh · δ T f + (2 π i ) log( h ) δ T f ∩ T g , where T f = f − ( R < ) (oriented from −∞ to 0) and log( f ) is the (discontinuous) branch with imaginary part in [ − π, π ). Otherwise there would be a contribution from
Res v =0 R { φ, u, v } , and not just theone shown (from Res φ =0 ); the detailed argument is exactly as in [DK, §4.2]. (6.3) = −
15 log( z z ) − X m,r ≥ (5 m + 3 r )!( − z ) r ( − z z ) m ((2 m + r )!) m ! r !(5 m + 3 r ) , where [ ] takes the constant term of a Laurent polynomial, and P means to omit ( m, r ) = (0 , α , the analogous computation is(6.4) π i AJ(Ξ z )( α ) ≡ Q (1) 1(2 π i ) Z τ log( x + φ ) d ˜ u ˜ u ∧ d ˜ v ˜ v (6.5) ≡ Q (1) log( x ) − X n> ( − n x − n n h(cid:16) x ˜ u ˜ v + x ˜ v − + x ˜ u ˜ v + x ˜ u − ˜ v − (cid:17) n i (6.6) = −
15 log( z z ) − X m,r ≥ (5 m + 2 r )!( − z z ) m ( − z ) r (3 m + r )! r !( m !) (5 m + 2 r ) . The series in (6.3) and (6.6) converge absolutely on U , hence compute π i AJ(Ξ z )( α i ) ( i = 1 ,
2) there, and can be shown to converge to theirlimit at z = z (0) . Write N i = log T i for the monodromy logarithmsabout the 2 local components of the discriminant locus at z (0) , and N := N + N . Then α and α generate coker( N ) ∼ = (ker( N )) ∨ , hence(6.3) and (6.6) are sufficient to capture the limit of the normal function ν associated to Ξ at z (0) .Turning to the right-hand side of (5.6), we may write the formulafor the limiting curve X z (0) as(6.7) 0 = X + Y + X − Y − − X − Y − + 5 . The two singularities of this curve are q = ( − ϕ, − ϕ ) , q = ( − ˜ ϕ, − ˜ ϕ ) , where ϕ := (1 + √
5) and ˜ ϕ := (1 − √ γ , γ passingthrough these nodes q q γ γ -THEORY ELEVATOR 43 are the images of α and α in H ( X z (0) ) under r ∗ . Consider the twouniformizations of X z (0) by P :(6.8) X ( t ) = − ϕ (cid:16) − ζ t (cid:17) (cid:16) − ζ t (cid:17) (cid:16) − t (cid:17) , Y ( t ) = − ϕ (1 − ζ t ) (cid:16) − tζ (cid:17) (1 − t ) , and(6.9) X ( t ) = − ˜ ϕ (cid:16) − ζt (cid:17) (cid:16) − ζt (cid:17) (cid:16) − t (cid:17) , Y ( t ) = − ˜ ϕ (1 − ζt ) (cid:16) − tζ (cid:17) (1 − t ) , where ζ := e π i . The first one t ( X ( t ) , Y ( t )) maps t = 0 , ∞ to q ;the second maps 0 , ∞ 7→ q : so they send the path from “ −∞ to 0”to γ resp. γ . This allows us to “plug in” to the formula from [DK,§6.2], which assigns a divisor N on P \{ , ∞} to each uniformization.In the present case, N = − ζ ] + 9[ ζ ] + 4[ ζ ] + 4[ ζ ]and N = − ζ ] + 9[ ζ ] + 4[ ζ ] + 4[ ζ ] . Working modulo the scissors congruence relations[ ξ ] + [ ξ ] = 0 , [ ξ ] + [ ¯ ξ ] = 0 , [ ξ ] + [1 − ξ ] = 0 , and[ x ] + [ y ] + [ − x − xy ] + [1 − xy ] + [ − y − xy ] = 0 , we have(6.10) ( N ≡ − ζ ] + 5[ ζ ] ≡ − ζ ˜ ϕ ] ≡ ζϕ ] N ≡ − ζ ] + 5[ ζ ] ≡ − ζ ϕ ] ≡ e π i ϕ ] . But according to [loc.cit.] we then have (using (6.10))(6.11) Re (cid:16) π i AJ(Ξ z (0) )( γ ) (cid:17) = π D ( N ) = π D ( ζϕ ) , (6.12) Re (cid:16) π i AJ(Ξ z (0) )( γ ) (cid:17) = π D ( N ) = π D ( e π i ϕ ) . Here Ξ z (0) denotes the pullback motivic cohomology class on X z (0) , and D ( z ) = Im(Li ( z )) + arg(1 − z ) log | z | is the Bloch-Wigner function.By (5.6), we have that (6.11) [resp. (6.12)] is equal to the real partof the z → z (0) limit of (6.3) [resp. (6.6)], which yields precisely therelations(6.13) π D ( e π i ϕ ) = log(5) − X m,r ≥ ( − m (5 m + 3 r )!((2 m + r )!) m !(5 m + 3 r )5 m +2 r and(6.14) π D ( e π i ϕ ) = log(5) − X m,r ≥ ( − r (5 m + 2 r )!(3 m + r )! r !( m !) (5 m + 2 r )5 m + r conjectured by Codesido, Grassi and Marino [CGM, (4.106)] as a test(for the mirror C / Z geometry) of the correspondence between spec-tral theory and enumerative geometry proposed in [GHM]. References [As] M. Asakura,
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