Analytic torsion for log-Enriques surfaces and Borcherds product
aa r X i v : . [ m a t h . DG ] S e p ANALYTIC TORSION FOR LOG-ENRIQUES SURFACES ANDBORCHERDS PRODUCT
XIANZHE DAI AND KEN-ICHI YOSHIKAWA
Abstract.
We introduce a holomorphic torsion invariant of log-Enriques sur-faces of index two with cyclic quotient singularities of type (1 , k singular points is a modular varietyof orthogonal type associated with a unimodular lattice of signature (2 , − k ).We prove that the invariant, viewed as a function on the modular variety, isgiven by the Petersson norm of an explicit Borcherds product. We note thatthis torsion invariant is essentially the BCOV invariant in the complex dimen-sion 2. As a consequence, the BCOV invariant in this case is not a birationalinvariant, unlike the Calabi-Yau case. Contents
1. Introduction 12. log-Enriques surfaces 43. Analytic torsion for K K τ k and the BCOV invariant 47References 521. Introduction
The analytic torsion, which is a certain combination of the determinants of HodgeLaplacians on differential forms, is an invariant of Riemannian manifolds defined byRay and Singer [40] as an analytic analog of the Reidemeister torsion, the first topo-logical invariant which is not a homotopy invariant. It was proved independentlyby Cheeger [11] and M¨uller [38] that the analytic torsion and the Reidemeister tor-sion agree on closed manifolds (Ray-Singer conjecture). Ray and Singer [41] alsointroduced a version of the analytic torsion for complex manifolds, usually referredas the holomorphic torsion. The holomorphic torsion has found significant applica-tions in Arakelov theory, canonical metrics, and mirror symmetry. Unlike its realanalogue, it depends on the geometry and complex structure of the underlying com-plex manifold [6] (the anomaly formula), which gives rise to interesting functionson moduli spaces. In this paper we focus on this aspect of holomorphic torsion,i.e., its connection with modular forms.
In fact, Ray and Singer already noticed the remarkable connection. Using Kro-necker’s first limit formula, Ray and Singer [41] computed the analytic torsion forelliptic curves and found it to be given in terms of the Jacobi ∆-function, a modu-lar form of weight 12 on H / SL(2 , Z ). Their result has since then been extended tohigher genus Riemann surfaces by Zograf [51], McIntyre-Takhtajan [35], Kokotov-Korotkin [28] and McIntyre-Park [36]; Zograf and McIntyre-Takhtajan studied theanalytic torsion of Riemann surfaces with respect to the hyperbolic metric, whileKokotov-Korotkin and McIntyre-Park studied it with respect to the (degenerate)flat metric attached to an abelian differential of the Riemann surface.In dimension two, motivated by string duality, the second author [43] studiedthe case of 2-elementary K K X and a holomorphic involution ι : X −→ X (acting nontrivially on holomorphictwo forms) and introduced a (equivariant) holomorphic torsion invariant for thosesurfaces. By the global Torelli theorem for K K ι is fixed point free, then the quotient Y = X/ι is an Enriques surface, whoseholomorphic torsion invariant is given by one of the most remarkable Borcherdsproducts, the Borcherds Φ-function of rank 10. In this paper we extend this resultto a class of singular rational surfaces called log-Enriques surfaces introduced byD.-Q. Zhang [47]. As in the case of Enriques surfaces, a log-Enriques surface Y isexpressed as a quotient Y = X/ι , where X is a K Y , and ι is an anti-symplectic involutionon X free from fixed points outside the singular points. To be precise, our log-Enriques surfaces are those of index two in the sense of Zhang [47]. To obtain anice moduli space, we restrict ourself to the case where X has only nodes as itssingular points. A log-Enriques surface with this property is called good in thispaper. Then a good log-Enriques surface can admit at most 10 singular points,any of which is a cyclic quotient singularity of type (1 , k singular points is again a Zariskiopen subset of a modular variety of orthogonal type of dimension 10 − k attachedto a unimodular lattice of signature (2 , − k ). Let us write M k for this modularvariety. When k = 2, we have M odd2 and M even2 , according to the parity of theunimodular lattice of signature (2 , M for M odd2 and M even2 when there is no possibility of confusion. For a good log-Enriques surface Y with k singular points, we write ̟ ( Y ) ∈ M k for the isomorphism class of Y .Interestingly enough, M k can be identified with a Zariski open subset of the K¨ahlermoduli of a Del Pezzo surface V of degree deg V = k , the modular variety given by KM ( V ) = Ω H ( V, Z ) /O + ( H ( V, Z )), where H ( V, Z ) is the total cohomology lattice of V , O + ( H ( V, Z )) is its automorphism group, and Ω H ( V, Z ) is the domain of type IVattached to H ( V, Z ). (See Theorem 2.10.)Analogously to the Enriques lattice, the Del Pezzo lattice H ( V, Z ) admits a re-flective modular form Φ V on Ω H ( V, Z ) for O + ( H ( V, Z )) of weight deg V + 4, which NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 3 is nowhere vanishing on the Zariski open subset corresponding to M k and char-acterizes the Heegner divisor of norm ( − V is thedenominator function of a generalized Kac-Moody algebra with explicit Fourierseries expansion by Gritsenko and Nikulin [22], [23]. (See § V .)On the other hand, even though they are rational surfaces, every log-Enriquessurface Y admits a Ricci flat K¨ahler orbifold metric [27]. Let τ ( Y ) denote theanalytic torsion of Y in the sense of X. Ma [32] (suitably normalized by volume,see Section 8.1, especially Theorem 8.3 and Theorem 8.4 for the precise definition).Then our main result says that τ ( Y ) is given by some power of the Petersson normof the Borcherds product Φ V . Theorem 1.1.
There exists a constant C k > depending only on k such that forevery good log-Enriques surface Y with k singular points, τ ( Y ) = C k k Φ V ( ̟ ( Y )) k − / , where V is a Del Pezzo surface of degree k . It is important to note that our torsion invariant is essentially the complex 2-dimensional analogue of the BCOV invariant (See [3], [19], [17], [20]). In higherdimensions, Bershadsky, Cecotti, Ooguri and Vafa [3] introduced a certain combi-nation of holomorphic torsions, called the BCOV torsion, and predicted the mirrorsymmetry at genus one as an equivalence of the BCOV torsion and certain curvecounting invariants at genus one. The corresponding holomorphic torsion invari-ant of Calabi-Yau threefolds, called the BCOV invariant, was introduced by Fang,Lu and the second author [19], who verified some prediction in [3]. Very recently,the BCOV invariant is extended to Calabi-Yau manifolds of arbitrary dimensionby Eriksson, Freixas i Montplet and Mourougane [17], who have established themirror symmetry at genus one for the Dwork family in arbitrary dimension [18].The notion of BCOV invariant is further extended to certain class of pairs by Y.Zhang [49], who, together with L. Fu, has established the birational invariance ofthe BCOV invariants [50], [20]. According to mirror symmetry, the BCOV invari-ants correspond to the topological string amplitudes whose modular properties areimportant features. In the final section, we will interpret Theorem 1.1 in termsof the BCOV torsion, so that the BCOV invariant of good log-Enriques surfacesis expressed as the Borcherds product Φ V , an infinite product of expected type inmirror symmetry. As log-Enriques surfaces are rational, the BCOV invariant is not a birational invariant in this case.We remark that the equivalence of the analytic torsion of Ricci flat Enriquessurfaces and the Borcherds Φ-function [43] may be viewed as the limiting case k = 0.Since τ ( Y ) is the analytic torsion of a resolution of Y with respect to a degenerateRicci flat metric, our theorem may be viewed as a two-dimensional analogue of thetheorems of Kokotov-Korotkin [28] and McIntyre-Park [36] as mentioned above.Because of the isomorphism between the complex structure moduli of good log-Enriques surfaces and the K¨ahler moduli of Del Pezzo surfaces, in view of mirrorsymmetry at genus one as mentioned above, it may be worth asking if the Fouriercoefficients of the elliptic modular form appearing in the infinite product expansionof Φ V are interpreted as some counting invariants of Del Pezzo surfaces. We alsoremark that by Theorem 1.1 and the recent result of S. Ma [29], the analytic torsionof good log-Enriques surfaces is obtained from the Borcherds Φ-function of rank 10by manipulating quasi-pullbacks successively. See Section 8.3 for the details. XIANZHE DAI AND KEN-ICHI YOSHIKAWA
Our method of proof, which should have independent interest and which carriesout the program proposed in [44, Question 5.18] for 2-elementary K Y via the Eguchi-Hanson instanton to obtain a2-elementary K e X, θ ) and study the limiting behavior of the (equivariant)analytic torsion of ( e X, θ ), as well as other constituents of the invariant τ ( e X, θ ) of( e X, θ ), as e X degenerates into the orbifold double covering X of Y . As a result, theratio τ ( Y ) /τ ( e X, θ ) / may be viewed as the (equivariant) analytic torsion of theEguchi-Hanson instanton (cf. Theorem 7.12). In [5], Bismut computed the behaviorof Quillen metrics when the exceptional divisor is blown down to a smooth point.In this paper, we study the same type of problem, where the blowing-up of C will be replaced by the Eguchi-Hanson instanton. We remark that Theorem 1.1would be proved in the same way as in [43] by making use of the fundamentaltheorems for Quillen metrics such as the curvature formula, anomaly formula, andthe embedding formula [4], [6], [7], [31], whose extension to orbifolds were obtainedby X. Ma [32], [33], if we could understand degenerations of log Enriques surfaces.On the other hand, it would be difficult to understand the geometric meaning ofthe ratio τ ( Y ) /τ ( e X, θ ) / by this method. In the final section, we will observe that τ ( Y ) /τ ( e X, θ ) / is the key factor in the exact comparison formula for the BCOVinvariants for certain Calabi-Yau orbifolds.This paper is organized as follows.In Section 2, we recall log-Enriques surfaces and study their moduli space. InSection 3, we recall the notion of analytic torsion and also the holomorphic torsioninvariant τ ( e X, θ ) for 2-elementary K K { γ ǫ,δ } on e X converging to an orbifold metric with uniformlybounded Ricci curvature. In Section 5, we study the behavior of some constituentsof the invariant τ ( e X, θ ) with respect to the metric γ ǫ,δ as ǫ →
0. In Section 6, wederive some estimates for the heat kernels of ( e X, γ ǫ,δ ). In Section 7, we determinethe behavior of (equivariant) analytic torsion of ( e X, θ ) with respect to the metric γ ǫ,δ as ǫ → δ →
0. In Section 8, we introduce a holomorphic torsion invariantfor good log-Enriques surfaces and prove the main theorem. In Section 9, we studythe relation between the invariant τ ( Y ) and the BCOV invariant. Acknowledgements
The first author is partially supported by NSF-DMS1611915 and the SimonsFoundation. The second author is partially supported by JSPS KAKENHI GrantNumbers 16H03935, 16H06335. This research was initiated when the authors metin the conference “Analytic torsion and its applications” held in the department ofmathematics of Paris-Sud University in June 2012. The authors thank the organizerof this conference. 2. log-Enriques surfaces log-Enriques surfaces.
Following D.-Q. Zhang [47], [48], we recall the notionof log-Enriques surfaces (of index 2) and its basic properties.
Definition 2.1.
An irreducible normal projective complex surface Y is called a log-Enriques surface if the following conditions are satisfied: NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 5 (1) Y is singular and has at most quotient singularities except rational doublepoints. In particular, Y has the structure of a compact complex orbifold.(2) The irregularity of Y vanishes, i.e., H ( Y, O Y ) = 0.(3) Let K Y be the canonical line bundle of Y in the sense of orbifolds. Then K Y = O Y , K ⊗ Y = O Y . Remark . For p ∈ Sing Y , there exist a neighborhood U p of p in Y , a finitegroup G p ⊂ GL( C ) and a G p -invariant neighborhood V of 0 in C such that( U p , p ) ∼ = ( V /G p , K Y | U p is defined as ( V × C ) /G p , where the G p -actionis given by g · ( z, ζ ) = ( g · z, det( g ) ζ ). Remark . Logarithmic Enriques surfaces in this paper are those of index two inZhang’s papers [47], [48]. We only deal with log-Enriques surfaces of index two inthis paper.If a smooth complex surface satisfies conditions (2), (3), then it is an Enriquessurface. For this reason, we impose that log-Enriques surfaces are singular . Then alog-Enriques surface is rational [47, Lemma 3.4]. By Zhang [47, Lemma 3.1], everysingularity of a log-Enriques surface Y is the quotient of a rational double point by Z / Z and hence non-Gorenstein. Indeed, if p ∈ Sing Y , then there exists by (1) anisomorphism of germs of analytic spaces ( Y, p ) ∼ = ( C /G, G ⊂ GL( C ) isa finite group. By (3), the image of the homomorphism det : G ∋ g → det g ∈ C ∗ is ±
1. If G := ker det ⊂ G , then G ⊂ SL( C ) is a normal subgroup of G of index2, so that ( X,
0) = ( C /G ,
0) is a rational double point. If p : ( X, → ( Y, G ⊂ G , then p inducesan isomorphism of germs ( X/ ( G/G ) , → ( Y, G/G ∼ = {± } ∼ = Z / Z .By [47, Lemma 3.1], ( X,
0) is a rational double point of type A n − for some n .Since the homomorphism det : G → C ∗ is trivial, K ⊗ Y is a holomorphic line bundleon Y in the ordinary sense.2.2. The canonical double covering.
Let Y be a log-Enriques surface and let Ξ ∈ H ( Y, K ⊗ Y ) \ { } be a nowhere vanishing bicanonical form on Y in the senseof orbifolds. The canonical double covering of Y is defined as X := { ( y, ξ ) ∈ K Y ; ξ ⊗ ξ = Ξ } ⊂ K Y , which is equipped with the projection p : X → Y induced from the projection K Y → Y . Then p : X → Y is a double covering, which ramifies only over Sing Y .(Since K Y, p = C / ± p ∈ Sing( Y ), p − ( p ) consists of a single point.) The canonical involution ι : X → X is defined as the non-trivial covering transformation: ι ( y, ξ ) = ( y, − ξ ) . Since the ramification locus of p : X → Y is Sing X , we have X ι = Sing X and that ι has no fixed points on X \ Sing X .Let π : e X → X be the minimal resolution and let θ : e X → e X be the involutioninduced by the canonical involution ι . The involution θ is also called the canonicalinvolution on e X . We have the following commutative diagram:(2.1) e X π −−−−→ X p −−−−→ Y = X/ι θ y ι y y id e X −−−−→ π X −−−−→ p Y = X/ι
XIANZHE DAI AND KEN-ICHI YOSHIKAWA
Here the projection p : X → Y ramifies only at Sing Y . In what follows, we denoteby X ι and e X θ the sets of fixed points of ι and θ , respectively. Since ι has nofixed points on X \ Sing X , θ has no fixed points on e X \ π − (Sing X ). Hence e X \ π − (Sing X ) ⊂ e X \ e X θ . In other words, e X θ ⊂ π − (Sing X ). Lemma 2.4.
In the commutative diagram (2.1) , the following hold: (1) X is a K surface with rational double points and X ι = Sing X = p − (Sing Y ) , ι ∗ | H ( X,K X ) = − . (2) ( e X, θ ) is a -elementary K surface. Namely, θ acts non-trivially on holo-morphic -forms on e X . Moreover, there exists an integer k ∈ { , . . . , } such that e X θ = E ∐ . . . ∐ E k , E i ∼ = P . The pair ( e X, θ ) is called the -elementary K surface associated to Y .Proof. See [47, Lemma 3.1, Th. 3.6] for (1) and [48, Lemma 2.1] for (2). (cid:3)
Lemma 2.5.
Let Y , Y ′ be log Enriques surfaces with canonical double coverings p ′ : X ′ → Y ′ and p : X → Y , respectively. Let ϕ : Y ′ → Y be a birational holomor-phic map. Then the following hold: (1) ϕ ∗ induces an isomorphism from H ( Y, K ⊗ Y ) to H ( Y ′ , K ⊗ Y ′ ) . (2) ϕ (Sing Y ′ ) ⊂ Sing Y . (3) ϕ lifts to a holomorphic map f : X ′ → X of canonical double coverings.Proof. (1) Let Ξ ∈ H ( Y, K ⊗ Y ) \ { } and Ξ ′ ∈ H ( Y ′ , K ⊗ Y ′ ) \ { } . Then ϕ ∗ Ξ isa bicanonical from on Y ′ \ (Sing Y ′ ∪ ϕ − (Sing Y )), and Ξ ′ is nowhere vanishing.We get ϕ ∗ Ξ / Ξ ′ ∈ O ( Y ′ \ (Sing Y ′ ∪ ϕ − (Sing Y ))) = O ( Y ′ \ ϕ − (Sing Y )) = O ( Y \ Sing Y ) = O ( Y ) = C , where the first and the third equalities follow from thenormality of Y ′ and Y and the second equality follows from the Zariski MainTheorem. Hence ϕ ∗ Ξ = c Ξ ′ with some c ∈ C \ { } , and ϕ ∗ is an isomorphism.(2) Let o ∈ Sing Y ′ . Assume ϕ ( o ) ∈ Y \ Sing Y . There exist a neighborhood U of ϕ ( o ) and a nowhere vanishing canonical form η ∈ H ( U, K Y ). We can expressΞ | U = F · η ⊗ , F ∈ O ∗ ( U ). Since ϕ ∗ Ξ and ϕ ∗ F are nowhere vanishing on ϕ − ( U ),so is ϕ ∗ η ⊗ . Hence ϕ ∗ η is nowhere vanishing. Since any singular point of Y ′ isnon-Gorenstein, we get a contradiction. Thus ϕ ( o ) ∈ Sing Y .(3) Since ϕ ∗ Ξ is nowhere vanishing on Y ′ \ ϕ − (Sing Y ), ϕ has no critical pointson Y ′ \ ϕ − (Sing Y ). Since the restriction of ϕ to Y ′ \ ϕ − (Sing Y ) is a closedmap, ϕ : Y ′ \ ϕ − (Sing Y ) → Y \ Sing Y is an ´etale covering of degree one, i.e.,an isomorphism. ϕ induces a holomorphic map f : X ′ \ ( p ′ ) − ϕ − (Sing Y ) → X \ p − (Sing Y ) such that p ◦ f = ϕ ◦ p ′ . Since p − ( y ) consists of a unique point for any y ∈ Sing Y , f extends to a map from X ′ to X by setting f ( x ′ ) := p − ( ϕ ( p ′ ( x ′ ))) for x ′ ∈ ( p ′ ) − ϕ − (Sing Y ). By construction, p ◦ f = ϕ ◦ p ′ . By this equality and thebijectivity of the map p : Sing X → Sing Y , f is continuous. Since f is holomorphicon a Zariski open subset, f : X ′ → X is holomorphic by the normality of X ′ . (cid:3) The good model of a log-Enriques surface.
The group Z / Z acts on C as the multiplication by i = √−
1, i.e., i ( z , z ) := ( iz , iz ). We define the cyclicquotient singularity of type (1 ,
1) by( C / h i i , . NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 7
Its minimal resolution is the total space of the line bundle O P ( − ̟ : ( O P ( − , E ) → ( C / h i i , , where the exceptional divisor E = ̟ − (0) is a ( − E = − Definition 2.6.
A log-Enriques surface Y is good if Y has only cyclic quotientsingularities of type (1 , Y be a log-Enriques surface, p : X → Y be its canonical double covering,and π : e X → X be the minimal resolution. Then X and e X are equipped with thecanonical involutions ι and θ , respectively. Let E = π − (Sing X ) be the exceptionaldivisor of π : e X → X . Then E ⊃ e X θ = ∐ ki =1 E i with 1 ≤ k ≤
10. Since E i is a( − e X , it is a ( − e X/θ and its contraction produces a cyclicquotient singularity of type (1 , Definition 2.7.
The good model of a log-Enriques surface Y , denoted by Y ♮ , isdefined as the contraction of the disjoint union of ( − e X θ in e X/θ , where( e X, θ ) is the 2-elementary K Y .Another construction of Y ♮ from Y is as follows [47, Th. 3.6], [48, Lemmas 1.4and 2.1]. Let e Y be the minimal resolution of Y with exceptional divisor D ⊂ e Y . Let Y be the blowing-up of e Y at Sing D . Then the proper transform of D consistsof disjoint ( − e D , . . . , e D k . Then Y ∼ = e X/θ and Y ♮ is obtainedfrom Y by contracting the e D i ’s. (Notice that e Y and Y are not log-Enriquessurfaces.) As is verified easily, the composition of the rational map Y ♮ Y andthe blowing-down Y → Y extends to a holomorphic map from Y ♮ to Y .By construction, Y ♮ has at most cyclic quotient singularities of type (1 , Y is a good log-Enriques surface, then Y = Y ♮ . Proposition 2.8.
Let Y be a log-Enriques surface. If there is a birational holo-morphic map from a good log-Enriques surface Y ′ to Y , then Y ′ ∼ = Y ♮ .Proof. Let X ♮ (resp. X ′ ) be the canonical double covering of Y ♮ (resp. Y ′ ) and let e X ♮ (resp. f X ′ ) be the minimal resolution of X ♮ (resp. X ′ ). The birational morphism Y ′ → Y induces a birational morphism ψ : ( X ′ , ι ′ ) → ( X, ι ) by Lemma 2.5 (3), andthis ψ induces an isomorphism f : ( e X ′ , θ ′ ) → ( e X, θ ) = ( e X ♮ , θ ), by the minimality of K e X ′ /θ ′ , ( e X ′ ) θ ′ ) ∼ = ( e X ♮ /θ, ( e X ♮ ) θ ). Since the projection e X ′ /θ ′ → Y ′ (resp. e X ♮ /θ → Y ♮ ) is obtained by contracting every component of ( e X ′ ) θ ′ (resp.( e X ♮ ) θ ) to a cyclic quotient singularity of type (1 , f induces an isomorphismfrom Y ′ to Y . (cid:3) By Proposition 2.8, every log-Enriques surface has a unique good model. ByZhang [47, Th. 3.6], [48, Th.4, Cor. 5, Lemma 2.3], one can associate to a log-Enriques surface another log-Enriques surface with a unique singular point in thecanonical way. So log-Enriques surfaces of this type form another class to be stud-ied. Because of the uniqueness (up to a scaling) of the Ricci-flat ALE hyperk¨ahlermetric on the minimal resolution of A -singularity, in this paper, we focus on goodlog-Enriques surfaces.In the rest of this section, we study the moduli space of good log-Enriquessurfaces. Throughout this paper, we mean by lattice a free Z -module of finite rankequipped with a non-degenerate integral symmetric bilinear form. We often identifya lattice with its Gram matrix. XIANZHE DAI AND KEN-ICHI YOSHIKAWA -elementary K surfaces and log-Enriques surfaces. A pair (
Z, ι ) iscalled a 2-elementary K Z is a K ι : Z → Z is a holomor-phic anti-symplectic involution. For a 2-elementary K Z, ι ), we define H ( Z, Z ) ± = { l ∈ H ( Z, Z ); ι ∗ ( l ) = ± l } , which is equipped with the integral bilinear form induced from the intersectionpairing on H ( Z, Z ). Then H ( Z, Z ) is isometric to the K L K := U ⊕ U ⊕ U ⊕ E ( − ⊕ E ( − , where U = (cid:0) (cid:1) and E ( −
1) is the negative-definite even unimodular lattice ofrank 8 whose Gram matrix is given by the Cartan matrix of type E . If r denotesthe rank of H ( Z, Z ) + , then H ( Z, Z ) + (resp. H ( Z, Z ) − ) has signature (1 , r − , − r )). For a 2-elementary K Z, ι ), the topological type of Z ι is determined by the isometry class of the lattice H ( Z, Z ) − .Let Y be a good log-Enriques surface and let ( e X, θ ) be the corresponding 2-elementary K e X/θ, e X θ ) → ( Y, Sing( Y )) is the minimal resolutionof the cyclic quotient singularities of type (1 ,
1) of Y . We set k := Y )and define Λ k as the unimodular lattice of signature (2 , − k ). Under the identi-fication with a lattice with its Gram matrix, we haveΛ k = (cid:18) I − I − k (cid:19) ( k = 2) , Λ = (cid:18) I − I (cid:19) or U ⊕ U ( k = 2) . According to the parity of Λ , we set Λ odd2 := I ⊕ − I and Λ even2 := U ⊕ U . Since e X θ consists of smooth rational curves, we deduce from Nikulin [39, Th. 4.2.2] thatthere is an isometry of lattices α : H ( e X, Z ) ∼ = L K with(2.2) α : H ( e X, Z ) − ∼ = Λ k (2) . Here Λ k (2) stands for the rescaling of Λ k , whose bilinear form is the double ofthat of Λ k . An isometry of lattices α : H ( e X, Z ) ∼ = L K satisfying (2.2) is called amarking of ( e X, θ ). We set M k := Λ k (2) ⊥ , where the orthogonal complement is considered in the K L K . A 2-elementary K M k if its invariant lattice is isometric to M k .We define Ω k := { [ η ] ∈ P (Λ k ⊗ C ); h η, η i = 0 , h η, η i > } . Then Ω k consists of two connected components Ω + k and Ω − k , each of which is iso-morphic to bounded symmetric domain of type IV of dimension 10 − k . Let O (Λ k )be the automorphism group of Λ k and let O + (Λ k ) ⊂ O (Λ k ) be the subgroup ofindex 2 consisting of elements preserving Ω ± k . We define the orthogonal modularvariety associated with Λ k by M k := Ω k /O (Λ k ) = Ω + k /O + (Λ k ) . When k = 2, we define M odd2 := Ω /O (Λ odd2 ) and M even2 := Ω /O (Λ even2 ). Whenthere is no possibility of confusion, we write M for M odd2 and M even2 . NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 9
Since θ acts non-trivially on H ( e X, Ω e X ), we deduce the inclusion from the Hodgedecomposition H ( e X, Ω e X ) ⊂ H ( e X, C ) − . Since H ( e X, Ω e X ) is a complex line, itfollows from the Riemann-Hodge bilinear relations that ̟ ( e X, θ, α ) := [ α ( H ( e X, Ω e X )] ∈ Ω k . The point ̟ ( e X, θ, α ) ∈ Ω k is called the period of ( e X, θ, α ). We define the periodof ( e X, θ ) as the O (Λ k )-orbit of ̟ ( e X, θ, α ), i.e., ̟ ( e X, θ ) := O (Λ k ) · [ α ( H ( e X, Ω e X )] ∈ M k . By [43, Th. 1.8], the coarse moduli space of 2-elementary K M k is isomorphic via the period map to the analytic space M ok := M k \ D k , where D k is the discriminant divisor D k = ( [ d ∈ Λ k , d = − d ⊥ ) /O (Λ k ) , d ⊥ := { [ η ] ∈ Ω k ; h η, d i = 0 } . The period mapping for log-Enriques surfaces.Definition 2.9.
The period of a good log-Enriques surface Y with k singular pointsis defined as the period of the corresponding 2-elementary K e X, θ ): ̟ ( Y ) := ̟ ( e X, θ ) ∈ M k . When k = 2, we define the parity of Y as that of the lattice Λ defined by (2.2). Theorem 2.10.
The period mapping induces a bijection between the isomorphismclasses of good log-Enriques surfaces with k singular points (and fixed parity when k = 2 ) and M ok .Proof. Let N k be the isomorphism classes of good log-Enriques surfaces with k singular points (and fixed parity when k = 2). By [43, Th. 1.8], we can identify M ok with the isomorphism classes of 2-elementary K M k via theperiod mapping. We define a map f : N k → M ok by setting f ( Y ) = ( e X, θ ), where( e X, θ ) is the 2-elementary K Y . Similarly, we define a map g : M ok → N k by sending ( Z, σ ) ∈ M ok to the surface obtained from Z/σ by blowingdown Z σ . Since Z σ consists of k disjoint ( − Z/σ consists of k disjoint ( − g ( Z, σ ) is a good log-Enriques surface with k singularpoints. Since g = f − by [48, Lemmas 1.4 and 2.1], f is a bijection. (cid:3) Since the (locally defined) family of 2-elementary K M k asso-ciated to a holomorphic family of good log-Enriques surfaces with k -singular pointsis again holomorphic, the period mapping for any holomorphic family of good log-Enriques surfaces with k -singular points is holomorphic. In what follows, we regard M ok as a coarse moduli space of good log-Enriques surfaces with k singular points(and fixed parity when k = 2).3. Analytic torsion for K surfaces and -elementary K surfaces Analytic torsion.
Let Z be a compact complex orbifold of dimension n andlet γ be a K¨ahler form on Z in the sense of orbifolds. Let ι : Z → Z be a holomorphic involution and assume that ι preserves γ . Let A ,qZ be the space of smooth (0 , q )-forms on Z in the sense of orbifolds. Let (cid:3) q = ( ¯ ∂ + ¯ ∂ ∗ ) be the Hodge-KodairaLaplacian acting on A ,qZ . Let ζ q ( s ) := X λ ∈ σ ( (cid:3) q ) \{ } λ − s dim E ( λ ; (cid:3) q )be the spectral zeta function of (cid:3) q , where E ( λ ; (cid:3) q ) is the eigenspace of (cid:3) q corre-sponding to the eigenvalue λ . Similarly, let ζ q ( s )( ι ) := X λ ∈ σ ( (cid:3) q ) \{ } λ − s Tr (cid:2) ι ∗ | E ( λ ; (cid:3) q ) (cid:3) be the equivariant spectral zeta function of (cid:3) q . Since ( Z, γ ) is a K¨ahler orbifold, ζ q ( s ) and ζ q ( s )( ι ) converge absolutely when ℜ s > dim Y , extend to meromorphicfunctions on C , and are holomorphic at s = 0. After Ray-Singer [41] and Bismut[4], we make the following: Definition 3.1.
The analytic torsion of the K¨ahler orbilod (
Z, γ ) is defined as τ ( Z, γ ) := exp[ − n X q =0 ( − q q ζ ′ q (0)] . The equivariant analytic torsion of (
Z, ι, γ ) is defined as τ Z ( Z, γ )( ι ) := exp[ − n X q =0 ( − q q ζ ′ q (0)( ι )] . Analytic torsion for K surfaces.Theorem 3.2. Let Z be a K surface and let η ∈ H ( Z, K Z ) \ { } and let γ be aK¨ahler form on Z . Then the following formula holds: τ ( Z, γ ) = exp (cid:20) − Z Z log (cid:26) η ∧ ηγ / · Vol(
Z, γ ) k η k L (cid:27) c ( Z, γ ) (cid:21) , where c i ( Z, γ ) denotes the i -th Chern form of ( T Z, γ ) and k η k L := R Z η ∧ η .Proof. Let ω be a Ricci-flat K¨ahler form on Z such that(3.1) ω
2! = η ∧ η. Since the L -metric on H ( Z, O Z ) = H ( Z, K Z ) ∨ is independent of the choice of aK¨ahler metric on Z , we get by the anomaly formula for Quillen metrics [6](3.2) log (cid:18) τ ( Z, γ ) Vol(
Z, γ ) τ ( Z, ω ) Vol(
Z, ω ) (cid:19) = 124 Z Z g c c ( T Z ; γ, ω ) , where g c c ( T Z ; γ, ω ) is the Bott-Chern secondary class [6] such that − dd c g c c ( T Z ; γ, ω ) = c ( Z, γ ) c ( Z, γ ) − c ( Z, ω ) c ( Z, ω ) . Since c ( Z, ω ) = 0 by the Ricci-flatness of ω , and e c ( L ; h, h ′ ) = log( h/h ′ ) for aholomorphic line bundle L and Hermitian metrics h and h ′ on L , and since g c c ( T Z ; γ, ω ) = e c ( T Z ; γ, ω ) c ( Z, γ ) + c ( Z, ω ) e c ( T Z ; γ, ω ) NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 11 by [21], we get by (3.1)(3.3) g c c ( T Z ; γ, ω ) = e c ( T Z ; γ, ω ) c ( Z, γ ) = log (cid:18) γ ω (cid:19) c ( Z, γ ) = log (cid:18) γ / η ∧ η (cid:19) c ( Z, γ ) . Since Vol(
Z, γ ) / Vol(
Z, ω ) = Vol(
Z, γ ) / k η k L , we get by substituting (3.3) into (3.2)(3.4) log (cid:18) τ ( Z, γ ) τ ( Z, ω ) (cid:19) = − log (cid:18) Vol(
Z, γ ) k η k L (cid:19) − Z Z log (cid:18) η ∧ ηγ / (cid:19) c ( Z, γ )= − Z Z log (cid:26) η ∧ ηγ / · Vol(
Z, γ ) k η k L (cid:27) c ( Z, γ ) , where we used the Gauss-Bonnet-Chern formula for Z to get the second equality.Since ω is Ricci-flat, the Laplacians (cid:3) and (cid:3) are isospectral via the map A , Y ∋ f f η ∈ A , Y . Hence, for the Ricci-flat metric ω , we get the equality ofmeromorphic functions(3.5) ζ ( s ) = ζ ( s )Since the Dolbeault complex is exact on the orthogonal complement of harmonicforms, we get the equality of meromorphic functions(3.6) ζ ( s ) − ζ ( s ) + ζ ( s ) = 0 . By (3.5) and (3.6), we get(3.7) τ ( Z, ω ) = 1 . The result follows from (3.4) and (3.7). (cid:3)
Equivariant analytic torsion for -elementary K surfaces. Let Z bea K ι : Z → Z be an anti-symplectic holomorphic involution. Let Z ι = ∐ α C α be the decomposition into the connected components. By Nikulin [39,Th. 4.2.2], every C α is a compact Riemann surface unless Z ι = ∅ .Let γ be an ι -invariant K¨ahler form on Z and let η ∈ H ( Z, K Z ) \ { } . Let M := H ( Z, Z ) + be the invariant sublattice of H ( Z, Z ) with respect to the ι -action. We define τ M ( Z, ι ) := Vol(
Z, γ ) − r ( M )4 τ Z ( Z, γ )( ι ) A M ( Z, ι, γ ) Vol( Z ι , γ | Z ι ) τ ( Z ι , γ | Z ι ) , where we define τ ( Z ι , γ | Z ι ) := Y α τ ( C α , γ | C α ) , Vol( Z ι , γ | Z ι ) := Y α Vol( C α , γ | C α )and A M ( Z, ι, γ ) := exp (cid:20) Z Z ι log (cid:26) η ∧ ηγ / · Vol(
Z, γ ) k η k L (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) Z ι c ( Z ι , γ | Z ι ) (cid:21) . As before, c ( Z ι , γ | Z ι ) is the first Chern form of ( T Z ι , γ | Z ι ). Theorem 3.3.
The number τ M ( Z, ι ) is independent of the choice of an ι -invariantK¨ahler form on Z .Proof. See [43, Th. 5.7]. (cid:3)
For an explicit formula for τ M as a function on the moduli space of 2-elementary K τ M ( Z, ι ) as follows(3.8) τ M ( Z, ι ) = Vol(
Z, γ ) − r ( M )4 τ ( Z, γ ) τ Z ( Z, γ )( ι ) Vol( Z ι , γ | Z ι ) τ ( Z ι , γ | Z ι ) × A M ( Z, ι, γ ) exp (cid:20) Z Z log (cid:26) η ∧ ηγ / · Vol(
Z, γ ) k η k L (cid:27) c ( Z, γ ) (cid:21) . A degenerating family of K¨ahler metrics
Let Y be a good log-Enriques surface. For an orbifold K¨ahler form γ on Y , wewrite Vol( Y, γ ) = R Y γ /
2! for the volume of (
Y, γ ). We set k := Y ) ∈ { , . . . , } . Let ( e X, θ ) be the 2-elementary K Y such that e X θ = ∐ p ∈ Sing( Y ) E p , E p ∼ = P . Let π : ( e X, e X θ ) → ( X, Sing X )be the blowing-down of the disjoint union of ( − p = π ( E p ) . In this section, we construct a two parameter family of K¨ahler metrics { γ ǫ,δ } on e X converging to an orbifold K¨ahler metric on X , which is obtained by gluing theEguchi-Hanson instanton at each p and a K¨ahler metric on X . In the subsequentsections, we study the limiting behavior of various geometric quantities of ( e X, γ ǫ,δ )to construct an invariant of the log-Enriques surface Y .4.1. Eguchi-Hanson instanton.
For ǫ ≥
0, let F ǫ ( z ) be the function on C \ { } defined by F ǫ ( z ) := p k z k + ǫ + ǫ log k z k p k z k + ǫ + ǫ ! . On every compact subset of C \ { } , we have lim ǫ → F ǫ ( z ) = k z k . For all ǫ ≥ δ > F ǫ ( δz ) = δ F ǫδ − ( z ) . Let T ∗ P be the holomorphic cotangent bundle of the projective line and let E ⊂ T ∗ P be its zero section. Let Π : ( T ∗ P , E ) → ( C / {± } , i∂ ¯ ∂F ǫ ( z ) = i ǫ ∂ k z k ∧ ¯ ∂ k z k p ( k z k + ǫ + ǫ ) p k z k + ǫ + k z k ∂ ¯ ∂ k z k p k z k + ǫ + ǫ + ǫ ∂ ¯ ∂ log k z k ! is a positive (1 , C \ { } ) / ± i∂ ¯ ∂F ǫ )
2! = ( √− dz ∧ d ¯ z ∧ dz ∧ d ¯ z , its pull-back to T ∗ P γ EH ǫ := Π ∗ ( i∂ ¯ ∂F ǫ ) NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 13 extends to a Ricci-flat K¨ahler form on T ∗ P for ǫ >
0, called the
Eguchi-Hansoninstanton . We write γ EH for γ EH1 . The coordinate change z
7→ √ ǫz on C inducesan isometry of K¨ahler manifolds(4.1) ( T ∗ P , γ EH ǫ ) ∼ = ( T ∗ P , ǫγ EH ) . When ǫ = 0, i∂ ¯ ∂F = i∂ ¯ ∂ k z k is the Euclidean K¨ahler form on C / {± } , and γ EH0 = Π ∗ { i∂ ¯ ∂F ( z ) } = π ∗ ( i∂ ¯ ∂ k z k )is a degenerate K¨ahler form on T ∗ P .Let ω FS be the Fubini-Study form on P such that[ ω FS ] = c ( O P (1)) . By the definition of F ǫ , we get γ EH ǫ | E = ǫΠ ∗ ( i∂ ¯ ∂ log k z k ) | E = 2 πǫ ω FS . Glueing of the Eguchi-Hanson instanton.
A modification of the Eguchi-Hanson instanton.
Let B ( r ) ⊂ C be the ballof radius r > ∈ C and set V ( r ) := B ( r ) / {± } . Let Π : ( e V ( r ) , E ) → ( V ( r ) ,
0) be the blowing-up at the origin. Then V ( ∞ ) = C / ± e V ( ∞ ) = T ∗ P . For z ∈ ( C \ { } ) / ± ǫ ≥
0, we define E ( z, ǫ ) := F ǫ ( z ) − k z k . Since the error term E ( z, ǫ ) is a C ω function on ( V (4) \ V (1)) × [0 ,
1] with E ( z,
0) = 0,there is a constant C k for all k ≥ z ∈ V (4) \ V (1) | ∂ kz E ( z, ǫ ) | ≤ C k ǫ. Let ρ ( t ) be a C ∞ function on R such that 0 ≤ ρ ( t ) ≤ R , ρ ( t ) = 1 for t ≤ ρ ( t ) = 0 for t ≥
2. We set φ ǫ ( z ) := ρ ( k z k ) F ǫ ( z ) + { − ρ ( k z k ) } k z k = k z k + ρ ( k z k ) E ( z, ǫ )and we define a (1 ,
1) form on V ( ∞ ) \ { } by κ ǫ := i∂ ¯ ∂φ ǫ . Since φ ǫ ( z ) = F ǫ ( z ) on V (1), κ ǫ extends to a real (1 , T ∗ P , which ispositive on e V (1). Since φ ǫ ( z ) = k z k + ρ ( k z k ) E ( z, ǫ ) on V (2) \ V (1), there exists by(4.2) a constant ǫ ( ρ ) ∈ (0 ,
1) depending only on the choice of the cut-off function ρ such that κ ǫ is a positive (1 , V (2) \ V (1) for 0 < ǫ ≤ ǫ ( ρ ). As aresult, { κ ǫ } <ǫ ≤ ǫ ( ρ ) is a family of K¨ahler forms on T ∗ P such that κ ǫ = i∂ ¯ ∂ k z k on T ∗ P \ e V (2).We have the following slightly refined estimate for the error term E ( z, ǫ ). Set E ( z ) := E ( z,
1) = E ( z ) + E ( z ) , where E ( z ) = p k z k + 1 − k z k = 1 p k z k + 1 + k z k , E ( z ) = log k z k p k z k + 1 + 1 . Then, for any nonnegative integer k , there exists a constant C k > | ∂ kz E ( z ) | ≤ C k (1 + k z k ) − (2+ k ) for all z ∈ V ( ∞ ) \ { } ;(ii) | ∂ kz E ( z ) | ≤ C k (1 + k z k ) − (2+ k ) for all z ∈ V ( ∞ ) \ V (2);(iii) | ∂ kz E ( z ) | ≤ C k k z k − k for all z ∈ V (2) \ { } k ≥ C log k z k for k = 0.From these inequalities, we get(4.3) | ∂ kz E ( z ) | ≤ ( C k k z k − k ( k ≥ C log k z k , k = 0) ( ∀ z ∈ V (2) \ { } ) ,C k (1 + k z k ) − (2+ k ) ( ∀ z ∈ V ( ∞ ) \ V (2)) . Since E ( z, ǫ ) = ǫ E ( z √ ǫ ,
1) = ǫ E ( z √ ǫ ) and hence ∂ kz E ( z, ǫ ) = ǫ − k ( ∂ kz E )( z √ ǫ ), weget by (4.3)(4.4) | ∂ kz E ( z, ǫ ) | ≤ ( C k ǫ k z k − k ( k ≥ C ǫ (log k z k + log ǫ ) , k = 0) ( ∀ z ∈ V (2) \ { } ) ,C k ǫ ( √ ǫ + k z k ) − (2+ k ) ( ∀ z ∈ V ( ∞ ) \ V (2)) . Here, to get the estimate on V (2) \{ } , we used the fact ǫ ( √ ǫ + k z k ) − (2+ k ) < ǫ k z k − k on V (2) \ V (2 √ ǫ ). Replacing ǫ ( ρ ) by a smaller constant if necessary, we may assumeby (4.4) the following inequality of Hermitian matrices for all 0 < ǫ ≤ ǫ ( ρ ) and z ∈ V ( ∞ ) \ V (2):(4.5) 12 ( δ ij ) ≤ ( δ ij + ∂ E ( z, ǫ ) ∂z i ∂ ¯ z j ) ≤ δ ij ) . Moreover, for k z k ≤ | ∂ ǫ E ( z, ǫ ) | ≤ C ǫ k z k − . Lemma 4.1.
There exist constants C , C > such that the following inequalityof (1 , -forms on T ∗ P hold for all < ǫ ≤ ǫ ( ρ ) : C γ EH ǫ ≤ κ ǫ ≤ C γ EH ǫ . Proof. (Step 1) On e V (1), we have κ ǫ = γ EH ǫ . On e V (2) \ e V (1), it follows from (4.2)that there exist constants C , C > ǫ ∈ (0 , ǫ ( ρ )] with C γ EH ǫ ≤ κ ǫ ≤ C γ EH ǫ . Combining these two estimates, we get C γ EH ǫ ≤ κ ǫ ≤ C γ EH ǫ on e V (2). (Step 2) We compare κ ǫ and γ EH ǫ on T ∗ P \ e V (2). On T ∗ P \ e V (2), we have κ ǫ = γ EH0 . By (4.5), we have γ EH ǫ ≤ γ EH0 ≤ γ EH ǫ on T ∗ P \ e V (2). Since κ ǫ = γ EH0 on T ∗ P \ e V (2), We get the desired estimate on T ∗ P \ e V (2). This completes theproof. (cid:3) A family of K¨ahler metrics on e X . Since E p is a ( − e X , there exista neighborhood U p of E p in e X and an isomorphism of pairs ψ p : ( U p , E p ) ∼ = ( e V (1) , E ) . We may and will assume that ψ p extends to an isomorphism between an open subsetof e X containing U p and e V (4). We write V ( r ) p for V ( r ) viewed as a neighborhoodof p ∈ Sing( X ). In what follows, we identify e V ( r ) p with ψ − p ( e V ( r ) p ).Let γ be a θ -invariant K¨ahler form on X in the sense of orbifolds, which has apotential function on every V (4) p . By modifying the potential of γ on each V (4) p NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 15 (cf. [43, Proof of Lemma 6.2]), there exists a K¨ahler form γ on X in the sense oforbifolds such that(4.7) γ | X \ S p ∈ Sing( X ) V (2) p = γ, γ | V (2) p = i∂ ¯ ∂ k z k ( ∀ p ∈ Sing( X )) . In particular, k z k ∈ C ω ( V (2) p ) is a potential function of γ on every V (2) p . Since φ ǫ ( z ) = k z k near ∂V (2) p , we can glue the K¨ahler form κ ǫ on S p ∈ Sing( X ) e V (2) p andthe K¨ahler form γ on X \ S p ∈ Sing( X ) e V (2) p by setting(4.8) γ ǫ := ( κ ǫ on S p ∈ Sing( X ) e V (2) p ,γ on X \ S p ∈ Sing( X ) e V (2) p . By construction, { γ ǫ } <ǫ ≤ ǫ ( ρ ) is a family of θ -invariant K¨ahler forms on e X . Lemma 4.2.
The family of K¨ahler forms { γ ǫ } <ǫ ≤ ǫ ( ρ ) on e X satisfies the following: (1) For all p ∈ Sing( X ) , γ | V (2) p = i∂ ¯ ∂ k z k . (2) For all p ∈ Sing( X ) , γ ǫ | e V (1) p = ψ ∗ p γ EH ǫ . (3) On e X , γ ǫ converges to π ∗ γ in the C ∞ -topology. (4) There exist constants
C, C ′ > independent of ǫ (but depending on ρ ) suchthat | Ric( γ ǫ ) | γ ǫ ≤ C · ǫ on S p ∈ Sing X e V (2) p and | Ric( γ ǫ ) | γ ǫ ≤ C ′ on e X .Proof. By construction, (1), (2), (3) are obvious. Let us see (4). Since γ EH ǫ is Ricci-flat and since κ ǫ = γ EH ǫ on e V (1) p , we get Ric( κ ǫ ) = Ric( γ EH ǫ ) = 0 on e V (1) p . On e V (2) p \ e V (1) p , we get | Ric( γ ǫ ) | γ ǫ = | Ric( κ ǫ ) | κ ǫ ≤ C · ǫ by (4.2). This proves the firstestimate. Since γ ǫ = γ on X \ S p ∈ Sing( X ) e V (2) p , we get the second estimate. (cid:3) A two parameter family of K¨ahler metrics on T ∗ P . For later use we in-troduce another small parameter δ >
0. Instead of gluing in the Eguchi-Hansoninstanton in the region e V (2) − e V (1) we now do it in the region e V (2 δ ) − e V ( δ ). This iseffected by replacing the cut-off function ρ ( t ) by ρ δ ( t ) = ρ ( tδ ) in defining the K¨ahlerpotential φ ǫ for the K¨ahler metric γ ǫ such that ρ δ ( t ) = 1 for t ≤ δ and ρ δ ( t ) = 0for t ≥ δ . This gives us the family of real (1 , T ∗ P κ ǫ,δ := i∂∂φ ǫ,δ , where φ ǫ,δ ( z ) := k z k + ρ δ ( k z k ) E ( z, ǫ ) . To verify the positivity of κ ǫ,δ , we see the relation between φ ǫ and φ ǫ,δ . Since F ǫ ( δz ) = δ F ǫ/δ ( z ), we get E ( δz, ǫ ) = δ E ( z, ǫ/δ ). Since φ ǫ, ( z ) = φ ǫ ( z ) and φ ǫ,δ ( z ) = k δ · zδ k + ρ ( k z k δ ) E ( δ · zδ , ǫ ), this implies that φ ǫ,δ ( z ) = δ φ ǫ/δ ( z/δ ) . Hence if 0 < ǫ/δ ≤ ǫ ( ρ ), then κ ǫ,δ = i∂ ¯ ∂φ ǫ,δ is a positive (1 , T ∗ P .In what follows, we define φ ǫ,δ for ǫ, δ ∈ (0 ,
1] with 0 < ǫ/δ ≤ ǫ ( ρ ). Then { κ ǫ,δ } <ǫ/δ ≤ ǫ ( ρ ) , ǫ,δ ∈ (0 , is a family of K¨ahler forms on T ∗ P . Moreover, the rela-tion φ ǫ,δ ( z ) = δ φ ǫ/δ ( z/δ ) implies that the automorphism of T ∗ P induced fromthe one z z/δ on V ( ∞ ) yields an isometry of K¨ahler manifolds ( T ∗ P , κ ǫ,δ ) ∼ =( T ∗ P , δ κ ǫ/δ ) such that(4.9) ( e V (2 δ ) , κ ǫ,δ ) ∼ = ( e V (2) , δ κ ǫ/δ ) . Lemma 4.3.
There exist constants C , C > such that the following inequalityof (1 , -forms on T ∗ P holds for all ǫ, δ ∈ (0 , with < ǫ/δ ≤ ǫ ( ρ ) : C κ ǫ ≤ κ ǫ,δ ≤ C κ ǫ . Proof. (Step 1)
By Lemma 4.3 (Step 1), we get C γ EH ǫ ≤ κ ǫ ≤ C γ EH ǫ on e V (2). By(4.9) and the relation δ γ EH ǫ/δ = γ EH ǫ , this implies the inequality C γ EH ǫ ≤ κ ǫ,δ ≤ C γ EH ǫ on e V (2 δ ). Hence we get C C − κ ǫ ≤ κ ǫ,δ ≤ C C − κ ǫ on e V (2 δ ). (Step 2) Next we compare κ ǫ,δ and κ ǫ on T ∗ P \ e V (2 δ ). By definition, we have κ ǫ,δ = γ EH0 on T ∗ P \ e V (2 δ ). Let H ǫ be the automorphism of T ∗ P induced fromthe automorphism z
7→ √ ǫz of V ( ∞ ) = C / ±
1. Then H ǫ is an isomorphism from T ∗ P \ e V (2 δ/ √ ǫ ) to T ∗ P \ e V (2 δ ) inducing the isometries(4.10) ( T ∗ P \ e V (2 δ ) , γ EH ǫ ) ∼ = ( T ∗ P \ e V (2 δ/ √ ǫ ) , ǫγ EH ) , (4.11) ( T ∗ P \ e V (2 δ ) , γ EH0 ) ∼ = ( T ∗ P \ e V (2 δ/ √ ǫ ) , ǫγ EH0 ) . Since ǫ/δ ≤ ǫ ( ρ ) and hence δ/ √ ǫ > / p ǫ ( ρ ), we have the inclusion T ∗ P \ e V ( δ √ ǫ ) ⊂ T ∗ P \ e V (2 / p ǫ ( ρ )). By (4.4), there exist constants C ′ , C ′ > C ′ γ EH ≤ γ EH0 ≤ C ′ γ EH on T ∗ P \ e V (2 / p ǫ ( ρ )). This, together with (4.10), (4.11), yieldsthe inequality C ′ γ EH ǫ ≤ γ EH0 ≤ C ′ γ EH ǫ on T ∗ P \ e V (2 δ ) for all ǫ, δ ∈ (0 ,
1] with0 < ǫ/δ ≤ ǫ ( ρ ). Since κ ǫ,δ = γ EH0 on T ∗ P \ e V (2 δ ), we get C ′ γ EH ǫ ≤ κ ǫ,δ ≤ C ′ γ EH ǫ on T ∗ P \ e V (2 δ ). By Lemma 4.1, this implies the inequality C ′′ κ ǫ ≤ κ ǫ,δ ≤ C ′′ κ ǫ on T ∗ P \ e V (2 δ ), where C ′′ , C ′′ > ǫ, δ ∈ (0 ,
1] with0 < ǫ/δ ≤ ǫ ( ρ ). This completes the proof. (cid:3) A two parameter family of K¨ahler metrics on e X . Modifying the construction(4.8), we introduce a two parameter family of θ -invariant K¨ahler forms on e X by(4.12) γ ǫ,δ := ( κ ǫ,δ on S p ∈ Sing( X ) e V (2) p ,γ on X \ S p ∈ Sing( X ) e V (2) p for ǫ, δ ∈ (0 ,
1] with 0 < ǫ/δ ≤ ǫ ( ρ ). Lemma 4.4.
There exist constants C , C > such that the following inequalityof (1 , -forms on e X hold for all ǫ, δ ∈ (0 , with < ǫ/δ ≤ ǫ ( ρ ) : C γ ǫ ≤ γ ǫ,δ ≤ C γ ǫ . Proof. On S p ∈ Sing( X ) e V (2) p , the result follows from Lemma 4.3. On X \ S p ∈ Sing( X ) e V (2) p ,the result is obvious since γ ǫ,δ = γ ǫ = γ is independent of ǫ, δ there. (cid:3) Lemma 4.5.
There exists a constant C > such that the following estimate holdsfor all ǫ, δ ∈ (0 , with < ǫ/δ ≤ ǫ ( ρ ) : | Ric( γ ǫ,δ ) | γ ǫ,δ ≤ C ( ǫδ − + 1) . Proof.
Since γ ǫ,δ = γ on X \ S p ∈ Sing( X ) e V (2) p , it suffices to prove the estimate on S p ∈ Sing( X ) e V (2) p . Since γ ǫ,δ = i∂ ¯ ∂ k z k is a flat metric on S p ∈ Sing( X ) e V (2) p \ e V (2 δ ) p ,it suffices to prove the estimate on S p ∈ Sing( X ) e V (2 δ ) p . By (4.9), we get on each e V (2 δ ) p | Ric( γ ǫ,δ ) | γ ǫ,δ = | Ric( δ γ ǫ/δ ) | γ ǫ,δ = δ − | Ric( γ ǫ/δ ) | γ ǫ/δ ≤ δ − C ( ǫ/δ ) = Cǫδ − , NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 17 where we used Lemma 4.2 (4) to get the inequality | Ric( γ ǫ/δ ) | γ ǫ/δ ≤ C ( ǫ/δ ) on e V (2 δ ) p . This completes the proof. (cid:3) Fix a nowhere vanishing holomorphic 2-form η ∈ H ( e X, K e X ) \ { } . Since ( Π − ) ∗ ( η | V (1) p ) is a nowhere vanishing holomorphic 2-form on V (1) p \ { } ,there exists by the Hartogs extension theorem a nowhere vanishing holomorphicfunction f p ( z ) on B (1) such that( Π − ) ∗ ( η | V (1) p ) = f p ( z ) dz ∧ dz and f p ( − z ) = f p ( z ). Since γ ǫ,δ = γ ǫ on e V ( δ ) p and hence( Π − ) ∗ ( γ ǫ,δ / (cid:12)(cid:12) V ( δ ) p \{ } = ( i∂ ¯ ∂F ǫ ) /
2! = ( i ) dz ∧ d ¯ z ∧ dz ∧ d ¯ z , we get the equality of functions on e V ( δ ) p (4.13) η ∧ ηγ ǫ,δ / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e V ( δ ) p = π ∗ | f p ( z ) | . In particular, we have the following:(i) On each e V ( ǫ ) p , the volume form of γ ǫ,δ is independent of ǫ ∈ (0 , ǫ ( ρ )].(ii) f p (0) is independent of δ ∈ (0 ,
1] and the choice of the cut-off function ρ .Since γ ǫ,δ converges to γ outside S p ∈ Sing( X ) e V ( δ ) p , we get the continuity(4.14) lim ǫ → Vol( e X, γ ǫ,δ ) = Vol(
X, γ ) . Ricci-flat K¨ahler form on the blowing-down of e X θ . Recall that π : ( e X, e X θ ) → ( X, Sing X )is the blowing-down of the disjoint union of ( − e X θ = ∐ p ∈ Sing X E p . Then p = π ( E p ). Under the identification ψ p : ( U p , E p ) ∼ = ( e V (1) p , E ), π : e X → X isidentified with the blowing-down Π : T ∗ P → C / {± } on each V (1) p .By [26], there exists a Ricci-flat orbifold K¨ahler form ω η on X such that π ∗ ω η /
2! = η ∧ η. By (4.13), we have π ∗ ω η /γ ǫ,δ (cid:12)(cid:12) e V ( δ ) p = Π ∗ | f p ( z ) | . Since the right hand side is independent of ǫ ∈ (0 , ǫ → ω η /γ (cid:12)(cid:12) e V ( δ ) p = | f p | . Hence we get the following relation by regarding η as a nowhere vanishing holo-morphic 2-form on both e X and Xη ∧ ηγ ǫ,δ / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E p = | f p (0) | = η ∧ ηγ ( p ) . Behavior of some geometric quantities under the degeneration
In this section, we study the behavior of the second Chern form, the Bott-Chernterm, and the analytic torsion of the fixed curves when γ ǫ,δ converges to the orbifoldmetric γ .5.1. Behavior of the second Chern form as ǫ → .Proposition 5.1. For any δ ∈ (0 , , one has lim ǫ → π ∗ c ( e X, γ ǫ,δ ) = c ( X, γ ) + 32 X p ∈ Sing( X ) δ p as currents on X , where δ p is the Dirac δ -current supported at p . In particular, Z Y c ( Y, γ ) = 132 (16 − k ) . Proof.
Let h ∈ C ∞ ( X ). By the definition of the K¨ahler form γ ǫ,δ , we have(5.1) Z e X π ∗ h · c ( e X, γ ǫ,δ ) = Z X \ S p ∈ Sing( X ) e V ( δ ) p h · c ( X, γ ǫ,δ ) + X p ∈ Sing( X ) h ( p ) Z e V ( δ ) p c ( e X, γ ǫ,δ )+ X p ∈ Sing( X ) Z e V ( δ ) p π ∗ { h − h ( p ) } · c ( e X, γ ǫ,δ ) . For a >
0, let T a ( z ) := az be the homothety of C and let e T a : T ∗ P → T ∗ P be the biholomorphic map induced by T a . Then e T ǫ induces an isometry of K¨ahlermanifolds e T ǫ : ( e V ( ǫ − ) , ǫ γ EH ) ∼ = ( e V (1) , γ EH ǫ ) . Under the identification T ∗ P \ E ∼ = V ( ∞ ) \ { } , we have the following estimates (cid:13)(cid:13) γ EH ( z ) − i∂ ¯ ∂ k z k (cid:13)(cid:13) ≤ C (1 + k z k ) − , k c ( T ∗ P , γ EH )( z ) k ≤ C (1 + k z k ) − for k z k ≫ C > γ EH .Since there is a constant C ′ > (cid:12)(cid:12) h | V ( δ ) p ( z ) − h ( p ) (cid:12)(cid:12) ≤ C ′ k z k / (1 + k z k ) on V ( δ ) p , we get(5.2) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z e V ( δ ) p { π ∗ h − h ( p ) } · c ( e X, γ ǫ,δ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z e V ( δ ) p π ∗ { h | V ( δ ) p − h ( p ) } · c ( T ∗ P , γ EH ǫ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z e V ( δ √ ǫ − ) e T ∗ ǫ π ∗ { h | V ( δ ) p − h ( p ) } · c ( T ∗ P , γ EH ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z V ( δ √ ǫ − ) C ′ √ ǫ k z k √ ǫ k z k · C k z k ( γ EH ) ≤ C ′′ √ ǫ → ǫ → , NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 19 where C ′′ > ǫ → Z e X π ∗ h · c ( e X, γ ǫ,δ ) = Z X \ S p ∈ Sing( X ) V ( δ ) p h · c ( X, γ )+ X p ∈ Sing( X ) h ( p ) Z T ∗ P c ( T ∗ P , γ EH )= Z X h · c ( X, γ ) + X p ∈ Sing( X ) h ( p ) Z T ∗ P c ( T ∗ P , γ EH ) , where we used the vanishing of c ( X, γ ) on V ( δ ) p to get the second equality.Setting h = 1 in (5.3) and comparing it with the formula [27, p.396 l.5], we get(5.4) Z T ∗ P c ( T ∗ P , γ EH ) = χ ( P ) − | Z | = 32 . The first assertion follows from (5.3) and (5.4).Since Y ) = k , we get by the first assertion2 Z Y c ( Y, γ ) = Z X c ( X, γ ) = Z e X c ( e X, γ ǫ,δ ) − X p ∈ Sing( X ) Z X δ p = 24 − k. This proves the second assertion. (cid:3)
Behavior of the Bott-Chern terms as ǫ → .Proposition 5.2. For any δ ∈ (0 , , one has lim ǫ → Z e X log ( η ∧ ηγ ǫ,δ / · Vol( e X, γ ǫ,δ ) k η k L ) c ( e X, γ ǫ,δ )= Z X log (cid:26) η ∧ ηγ / · Vol(
X, γ ) k η k L (cid:27) c ( X, γ ) + 32 X p ∈ Sing( X ) log (cid:26) | f p (0) | Vol(
X, γ ) k η k L (cid:27) . Proof.
Since γ ǫ,δ converges to γ outside S p ∈ Sing( X ) e V ( δ ) p and since Vol( e X, γ ǫ,δ )converges to Vol(
X, γ ) as ǫ →
0, we get the convergence Z e X \ S p ∈ Sing( X ) e V ( δ ) p + X p ∈ Sing( X ) Z e V ( δ ) p log ( η ∧ ηγ ǫ,δ / · Vol( e X, γ ǫ,δ ) k η k L ) c ( e X, γ ǫ,δ ) → Z e X \ S p ∈ Sing( X ) e V ( δ ) p log (cid:26) η ∧ ηγ / · Vol(
X, γ ) k η k L (cid:27) c ( e X, γ )+ lim ǫ → X p ∈ Sing( X ) Z e V ( δ ) p ( log π ∗ | f p ( z ) | c ( e X, γ ǫ,δ ) + log Vol( e X, γ ǫ,δ ) k η k L c ( e X, γ ǫ,δ ) ) = Z X \ S p ∈ Sing( X ) V ( δ ) p log (cid:26) η ∧ ηγ / · Vol(
X, γ ) k η k L (cid:27) c ( X, γ )+ 32 X p ∈ Sing( X ) log (cid:18) | f p (0) | Vol(
X, γ ) k η k L (cid:19) as ǫ →
0, where the last equality follows from Proposition 5.1. Since c ( X, γ ) = 0on S p ∈ Sing( X ) V ( δ ) p , we get the result. (cid:3) Corollary 5.3.
For any δ ∈ (0 , , one has lim δ → lim ǫ → τ ( e X, γ ǫ,δ ) = Y p ∈ Sing( X ) (cid:26) | f p (0) | Vol(
X, γ ) k η k L (cid:27) − × exp (cid:18) − Z X log (cid:26) η ∧ ηγ / · Vol(
X, γ ) k η k L (cid:27) c ( X, γ ) (cid:19) . In particular, the limit lim δ → lim ǫ → τ ( e X, γ ǫ,δ ) is independent of the choice of ρ .Proof. By Theorem 3.2 and Proposition 5.2, we get the desired equality. Theindependence of the double limit lim δ → lim ǫ → τ ( e X, γ ǫ,δ ) from ρ is obvious, becausethe right hand side is independent of the choice of ρ . (cid:3) Define the Fubini-Study form on E p by ω FS ( E p ) := Π ∗ (cid:18) i π ∂ ¯ ∂ log k z k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) E p Then for p ∈ Sing( X ), we have γ ǫ,δ | E p = ǫ ω FS ( E p )and an isomorphism of K¨ahler manifolds ( E p , ω FS ( E p )) ∼ = ( P , ω FS ). Proposition 5.4.
For any δ ∈ (0 , , one has lim ǫ → A M ( e X, θ, γ ǫ,δ ) = Y p ∈ Sing( X ) (cid:26) | f p (0) | Vol(
X, γ ) k η k L (cid:27) . Proof.
Since γ ǫ,δ | E p = ǫ ω FS ( E p ) and since ω FS ( E p ) is K¨ahler-Einstein, we get c ( e X θ , γ ǫ,δ | e X θ ) | E p = χ ( P ) ω FS ( E p ) = 2 ω FS ( E p ) . Since ( η ∧ η ) / ( γ ǫ,δ / | E p = | f p (0) | by (4.13), we get A M ( e X, θ, γ ǫ,δ ) = exp " Z e X θ log ( η ∧ ηγ ǫ,δ / · Vol( e X, γ ǫ,δ ) k η k L )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e X θ c ( e X θ , γ ǫ,δ | e X θ ) = exp X p ∈ Sing( X ) Z E p log ( | f p (0) | Vol( e X, γ ǫ,δ ) k η k L )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E p ω FS ( E p ) = exp X p ∈ Sing( X ) log ( | f p (0) | Vol( e X, γ ǫ,δ ) k η k L ) → Y p ∈ Sing( X ) (cid:26) | f p (0) | Vol(
X, γ ) k η k L (cid:27) as ǫ →
0, where we used (4.14) to get the last limit. This completes the proof. (cid:3)
Behavior of the analytic torsion of the exceptional divisors.Proposition 5.5.
For any δ ∈ (0 , and p ∈ Sing( X ) , the following equality holdsfor all ǫ ∈ (0 , δ ǫ ( ρ )] Vol( E p , γ ǫ,δ | E p ) τ ( E p , γ ǫ,δ | E p )Vol( P , ω FS ) τ ( P , ω FS ) = ǫ / . NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 21
Proof.
We recall a formula of Bost [9, Prop. 4.4]. Let (
Z, g ) be a compact K¨ahlermanifold of dimension d and let λ > (cid:18) τ ( Z, λg ) τ ( Z, g ) (cid:19) = − d X i =0 ( − i ( d − i ) h ,i ( Z ) + Z Z Td ′ ( T Z ) ! log λ, where the characteristic class Td ′ ( E ) is defined as follows (cf. [9, Prop. 4.4]). If ξ i ( i = 1 , . . . , r = rk( E )) are the Chern roots of a vector bundle E , thenTd ′ ( E ) := Td( E ) · r X i =1 (cid:18) ξ i − e − ξ i − e − ξ i (cid:19) . Since Td ′ ( x ) = x − e − x (cid:18) x − e − x − e − x (cid:19) = 12 + 16 x + O ( x )and hence R P Td ′ ( T P ) = 1 /
3, we get by (5.5) applied to (
Z, g ) = ( P , ω FS )(5.6) τ ( E p , γ ǫ,δ | E p ) /τ ( P , ω FS ) = τ ( P , ǫ ω FS ) /τ ( P , ω FS ) = ǫ − / . Since(5.7) Vol( E p , γ ǫ,δ | E p ) / Vol( P , ω FS ) = Vol( P , ǫ ω FS ) / Vol( P , ω FS ) = ǫ, the result follows from (5.6) and (5.7). (cid:3) Spectrum and heat kernels under the degeneration
In this section, we prove a uniform lower bound of the k -th eigenvalue of theLaplacian and also a certain uniform exponential decay of the heat kernel for thedegenerating family of metrics γ ǫ,δ .6.1. Uniformity of Sobolev inequality.
In order to study the limit of the ana-lytic torsions τ ( e X, γ ǫ,δ ) and τ Z ( e X, γ ǫ,δ )( θ ) in the next section, we need to establisha uniform Sobolev inequality. First, we consider our model space ( T ∗ P , γ EH ǫ ), theEguchi-Hanson instanton. Here γ EH ǫ is the Ricci-flat K¨ahler metric constructedin Section 5.1 on e V ( ∞ ) = T ∗ P . Note that, for 0 < ǫ ≤
1, under the iden-tification Φ : ( R − B ( ρ )) / {± } ≃ e V ( ∞ ) − K outside a compact neighborhood K = e V ( ρ ) ⊂ e V ( ∞ ) of the zero section of T ∗ P induced by the identification( C − B ( ρ )) / {± } = V ( ∞ ) − V ( ρ ) = e V ( ∞ ) − e V ( ρ ), one hasΦ ∗ ( γ EH ǫ ) ij = δ ij + O ( r − )uniformly in ǫ by (4.4). Lemma 6.1.
There is a constant C such that for all < ǫ ≤ the following holds. (1) For all f ∈ C ∞ ( e V ( ∞ )) , k f k L ( e V ( ∞ ) ,γ EH ǫ ) ≤ C k df k L ( e V ( ∞ ) ,γ EH ǫ ) . (2) Similarly, for all α ∈ A , ( e V ( ∞ )) , k α k L ( e V ( ∞ ) ,γ EH ǫ ) ≤ C k dα k L ( e V ( ∞ ) ,γ EH ǫ ) . (3) For all α ∈ A , ( e V ( ∞ )) , k α k L ( e V ( ∞ ) ,γ EH ǫ ) ≤ C (cid:16) k ∂α k L ( e V ( ∞ ) ,γ EH ǫ ) + k ∂ ∗ α k L ( e V ( ∞ ) ,γ EH ǫ ) (cid:17) . Here all norms are defined with respect to the metric γ EH ǫ .Proof. Since ( e V ( ∞ ) , γ EH ǫ ) ∼ = ( e V ( ∞ ) , ǫγ EH ) by (4.1) and since the inequalities (1),(2), (3) above are invariant under the scaling of metrics γ EH ǫγ EH , it suffices toprove (1), (2), (3) for γ EH . In the rest to proof, all norms are defined with respect to γ EH . Identifying a function in C ∞ ( e V ( ∞ ) − K ) with the corresponding ± R with compact support via Φ, we deduce from the Sobolev inequalityfor R that k f k L ( e V ( ∞ )) ≤ C k df k L ( e V ( ∞ )) , ∀ f ∈ C ∞ ( e V ( ∞ ) − K ) , where C is the Sobolev constant for R . By an argument using partition of unity,there is a constant C K > k f k L ( e V ( ∞ )) ≤ C K ( k df k L ( e V ( ∞ )) + k f k L ( K ) ) , ∀ f ∈ C ∞ ( e V ( ∞ )) . Assume that there is no constant
D > k f k L ( K ) ≤ D k df k L ( e V ( ∞ )) , ∀ f ∈ C ∞ ( e V ( ∞ )) . Then for any n ∈ N , there is a function f n ∈ C ∞ ( e V ( ∞ )) such that k f n k L ( K ) = 1 , k df n k L ( e V ( ∞ )) ≤ n . Therefore, we have k f n k L ( e V ( ∞ )) ≤ C K (1 + 1 /n ) ≤ C K . Passing to a subsequence if necessary, it follows that the sequence f n has a weaklimit f ∞ ∈ L ( e V ( ∞ )) with df ∞ = 0 as currents on e V ( ∞ ). This implies that in L ( e V ( ∞ )), f ∞ = 0. On the other hand, let K ′ be a sufficiently big compact subsetof e V ( ∞ ), whose open subset contains K . Now, for any compact subset K ′ ⊂ e V ( ∞ ),there is a constant C K ′ > k f n k L ( K ′ ) ≤ Vol( K ′ ) / k f n k / L ( K ′ ) ≤ C K ′ = p C K Vol( K ′ ) . Hence, by the Rellich lemma, we may assume (by passing to a subsequence ifnecessary again) that f n converges to f ∞ strongly in L ( K ′ ). Since K ⊂ K ′ andhence the convergence f n → f ∞ in L ( K ) is strong, we see that k f ∞ k L ( K ) =lim n →∞ k f n k L ( K ) = 1. This is a contradiction. Hence there exists a constant Dsuch that k f k L ( K ) ≤ D k df k L ( e V ( ∞ )) . By setting C = CK (1 + D ), we have k f k L ( e V ( ∞ )) ≤ C k df k L ( e V ( ∞ )) . This proves (1).(2) is an immediate consequence of (1) and the isomorphism C ∞ ( e V ( ∞ )) ∋ f f η ∈ A , ( e V ( ∞ )), which commutes with the operations involved. To see (3), let α ∈ A , ( e V ( ∞ )). Then, by (1) k α k L ( e V ( ∞ )) = Z e V ( ∞ ) | α | dx ! / ≤ C Z e V ( ∞ ) | d | α | | dx. Using Kato’s inequality, we have Z e V ( ∞ ) | d | α | | dx ≤ Z e V ( ∞ ) |∇ α | dx. NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 23
Now the Bochner formula [34, (1.4.63)] gives ( ∂∂ ∗ + ∂ ∗ ∂ ) α = ∇ ∗ ∇ α since ( e V ( ∞ ) , γ EH ǫ )is Ricci flat. Our result follows. (cid:3) Lemma 6.2.
There is a constant C such that for all ǫ, δ ∈ (0 , with ǫδ − ≤ ǫ ( ρ ) ,and all α ∈ A ,q ( e V ( ∞ )) , ≤ q ≤ , k α k L ( e V ( ∞ ) ,κ ǫ,δ ) ≤ C (cid:16) k ∂α k L ( e V ( ∞ ) ,κ ǫ,δ ) + k ∂ ∗ α k L ( e V ( ∞ ) ,κ ǫ,δ ) (cid:17) , where the norms and ¯ ∂ ∗ are defined with respect to the metric κ ǫ,δ .Proof. By Lemmas 4.1 and 4.3, there exist constants C , C > C γ EH ǫ ≤ κ ǫ,δ ≤ C γ EH ǫ . for all ǫ, δ ∈ (0 ,
1] with ǫδ − ≤ ǫ ( ρ ). Hence there is a constant C > C − k α k L ( e V ( ∞ ) ,γ EH ǫ ) ≤ k α k L ( e V ( ∞ ) ,κ ǫ,δ ) ≤ C k α k L ( e V ( ∞ ) ,γ EH ǫ ) , (6.3) C − k ∂α k L ( e V ( ∞ ) ,γ EH ǫ ) ≤ k ∂α k L ( e V ( ∞ ) ,κ ǫ,δ ) ≤ C k ∂α k L ( e V ( ∞ ) ,γ EH ǫ ) for all ǫ, δ ∈ (0 ,
1] with ǫδ − ≤ ǫ ( ρ ) and α ∈ A ,q ( e V ( ∞ )).Let Λ ǫ.δ (resp. Λ ǫ ) be the Lefschetz operator defined as the adjoint of themultiplication by κ ǫ,δ (resp. γ EH ǫ ). Since ∂ ∗ = ± i Λ ǫ,δ ∂ for (0 , q )-forms by theK¨ahler identity, there exists by (6.1) a constant C > C − k ∂ ∗ α k L ( e V ( ∞ ) ,γ EH ǫ ) ≤ k ∂ ∗ α k L ( e V ( ∞ ) ,κ ǫ,δ ) ≤ C k ∂ ∗ α k L ( e V ( ∞ ) ,γ EH ǫ ) . By Lemma 6.1 (3) and (6.2), (6.3), (6.4), we get the result. (cid:3)
For the minimal resolution e X and the family of K¨ahler metrics γ ǫ,δ constructedin Section 5.2 using the Eguchi-Hanson instanton, we have Proposition 6.3.
There is a constant C such that for all ǫ, δ ∈ (0 , with ǫδ − ≤ ǫ ( ρ ) , and all α ∈ A ,q ( e X ) , ≤ q ≤ , k α k L ( e X,γ ǫ,δ ) ≤ C (cid:16) k ∂α k L ( e X,γ ǫ,δ ) + k ∂ ∗ α k L ( e X,γ ǫ,δ ) + k α k L ( e X,γ ǫ,δ ) (cid:17) , where the norms are defined with respect to the metric γ ǫ,δ .Proof. Since γ ǫ,δ = κ ǫ,δ on S p ∈ Sing( X ) e V ( δ ) p , the result follows from Lemma 6.2and an easy partition of unity argument. (cid:3) A uniform lower bound of spectrum.
Let (cid:3) qǫ,δ = ( ¯ ∂ + ¯ ∂ ∗ ) (resp. (cid:3) q ) bethe Hodeg-Kodaira Laplacian of ( e X, γ ǫ,δ ) (resp. (
X, γ )) acting on (0 , q )-forms. Let λ qǫ,δ ( k ) (resp. λ q ( k )) be the k -th non-zero eigenvalue of the Laplacian (cid:3) qǫ,δ (resp. (cid:3) q ). Then the non-zero eigenvalues of (cid:3) qǫ,δ are given by0 < λ qǫ,δ (1) ≤ λ qǫ,δ (2) ≤ · · · ≤ λ qǫ,δ ( k ) ≤ λ qǫ,δ ( k + 1) ≤ · · · and the set of corresponding eigenforms { ϕ qk,ǫ,δ } k ∈ N . We set λ qǫ,δ (0) = 0 and listthe corresponding eigenforms ϕ q ,ǫ,δ (here we abuse the notation as there would bedim H ( ˜ X, Ω q ˜ X ) many of them) so that { ϕ qk,ǫ,δ } ∞ k =0 forms a complete orthonormal basis of L ,qǫ,δ ( e X ), the L -completion of A ,q ( e X ) with respect to the norm associatedto γ ǫ,δ . Since K qǫ,δ ( t, x, y ) = ∞ X k =0 e − tλ qǫ,δ ( k ) ϕ qk,ǫ,δ ( x ) ⊗ ϕ qk,ǫ,δ ( y ) ∗ , we get(6.5) (cid:12)(cid:12)(cid:12) K qǫ,δ ( t, x, y ) (cid:12)(cid:12)(cid:12) ≤ ∞ X k =0 e − tλ qǫ,δ ( k ) | ϕ qk,ǫ,δ ( x ) | · | ϕ qk,ǫ,δ ( y ) |≤ { ∞ X k =0 e − tλ qǫ,δ ( k ) | ϕ qk,ǫ,δ ( x ) | } / { ∞ X k =0 e − tλ qǫ,δ ( k ) | ϕ qk,ǫ,δ ( y ) | } / = q tr K qǫ,δ ( t, x, x ) q tr K qǫ,δ ( t, y, y ) . Proposition 6.4. If q = 0 , , then there are constants A, C > such that for all ǫ, δ ∈ (0 , with ǫδ − ≤ ǫ ( ρ ) , and x, y ∈ e X , t > , the following inequality holds: (6.6) 0 < (cid:12)(cid:12)(cid:12) K qǫ,δ ( t, x, y ) (cid:12)(cid:12)(cid:12) ≤ Ae C ( ǫδ − +1) ( t − + 1) . Moreover, for all ( ǫ, δ ) ∈ (0 , with ǫδ − ≤ ǫ ( ρ ) and for all t > , q ≥ , thefollowing inequality holds: (6.7) Tr e − t (cid:3) qǫ,δ ≤ Vol( e X, γ ǫ,δ ) Ae C ( ǫδ − +1) ( t − + 1) . Proof. (Case 1)
Let q = 0. By Proposition 6.3, the Sobolev constant is uniformfor ǫ, δ ∈ (0 ,
1] with ǫδ − ≤ ǫ ( ρ ). By [10, Thms. 2.1 and 2.16], there are constants A > B ≥ ǫ, δ ∈ (0 ,
1] with ǫδ − ≤ ǫ ( ρ ), and x, y ∈ e X , t > < K qǫ,δ ( t, x, y ) ≤ A e Bt t − . Let q = 2. By Lemma 4.5, the Lichnerowicz formula and [24, p.32 l.4-l.5], we have(6.9) | K qǫ,δ ( t, x, y ) | ≤ e t | Ric γ ǫ,δ | ∞ | K ǫ,δ ( t, x, y ) | ≤ e C ( ǫδ − +1) t Ae Bt ( t − + 1) . For t ≤
1, we get (6.6) by (6.8), (6.9). For t ≥
1, since tr K qǫ,δ ( t, x, x ) is a decreasingfunction in t , we deduce (6.6) from (6.5), (6.8), (6.9) and the inequality (cid:12)(cid:12)(cid:12) K qǫ,δ ( t, x, y ) (cid:12)(cid:12)(cid:12) ≤ q tr K qǫ,δ (1 , x, x ) q tr K qǫ,δ (1 , y, y ) ≤ e C ( ǫδ − +1) Ae B . Since Tr e − t (cid:3) qǫ,δ = R e X tr K qǫ,δ ( t, x, x ) dx , we get (6.7) from (6.6). (Case 2) Let q = 1. Since P q ( − q Tr e − t (cid:3) qǫ,δ = 0 for all t >
0, (6.7) for q = 1follows from (6.7) for q = 0 ,
2. This completes the proof. (cid:3)
Write λ qǫ ( k ) for λ qǫ, ( k ). Lemma 6.5.
There is a constant λ > such that for all ǫ ∈ (0 , ǫ ( ρ )] and q ≥ , (6.10) λ qǫ (1) ≥ λ > . Proof.
Since dim e X = 2 and hence λ ǫ (1) = λ ǫ (1) or λ ǫ (1) = λ ǫ (1), it suffices toprove (6.10) for q = 0 ,
2. Assume that there is a sequence { ǫ n } such that ǫ n → λ qǫ n (1) → n → ∞ for q = 0 or 2. By the same argument as in [42, p.434–p.436] using the uniformity of the Sobolev constant (cf. Proposition 6.3), there isa holomorphic q -form ψ on X \ Sing X , which is possibly meromorphic on e X , withthe following properties: NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 25 (i) The complex conjugation ϕ q ,ǫ n converges to ψ on every compact subset of X \ Sing( X ) as n → ∞ .(ii) k ψ k L = 1 and π ∗ ψ ⊥ H ( e X, Ω q e X ) with respect to the degenerate K¨ahlermetric π ∗ γ on e X .Since Sing X consists of isolated orbifold points, it follows from the Riemann exten-sion theorem that ψ extends to a holomorphic q -form on X in the sense of orbifolds.When q = 0, ψ is a constant. When q = 2, since X has canonical singularities, π ∗ ψ is a holomorphic 2-form on e X . In both cases, the condition π ∗ ψ ⊥ H ( e X, Ω q e X )implies ψ = 0, which contradicts the other condition k ψ k L = 1. This proves theresult. (cid:3) Lemma 6.6.
There is a constant λ ′ > such that for all ǫ, δ ∈ (0 , with ǫδ − ≤ ǫ ( ρ )] and q ≥ , λ qǫ,δ (1) ≥ λ ′ > . Proof.
Firstly we prove the inequality when q = 1. Since e X is a K (cid:3) ǫ,δ = 0, we get by (6.10)(6.11) λ k α k L ( e X,γ ǫ ) ≤ k ¯ ∂α k L ( e X,γ ǫ ) + k ¯ ∂ ∗ α k L ( e X,γ ǫ ) = k ∂α k L ( e X,γ ǫ ) for all α ∈ A , ( e X ), where we used the coincidence of the ¯ ∂ -Laplacian and the ∂ -Laplacian for K¨ahler manifolds to get the equality in (6.11). By Lemma 4.4, thereexist constants C > α ∈ A , ( e X ), C − k α k L ( e X,γ ǫ ) ≤ k α k L ( e X,γ ǫ,δ ) ≤ C k α k L ( e X,γ ǫ ) ,C − k ∂α k L ( e X,γ ǫ ) ≤ k ∂α k L ( e X,γ ǫ,δ ) ≤ C k ∂α k L ( e X,γ ǫ ) . Combining these inequalities and (6.11), we get for all α ∈ A , ( e X )(6.12) C − λ k α k L ( e X,γ ǫ,δ ) ≤ C k ∂α k L ( e X,γ ǫ,δ ) = C (cid:16) k ¯ ∂α k L ( e X,γ ǫ,δ ) + k ¯ ∂ ∗ α k L ( e X,γ ǫ,δ ) (cid:17) . The result for q = 1 follows from (6.12). Since ¯ ∂ϕ ǫ,δ (1) and ¯ ∂ ∗ ϕ ǫ,δ (1) are non-zero eigenforms of (cid:3) ǫ,δ , we get λ ′ ≤ λ ǫ,δ (1) ≤ λ ǫ,δ (1) and λ ′ ≤ λ ǫ,δ (1) ≤ λ ǫ,δ (1). (cid:3) Theorem 6.7.
There are constants Λ , C > such that for all k ∈ N , ǫ, δ ∈ (0 , with ǫδ − ≤ ǫ ( ρ ) and q ≥ , λ qǫ,δ ( k ) ≥ Λ e − C ( ǫδ − +1) k / . Proof.
By Proposition 6.4, we get for all ǫ, δ ∈ (0 ,
1] with ǫδ − ≤ ǫ ( ρ ) and t ∈ (0 , k X i =1 e − tλ qǫ,δ ( i ) ≤ h ,q ( e X ) + ∞ X i =1 e − tλ qǫ,δ ( i ) = Tr e − t (cid:3) qǫ,δ ≤ A ′ e C ( ǫδ − +1) t − , where A ′ is a constant such that A Vol( e X, γ ǫ,δ ) ≤ A ′ . Since λ ′ /λ qǫ,δ ( k ) ≤ t := λ ′ /λ qǫ,δ ( k ) in this inequality and using λ qǫ,δ ( i ) /λ qǫ,δ ( k ) ≤ i ≤ k , we get k e − λ ′ ≤ k X i =1 e − λ ′ λqǫ,δ ( i ) λqǫ,δ ( k ) ≤ A ′ e C ( ǫδ − +1) λ ′ λ qǫ,δ ( k ) ! − . We get the result by setting Λ := ( A ′ ) − / λ ′ e − λ ′ / . (cid:3) Corollary 6.8.
Let C and Λ be the same constants as in Theorem 6.7 and set Λ( R ) := Λ e − CR and Ψ( R ) := P ∞ k =1 e − Λ( R ) k / . Then, for all ǫ, δ ∈ (0 , with ǫδ − ≤ ǫ ( ρ ) and t ≥ , the following inequality holds < Tr e − t (cid:3) qǫ,δ − h ,q ( e X ) ≤ Ψ( ǫδ − + 1) e − Λ(1+ ǫδ − ) t . Proof.
Since λ qǫ,δ ( k ) ≥ Λ( ǫδ − +1)2 ( k / +1) by Theorem 6.7, we get P ∞ k =1 e − tλ qǫ,δ ( k ) ≤ e − t Λ( ǫδ − +1) / P ∞ k =1 e − t Λ( ǫδ − +1) k / ≤ Ψ( ǫδ − + 1) e − t Λ( ǫδ − +1) / for t ≥ (cid:3) We also need an estimate for the heat kernel K qǫ,δ, ∞ ( t, x, y ) of ( e V ( ∞ ) , κ ǫ,δ ). Proposition 6.9.
There are constants A ′ , C ′ > such that for all ǫ, δ ∈ (0 , with ǫδ − ≤ ǫ ( ρ ) , x ∈ e V ( ∞ ) , t > and q ≥ , the following inequality holds: (cid:12)(cid:12)(cid:12) K qǫ,δ, ∞ ( t, x, y ) (cid:12)(cid:12)(cid:12) ≤ A ′ e C ′ ( ǫδ − +1) ( t − + 1) . Proof.
When q = 0, the result follows from Lemma 6.2 and [10, Thms. 2.1 and2.16]. Let q >
0. Since | Ric( γ ǫ,δ ) | ≤ C ( ǫδ − + 1) by Lemma 4.5, we deduce from[24, p.32 l.4-l.5] and the Lichnerowicz formula for (cid:3) qǫ,δ that0 < (cid:12)(cid:12)(cid:12) K qǫ,δ, ∞ ( t, x, y ) (cid:12)(cid:12)(cid:12) ≤ e C ( ǫδ − +1) t K ǫ,δ, ∞ ( t, x, y ) ≤ A e C ( ǫδ − +1) t t − . This proves the result for t ≤
1. Since (6.5) remains valid for K qǫ,δ, ∞ ( t, x, y ) by thefact that K qǫ,δ, ∞ ( t, x, y ) is obtained as the limit R → ∞ of the Dirichlet heat kernelof e V ( R ), the result for t ≥ (cid:3) Behavior of (equivariant) analytic torsion
In the previous sections, the additional parameter δ is pretty harmless and theresults still hold in its presence. This parameter will play more essential role in thissection. Indeed, we shall prove the following: Theorem 7.1.
There exist constants C ( k ) , C ( k ) > depending only on k = Y ) such that lim δ → lim ǫ → τ ( e X, γ ǫ,δ ) = C ( k ) · τ ( X, γ ) , lim δ → lim ǫ → ǫ k/ τ Z ( e X, γ ǫ,δ )( θ ) = C ( k ) · τ Z ( X, γ )( ι ) . Existence of limit.
By Corollary 5.3, the first limit exists and is independentof the choice of a cut-off function ρ . For the second limit we have Proposition 7.2.
For any δ ∈ (0 , , the number ǫ k/ τ Z ( e X, γ ǫ,δ )( θ )Vol( e X, γ ǫ,δ ) is independent of ǫ, δ ∈ (0 , with < ǫδ − ≤ ǫ ( ρ ) . In particular, for any δ ∈ (0 , ,the following limit exists as ǫ → : lim ǫ → ǫ k/ τ Z ( e X, γ ǫ,δ )( θ ) and the limit is independent of δ ∈ (0 , and the choice of a cut-off function ρ . NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 27
Proof. (Step 1)
Let g , g be θ -invariant K¨ahler metrics on e X . Let f Td θ ( T e X ; g , g ) (1 , be the Bott-Chern class such that − dd c f Td θ ( T e X ; g , g ) = Td θ ( T e X, g ) − Td θ ( T e X, g ) . By Bismut [4, Th. 2.5],(7.1) log τ Z ( e X, g )( θ )Vol( e X, g ) τ Z ( e X, g )( θ )Vol( e X, g ) ! = Z e X θ f Td θ ( T e X ; g , g ) . SinceTd θ ( T e X ; g , g ) (1 , = 18 c ( T e X ) | e X θ c ( T e X θ )( g , g ) − c ( T e X θ ) ( g , g )by [43, Prop. 5.3], we have the following equality of Bott-Chern classes: f Td θ ( T e X ; g , g ) (1 , = 18 ^ c ( T e X ) | e X θ c ( T e X θ )( g , g ) − ^ c ( T e X θ ) ( g , g )= 18 e c ( T e X ; g , g ) | e X θ c ( T e X θ , g ) + 18 c ( T e X, g ) | e X θ e c ( T e X θ ; g , g ) − e c ( T e X θ ; g , g ) { c ( T e X θ , g ) + c ( T e X θ , g ) } , where [21, Eq. (1.3.1.2)] is used to get the second equality. For a holomorphic linebundle L and Hermitian metrics h , h on L , we have e c ( L ; h , h ) = log( h /h )by [21, Eq. (1.2.5.1)]. (Our sign convention is different from the one in Gillet-Soul´e[21]. Our e c ( L ; h , h ) is − e c ( L ; h , h ) in [21].) Hence(7.2) f Td θ ( T e X ; g , g ) (1 , ≡
18 log (cid:18) det g det g (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) e X θ c ( e X θ , g ) + 18 c ( e X, g ) log (cid:18) g g (cid:12)(cid:12)(cid:12)(cid:12) e X θ (cid:19) −
112 log (cid:18) g g (cid:12)(cid:12)(cid:12)(cid:12) e X θ (cid:19) { c ( e X θ , g ) + c ( e X θ , g ) } mod Im ∂ + Im ¯ ∂. (Step 2) We set g = γ ǫ,δ and g = γ ǫ ( ρ ) in Step 1. Since g = γ ǫ,δ is Ricci-flaton a neighborhood of e X θ , we have c ( e X, γ ǫ,δ ) | e X θ = 0 . Since the volume form of EH instanton i∂ ¯ ∂F ǫ is the standard Euclidean volumeform ( i∂ ¯ ∂F ǫ )
2! = i dz ∧ d ¯ z ∧ dz ∧ d ¯ z and since γ ǫ,δ = i∂ ¯ ∂F ǫ on e V ( δ ) p , we get (cid:18) det γ ǫ,δ det γ ǫ ( ρ ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) e X θ = γ ǫ,δ / γ ǫ ( ρ ) / !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e X θ = 1 . If E i ∼ = P is a component of e X θ , then ( E i , γ ǫ,δ | E i ) ∼ = ( P , ǫ ω FS ) and ( E i , γ ǫ ( ρ ) | E i ) ∼ =( P , ǫ ( ρ ) ω FS ). Hence γ ǫ,δ γ ǫ ( ρ ) (cid:12)(cid:12)(cid:12)(cid:12) e X θ = ǫǫ ( ρ ) . All together, we get(7.3) Z e X θ f Td θ ( T e X ; γ ǫ,δ , γ ǫ ( ρ ) ) (1 , = − Z e X θ
112 log (cid:18) γ ǫ,δ γ ǫ ( ρ ) (cid:12)(cid:12)(cid:12)(cid:12) e X θ ! { c ( e X θ , γ ǫ,δ ) + c ( e X θ , γ ǫ ( ρ ) ) } = − log( ǫ/ǫ ( ρ ))6 Z e X θ c ( e X θ ) = − log( ǫ/ǫ ( ρ ))6 χ ( e X θ ) = − k ǫǫ ( ρ ) , where we used the fact e X θ = E ∐ · · · ∐ E k , k = X , E i ∼ = P . This, togetherwith (7.1), yields that ǫ k/ τ Z ( e X, γ ǫ,δ )( θ )Vol( e X, γ ǫ,δ ) = ǫ ( ρ ) k/ Vol( e X, γ ǫ ( ρ ) ) τ Z ( e X, γ ǫ ( ρ ) )( θ )is independent of ǫ, δ ∈ (0 ,
1] with 0 < ǫδ − ≤ ǫ ( ρ ). (Step 3) Let χ be another cut-off function to glue Eguchi-Hanson instantonto the initial K¨ahler form γ on X (cf. Sections 5.2.1 and 5.2.3). Then thereexists ǫ ( χ ) ∈ (0 ,
1) such that the function φ ′ ǫ,δ ( z ) := k z k + χ δ ( k z k ) E ( z, ǫ ) on V ( ∞ ) \ { } is a potential of a K¨ahler form on T ∗ P = e V ( ∞ ) for any ǫ, δ ∈ (0 , < ǫδ − ≤ ǫ ( χ ). Let γ ′ ǫ,δ be the families of K¨ahler forms on e X constructed inthe same way as in (4.12) using κ ′ ǫ,δ := i∂ ¯ ∂φ ′ ǫ,δ instead of κ ǫ,δ . By Step 2, we get ǫ k/ τ Z ( e X, γ ′ ǫ,δ )( θ )Vol( e X, γ ′ ǫ,δ ) = ǫ ( χ ) k/ Vol( e X, γ ′ ǫ ( χ ) ) τ Z ( e X, γ ′ ǫ ( χ ) )( θ )for any ǫ, δ ∈ (0 ,
1] with 0 < ǫδ − ≤ ǫ ( χ ). To prove the independence of the limitlim ǫ → ǫ k/ τ Z ( e X, γ ǫ,δ )( θ )Vol( e X, γ ǫ,δ ) from the choice of ρ , we must prove(7.4) ǫ ( ρ ) k/ Vol( e X, γ ǫ ( χ ) ) τ Z ( e X, γ ǫ ( χ ) )( θ ) = ǫ ( χ ) k/ Vol( e X, γ ′ ǫ ( χ ) ) τ Z ( e X, γ ′ ǫ ( χ ) )( θ ) . We set g = γ ǫ ( ρ ) and g = γ ′ ǫ ( χ ) in (7.2). By the same computation as in (7.3), weget Z e X θ f Td θ ( T e X ; γ ǫ ( ρ ) , γ ′ ǫ ( χ ) ) (1 , = − k ǫ ( χ ) ǫ ( ρ ) . This, together with (7.1), yields (7.4). This completes the proof. (cid:3)
A comparison of heat kernels.
Recall that K qǫ,δ ( t, x, y ) denote the heatkernel of the Hodge-Kodaira Laplacian (cid:3) ǫ,δq for the K¨ahler metric γ ǫ,δ on e X , and K q ( t, x, y ) the heat kernel of the Hodge-Kodaira Laplacian (cid:3) q for the K¨ahler metric γ on X . For 0 < r ≤ e V r := [ p ∈ Sing( X ) e V ( r ) p , e X r := e X − e V r . Define e V ∞ to be e V extended by k copies of the infinite cone ( C − B (4)) / {± } .The metric γ ǫ,δ | e V similarly extends to a K¨ahler metric γ ∞ ǫ,δ on e V ∞ . We denoteby K qǫ,δ, ∞ ( t, x, y ) the corresponding heat kernel on e V ∞ . Similarly we have thecorresponding X r , V r , V ∞ on X , with X r identified with e X r . Note that V ∞ is just k copies of the infinite cone.We first established some uniform estimates on the heat kernel K qǫ,δ ( t, x, y ), im-proving on Proposition 6.4 when the points are in specific regions. NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 29
Theorem 7.3.
There are constants
A, C depending only on the Sobolev constantand dimension such that, for all ǫ, δ ∈ (0 , with ǫδ − ≤ ǫ ( ρ ) , and ≤ q ≤ , wehave | K qǫ ( t, x, z ) | ≤ Ae C (1+ ǫδ − ) δ − e − δ t , ∀ x ∈ e X δ , z ∈ e V δ , t > . Similarly we have, ∀ x ∈ e X δ , z ∈ e V δ , t > , | dK qǫ ( t, x, z ) | ≤ Ae C (1+ ǫδ − ) δ − e − δ t , | d ∗ ǫ,δ K qǫ ( t, x, z ) | ≤ Ae C (1+ ǫδ − ) δ − e − δ t , Here d, d ∗ ǫ,δ could act either on x or z variable. Finally, for < r < δ , x ∈ e X δ , z ∈ e V δ,r = e V δ − e V r , and i ∈ N , |∇ i K qǫ,δ ( t − s, x, z ) | ≤ C ( i, δ, r ) e − δ t , for a constant C ( i, δ, r ) depending on i, δ, r . Here ∇ i denotes the i -th covariantderivative with respect to the metric γ ǫ,δ , acting on either variable.Proof. Throughout the proof we fix x ∈ e X δ , z ∈ e V δ , t > . Since the Riccicurvature of γ ǫ,δ is bounded by Lemma 4.5, the Sobolev estimate together with theMoser iteration technique combined with the finite propagation speed argument asin Cheeger-Gromov-Taylor [13] gives the uniform estimate | K qǫ ( t, x, z ) | ≤ Ae C (1+ ǫδ − ) δ − e − δ t . Indeed the finite propagation speed technique gives us the L estimate k K qǫ ( t, · , · ) k L ( B δ/ ( x ) × B δ/ ( z )) ≤ ce − δ t for some uniform constant c . Now Moser iteration as in [13][pp.16-26], togetherwith semi-group domination [24] yields the desired estimate.For the estimate on dK qǫ ( t, x, z ) , d ∗ ǫ,δ K qǫ ( t, x, z ), let η ( r ) be a smooth cut-offfunction which is identically 1 for | r | ≤ δ/ | r | ≥ δ/
4, and | η ′ | ≤ δ . We will continue to denote by η its composition with a distance function(either d ( x, · ) or d ( z, · )). Note then k ( d + d ∗ ǫ,δ ) z [ ηK qǫ ( t, · , · )] k L ( B δ/ ( x ) × B δ/ ( z )) = k ( d ) z [ ηK qǫ ( t, · , · )] k L ( B δ/ ( x ) × B δ/ ( z )) + k ( d ∗ ǫ,δ ) z [ ηK qǫ ( t, · , · )] k L ( B δ/ ( x ) × B δ/ ( z )) , from which we deduce k ( d ) z [ K qǫ ( t, · , · )] k L ( B δ/ ( x ) × B δ/ ( z )) ≤ k ( d + d ∗ ǫ,δ ) z [ ηK qǫ ( t, · , · )] k L ( B δ/ ( x ) × B δ/ ( z )) + 16 δ k K qǫ ( t, · , · ) k L ( B δ/ ( x ) × B δ/ ( z )) ≤ k ( d + d ∗ ǫ,δ ) z [ K qǫ ( t, · , · )] k L ( B δ/ ( x ) × B δ/ ( z )) + 32 δ k K qǫ ( t, · , · ) k L ( B δ/ ( x ) × B δ/ ( z )) Now the same finite propagation speed technique gives k ( d + d ∗ ǫ,δ ) z [ K qǫ ( t, · , · )] k L ( B δ/ ( x ) × B δ/ ( z )) ≤ c ′ e − δ t , which in turn gives k ( d ) z [ K qǫ ( t, · , · )] k L ( B δ/ ( x ) × B δ/ ( z )) ≤ ( c ′ + c δ ) e − δ t , The same method as above then yields | ( d ) z [ K qǫ ( t, x, z )] | ≤ Ae C (1+ ǫδ − ) δ − e − δ t . The others can be proven in exactly the same way.Finally, for 0 < r < δ , we note that the curvature tensor and its derivatives of γ ǫ,δ are bounded in e V δ,r = e V δ − e V r by a constant depending on δ, r . Moreover theinjectivity radius of γ ǫ,δ in e V δ,r is bounded away from zero by a constant dependingon δ, r . Hence, by the elliptic estimate combined with the argument as before, wehave, for x ∈ e X δ , z ∈ e V δ,r = e V δ − e V r , and i ∈ N , |∇ i K qǫ,δ ( t − s, x, z ) | ≤ C ( i, δ, r ) e − δ t , for a constant C ( i, δ, r ) depending on i, δ, r . (cid:3) Theorem 7.4.
There are constants
A, C depending only on the Sobolev constantand dimension such that, for all ǫ, δ ∈ (0 , with ǫδ − ≤ ǫ ( ρ ) , and ≤ q ≤ , wehave | K qǫ,δ ( t, x, y ) − K q ( t, x, y ) | ≤ Ae C (1+ ǫδ − ) δ − e − δ t vol( ∂ e X δ ) , ∀ x, y ∈ e X δ , t > . Furthermore, ∀ x, y ∈ e X δ , t > , we have the pointwise (although not necessarilyuniform) convergence as ǫ → , K qǫ,δ ( t, x, y ) − K q ( t, x, y ) −→ . Proof.
For 0 < r ≤
4, we apply the Duhamel principle [11, (3.9)] to K qǫ,δ ( t, x, y ) − K q ( t, x, y ) on e X r to obtain K qǫ,δ ( t, x, y ) − K q ( t, x, y ) = − Z t Z e X r h ( ∂ t + (cid:3) q ) K qǫ,δ ( t − s, x, z ) i ∧ ∗ K q ( s, z, y )+ Z t Z ∂ e X r ∗ dK qǫ,δ ( t − s, x, z ) ∧ K q ( s, z, y )+( − q +1 Z t Z ∂ e X r K qǫ,δ ( t − s, x, z ) ∧ ∗ dK q ( s, z, y )+( − q +1 Z t Z ∂ e X r ∗ K qǫ,δ ( t − s, x, z ) ∧ d ∗ K q ( s, z, y )+ Z t Z ∂ e X r d ∗ K qǫ,δ ( t − s, x, z ) ∧ ∗ K q ( s, z, y ) . Now fix x, y ∈ e X δ . First we let r = 2 δ . Then the first term on the right handside goes away and we are left with only boundary terms. By Theorem 7.3, andnoticing that similar estimates hold for the orbifold heat kernel(7.5) | K q ( t, x, z ) | ≤ Cδ − e − δ t , ∀ x ∈ e X δ , z ∈ e V δ , t > , as well as its derivatives, we deduce then that | K qǫ,δ ( t, x, y ) − K q ( t, x, y ) | ≤ Ae C (1+ ǫδ − ) δ − e − δ t vol( ∂ e X δ ) . NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 31
To prove the pointwise convergence, we let r < δ , and denote e V δ,r = e V δ − e V r .Then the Duhamel principle becomes K qǫ,δ ( t, x, y ) − K q ( t, x, y ) = − Z t Z e V δ ,r h ( ∂ t + (cid:3) q ) K qǫ,δ ( t − s, x, z ) i ∧ ∗ K q ( s, z, y )+ Z t Z ∂ e X r ∗ dK qǫ,δ ( t − s, x, z ) ∧ K q ( s, z, y )+( − q +1 Z t Z ∂ e X r K qǫ,δ ( t − s, x, z ) ∧ ∗ dK q ( s, z, y )+( − q +1 Z t Z ∂ e X r ∗ K qǫ,δ ( t − s, x, z ) ∧ d ∗ K q ( s, z, y )+ Z t Z ∂ e X r d ∗ K qǫ,δ ( t − s, x, z ) ∧ ∗ K q ( s, z, y ) . Since ( ∂ t + (cid:3) q ) K qǫ ( t − s, x, z ) = ( (cid:3) q − (cid:3) qǫ ) K qǫ ( t − s, x, z ), and γ ǫ,δ = i∂∂φ ǫ,δ , φ ǫ,δ ( z ) = k z k + ρ δ ( z ) E ( z, ǫ ) on e V δ,r , by (4.2), (4.4), (4.6), we have, for x ∈ e X δ , z ∈ e V δ,r , | ( ∂ t + (cid:3) q ) K qǫ,δ ( t − s, x, z ) | ≤ ǫC ( δ, r ) e − δ t , for a constant C ( δ, r ) depending on δ, r but not on ǫ .Combining with the uniform estimates in Theorem 7.4, we obtain, for x, y ∈ e X δ , | K qǫ,δ ( t, x, y ) − K q ( t, x, y ) | ≤ ǫtC ′ ( δ, r ) e − δ t + C ′′ ( δ ) te − δ t vol( ∂ e X r ) . Now for any η >
0, we take r sufficiently small so that C ′′ ( δ ) te − δ t vol( ∂ e X r ) < η .Then we take ǫ sufficiently small such that ǫtC ′ ( δ, r ) e − δ t < η . Hence | K qǫ,δ ( t, x, y ) − K q ( t, x, y ) | < η. This proves the pointwise convergence. (cid:3)
Remark . Since we have the Ricci curvature lower bound, the pointwise con-vergence of the heat kernels should also be a consequence of some general spectralconvergence results due to Cheeger-Colding [12] for the case q = 0, Honda [25] forthe case q = 1, and Bei [2] for q = n = 2. See also [16].Our next task is to compare the heat kernel K q ( t, x, y ) for ( X, γ ) with the heatkernel K q , ∞ ( t, x, y ) of V ∞ when x, y ∈ V δ . Theorem 7.6.
There is a constant C depending only on the Sobolev constant anddimension such that, for δ ≤ , | K q ( t, x, y ) − K q , ∞ ( t, x, y ) | ≤ Ce − t vol( ∂V ) , ∀ x, y ∈ V δ , t > . Proof.
The Duhamel principle [11, (3.9)] applied to K q ( t, x, y ) − K q , ∞ ( t, x, y ) on V gives us K q ( t, x, y ) − K q , ∞ ( t, x, y ) = Z t Z ∂V ∗ dK q ( s, x, z ) ∧ K q , ∞ ( t − s, z, y )+( − q +1 Z t Z ∂V K q ( s, x, z ) ∧ ∗ dK q , ∞ ( t − s, z, y )+( − q +1 Z t Z ∂V ∗ K q ( s, x, z ) ∧ d ∗ K q , ∞ ( t − s, z, y )+ Z t Z ∂V d ∗ K q ( s, x, z ) ∧ ∗ K q , ∞ ( t − s, z, y ) . Thus we obtain, for x, y ∈ V δ , δ ≤
1, using the estimate 7.5, except with the δ there replaced by a fixed constant, say 1 /
4, as well as a similar estimate for K q , ∞ ( t, x, y ), | K q ( t, x, y ) − K q , ∞ ( t, x, y ) | ≤ Ce − t vol( ∂V ) . (cid:3) Our final task here is to compare the heat kernel K qǫ,δ ( t, x, y ) with K qǫ,δ, ∞ ( t, x, y ),the heat kernel on e V ∞ , when x, y ∈ e V δ . Theorem 7.7.
There are constants
A, C depending only on the Sobolev constantand dimension such that, for all ǫ, δ ∈ (0 , with ǫδ − ≤ ǫ ( ρ ) , and ≤ q ≤ , wehave | K qǫ,δ ( t, x, y ) − K qǫ,δ, ∞ ( t, x, y ) | ≤ Ae C (1+ ǫδ − ) e − t vol( ∂V ) , ∀ x, y ∈ e V δ , t > . Proof.
The proof follows the same line as above. We apply the Duhamel principle to K qǫ,δ ( t, x, y ) − K qǫ,δ, ∞ ( t, x, y ) on e V and use the heat kernel estimate in Theorem 7.3 aswell as the analogous estimate for K qǫ,δ, ∞ ( t, x, y ) to obtain the desired estimate. (cid:3) Partial analytic torsion.
Recall that in Section 4.1, for a compact K¨ahlerorbifold (
Z, γ ) of diemsnion n , ζ q ( s ) = X λ ∈ σ ( (cid:3) q ) \{ } λ − s dim E ( λ ; (cid:3) q ) = 1Γ( s ) Z ∞ t s − Tr( e − t (cid:3) q P ⊥ q ) dt with P ⊥ q the orthogonal projection onto the orthogonal compliment of ker (cid:3) q , and(the logarithm of) the analytic torsionln τ ( Z, γ ) = − n X q =0 ( − q q ζ ′ q (0) = − ζ ′ T (0) , where ζ T ( s ) = 1Γ( s ) Z ∞ t s − Tr s ( N e − t (cid:3) P ⊥ ) dt. Here (cid:3) denotes the Hodge-Kodaira Laplacian on A , ∗ ( Z ), P ⊥ the orthogonal pro-jection onto the orthogonal compliment of ker (cid:3) , Tr s the supertrace on A , ∗ ( Z ),i.e., the alternating sum of the traces on each degree, and N the so called numberoperator which simply multiply a differential form by its degree. NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 33
By Lidskii theoremTr s ( N e − t (cid:3) P ⊥ ) = Z Z tr s ( N K ( t, x, x ) P ⊥ ( x, x )) dx = n X q =0 ( − q q Z Z tr( K q ( t, x, x ) P ⊥ ( x, x )) dx where K ( t, x, y ), K q ( t, x, y ) denotes the heat kernel of (cid:3) , (cid:3) q , respectively, P ⊥ ( x, x )the Schwartz kernel of P ⊥ , and tr s (abusing notation) also the pointwise supertrace.At this point it is convenient to introduce what is called “partial analytic torsion”in [14]. For a domain D ⊂ Z , we define ζ D,ZT ( s ) = 1Γ( s ) Z ∞ t s − Z D tr s ( N K ( t, x, x ) P ⊥ ( x, x )) dx dt and(7.6) ln τ ( D, Z, γ ) = − (cid:16) ζ D,ZT (cid:17) ′ (0) . Clearly(7.7) ln τ ( Z, γ ) = ln τ ( D, Z, γ ) + ln τ ( Z − D, Z, γ ) . Similarly we can define the equivariant version τ Z ( D, Z, γ )( θ ) for θ -invariantdomain D ⊂ Z . That is, we define ζ D,ZT,θ ( s ) = 1Γ( s ) Z ∞ t s − Z D tr s ( N K ( t, x, θx ) P ⊥ ( x, θx )) dx dt and(7.8) ln τ Z ( D, Z, γ )( θ ) = − (cid:16) ζ D,ZT,θ (cid:17) ′ (0) . Then the discussion applies to the equivariant version as well.7.4.
Limit of partial analytic torsion I.Theorem 7.8.
For < δ ≤ , we have lim ǫ → ln τ ( e X δ , e X, γ ǫ,δ ) = ln τ ( X δ , X, γ ) , and lim ǫ → ln τ Z ( e X δ , e X, γ ǫ,δ )( θ ) = ln τ Z ( X δ , X, γ )( ι ) . Proof. (Step 1)
Let tr s ( N K ǫ,δ ( t, x, x )) ∼ ∞ X i =0 a ǫ,δi ( x ) t i − be the pointwise small time asymptotic expansion, and write ζ e X δ , e XT ( s ) = 1Γ( s ) (cid:20)Z ∞ t s − Z e X δ tr s ( N K ǫ,δ ( t, x, x ) P ⊥ ǫ,δ ( x, x )) dx dt + Z t s − Z e X δ [tr s ( N K ǫ,δ ( t, x, x )) − X i =0 a ǫ,δi ( x ) t i − ] dx dt + X i =0 Z e X δ a ǫ,δi ( x ) s + i − dx + 1 s Z e X δ [ a ǫ,δ ( x ) − tr s ( N P ǫ,δ ( x, x ))] dx , where P ǫ,δ ( x, x ) is the Schwartz kernel of P ǫ,δ , the orthogonal projection ontoker (cid:3) ǫ,δ . We obtainln τ ( e X δ , e X, γ ǫ,δ ) = − Z ∞ t − Z e X δ tr s ( N K ǫ,δ ( t, x, x ) P ⊥ ǫ,δ ( x, x )) dx dt − Z t − Z e X δ " tr s ( N K ǫ,δ ( t, x, x )) − X i =0 a ǫ,δi ( x ) t i − dx dt − X i =0 Z e X δ a ǫ,δi ( x ) i − dx + Γ ′ (1) Z e X δ h a ǫ,δ ( x ) − tr s ( N P ǫ,δ ( x, x )) i dx and similarly for ln τ ( X δ , X, γ ). Since the asymptotic expansion depends only onthe local data, we have a ǫ,δi ( x ) = a i ( x ) on e X δ . Hence(7.9)ln τ ( e X δ , e X, γ ǫ,δ ) − ln τ ( X δ , X, γ )= − Z ∞ t − Z e X δ tr s (cid:2) N K ǫ,δ ( t, x, x ) P ⊥ ǫ,δ ( x, x ) − N K ( t, x, x ) P ⊥ ( x, x ) (cid:3) dx dt − Z t − Z e X δ [tr s ( N K ǫ,δ ( t, x, x ) − N K ( t, x, x ))] dx dt − Γ ′ (1) Z e X δ tr s ( N P ǫ,δ ( x, x ) − N P ( x, x )) dx. We estimate each term of the right hand side. (Step 2)
Let Λ > Z e X δ (cid:12)(cid:12) tr s ( N K ǫ,δ ( t, x, x ) P ⊥ ǫ,δ ( x, x )) (cid:12)(cid:12) dx ≤ X q ≥ q (Tr e − t (cid:3) ǫ,δ − h ,q ( e X )) ≤ Ψ( ǫδ − + 1) exp[ − t Λ( ǫδ − + 1)]for all ǫ, δ ∈ (0 ,
1] with ǫδ − ≤ ǫ ( ρ ) and t ≥
1, where Ψ( R ) and Λ( R ) were definedin Corollary 6.8. Hence for any ν >
0, there is T ′ = T ′ ( ν ) > ν such that for all ǫ, δ ∈ (0 ,
1] with ǫ < min { ǫ ( ρ ) δ , δ } , and T > T ′ ,(7.10) Z ∞ T t − Z e X δ (cid:12)(cid:12) tr s ( N K ǫ,δ ( t, x, x ) P ⊥ ǫ,δ ( x, x )) (cid:12)(cid:12) dx dt ≤ Ψ(2) Z ∞ T e − Λ(2) t/ dtt < ν and similarly for the same term involving K . By Theorem 7.4 and Lebesguedominated convergence theorem, there exists ǫ > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T t − Z e X δ tr s [ N ( K ǫ,δ ( t, x, x )) − K ( t, x, x ))] dx dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ν, whenever ǫ < ǫ . Similarly,(7.12) (cid:12)(cid:12)(cid:12)(cid:12)Z t − Z e X δ tr s [ N ( K ǫ,δ ( t, x, x )) − K ( t, x, x ))] dx dt (cid:12)(cid:12)(cid:12)(cid:12) < ν, whenever ǫ < ǫ .On the other hand(7.13) tr s ( N K ǫ,δ ( t, x, x ) P ⊥ ǫ,δ ( x, x )) = tr s ( N K ǫ,δ ( t, x, x )) − tr s ( N P ǫ,δ ( x, x )) NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 35 and similarly for K . Recall ker (cid:3) = ker (cid:3) ǫ,δ = C · ⊕ C · ¯ η . For x ∈ e X δ , we get(7.14) tr s [ N ( P ǫ,δ ( x, x ) − P ( x, x ))] = 2 k η k L η ∧ ¯ ηγ ǫ,δ /
2! ( x ) − η ∧ ¯ ηγ /
2! ( x ) ! = 0 , because γ ǫ,δ = γ on e X δ . It follows from (7.11), (7.13), (7.14) that(7.15) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T t − Z e X δ tr s ( N [ K ǫ,δ ( t, x, x ) P ⊥ ǫ,δ ( x, x ) − K ( t, x, x ) P ⊥ ( x, x )]) dx dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ν. Substituting (7.10), (7.11), (7.12), (7.14), (7.15) into (7.9), we get (cid:12)(cid:12)(cid:12) ln τ ( e X δ , e X, γ ǫ,δ ) − ln τ ( X δ , X, γ ) (cid:12)(cid:12)(cid:12) < ν, whenever ǫ < ǫ . Since ν > θ has no fixed points in e X δ . Indeed,ln τ Z ( e X δ , e X, γ ǫ,δ )( θ ) = − Z ∞ t − Z e X δ tr s ( N K ǫ,δ ( t, x, θx ) P ⊥ ǫ,δ ( x, θx )) dx dt − Z t − Z e X δ tr s ( N K ǫ,δ ( t, x, θx )) dx dt − Γ ′ (1) Z e X δ tr s ( N P ǫ,δ ( x, θx ) dx and(7.16)ln τ Z ( e X δ , e X, γ ǫ,δ )( θ ) − ln τ Z ( X δ , X, γ )( ι )= − Z ∞ t − Z e X δ tr s (cid:2) N K ǫ,δ ( t, x, θx ) P ⊥ ǫ,δ ( x, θx ) − N K ( t, x, θx ) P ⊥ ( x, θx ) (cid:3) dx dt − Z t − Z e X δ [tr s ( N K ǫ,δ ( t, x, θx ) − N K ( t, x, θx ))] dx dt − Γ ′ (1) Z e X δ tr s ( N P ǫ,δ ( x, θx ) − N P ( x, θx )) dx. Now we proceed as before. (cid:3)
Limit of partial analytic torsion II.
To relate ln τ ( X δ , X, γ ) to ln τ ( X, γ ),by (7.7), it suffices to show Theorem 7.9.
We have lim δ → ln τ ( V δ , X, γ ) = 0 , lim δ → ln τ Z ( V δ , X, γ )( ι ) = 0 . Remark . This is closely related to [15] where analytic torsions on orbifoldsdefined from conical singularity pointview are shown to be the same as the onesdefined from orbifold singularity pointview.
Proof.
Again the proof for both formulas work the same so we only present thefirst one. Moreover, the argument works for any orbifold singularity but we willwork with the cyclic quotient singularity of type (1 ,
1) in our situation. First of all, by the same kind of argument as above, using Theorem 7.6 and vol( V δ ) → δ →
0, one has lim δ → ln τ ( V δ , X, γ ) = lim δ → ln τ ( V δ , V ∞ , κ ) . Now the right hand side can be explicitly computed since the heat kernel is explicitlyknown. Indeed, as V ∞ is just k copies of C / Z , the (orbifold) heat kernel of( V ∞ , κ ) on the (0 , q ) forms is k (cid:0) nq (cid:1) ( n = 2 in our case) copies of K ( t, x, x ′ ) = 1(4 πt ) n/ (cid:18) e − | x − x ′| t + e − | x + x ′| t (cid:19) . In terms of the polar coordinates x = ( r, y ) , y ∈ S n − , K ( t, x, x ) = 1(4 πt ) n/ (1 + e − r /t ) . Thus Z V δ K ( t, x, x ) dx = c n δ n t − n/ + d n Z δt / ξ n − e − ξ dξ, where c n = n ω n n (4 π ) n/ , d n = ω n (4 π ) n/ , ω n = vol( S n − ).The second term has different asymptotic behaviors for t → t → ∞ . Since Z δt / ξ n − e − ξ dξ = Z ∞ ξ n − e − ξ dξ − Z ∞ δt / ξ n − e − ξ dξ, by some elementary inequality, it is a constant d ′ n = d n R ∞ ξ n − e − ξ dξ plus an ex-ponentially decaying term as t → t → ∞ , the second term is O ( t − n − ).Set ζ δ ( s ) = 1Γ( s ) Z ∞ t s − (cid:18)Z V δ K ( t, x, x ) dx (cid:19) dt = 1Γ( s ) (cid:20)Z t s − (cid:18)Z V δ K ( t, x, x ) dx (cid:19) dt + Z ∞ t s − (cid:18)Z V δ K ( t, x, x ) dx (cid:19) dt (cid:21) , where the first term is defined through analytic continuation from a region wherethe real part of s is sufficiently large, whereas the second term defined throughanalytic continuation from a region where the real part of s is sufficiently negative.Therefore1Γ( s ) Z t s − (cid:18)Z V δ K ( t, x, x ) dx (cid:19) dt = 1Γ( s ) c n δ n s − n/ d ′ n Γ( s + 1) − d n Γ( s ) Z t s − Z ∞ δt / ξ n − e − ξ dξdt, and1Γ( s ) Z ∞ t s − (cid:18)Z V δ K ( t, x, x ) dx (cid:19) dt = − s ) c n δ n s − n/ d n Γ( s ) Z t s − Z δt / ξ n − e − ξ dξdt. NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 37
Thus, ζ ′ δ (0) = − d ′ n Γ ′ (1) − d n Z t − Z ∞ δt / ξ n − e − ξ dξdt + d n Z t − Z δt / ξ n − e − ξ dξdt. By a simple change of integration we arrive at ζ ′ δ (0) = − d ′ n Γ ′ (1) − d n Z ∞ δ δξ ξ n − e − ξ dξ + d n Z δ δξ ξ n − e − ξ dξ. This has a logarithmic divergence (2 d ′ n ln 3 δ ) as δ →
0, butln τ ( V δ , V ∞ , γ ) = − kζ ′ δ (0) n X q =0 ( − q q (cid:18) nq (cid:19) = 0by combinatorial formula since n ≥ θ into the heat kernel, which will result in only the d n termssimilar to the above formulas. (cid:3) Corollary 7.11.
We have lim δ → lim ǫ → ln τ ( e X δ , e X, γ ǫ,δ ) = ln τ ( X, γ ) , lim δ → lim ǫ → ln τ Z ( e X δ , e X, γ ǫ,δ )( ι ) = ln τ Z ( X, γ )( ι ) . Proof.
Since ln τ ( X, γ ) = ln τ ( X δ , X, γ ) + ln τ ( V δ , X, γ ) and ln τ Z ( X, γ )( ι ) =ln τ Z ( X δ , X, γ )( ι ) + ln τ Z ( V δ , X, γ )( ι ) by (7.7), we get by Theorem 7.9lim δ → ln τ ( X δ , X, γ ) = ln τ ( X, γ ) , lim δ → ln τ Z ( X δ , X, γ )( ι ) = ln τ Z ( X, γ )( ι ) , which, together with Theorem 7.8, yields the result. (cid:3) Limit of partial analytic torsion III.
On the other hand, we have
Theorem 7.12.
The following equalities hold: lim δ → lim ǫ → ln τ ( e V δ , e X, γ ǫ,δ ) = k ln C EH0 ( ρ ) , lim δ → lim ǫ → ln h ǫ k/ τ Z ( e V δ , e X, γ ǫ,δ )( θ ) i = k ln C EH1 ( ρ ) , where the constants C EH0 ( ρ ) , C EH1 ( ρ ) depend only on the cut-off function ρ . At this stage, the constants C EH0 ( ρ ), C EH1 ( ρ ) may depend on ρ . The fact thatthey are independent of ρ will be postponed to the next subsection.7.6.1. An integral expression of τ ( e V δ , e X, γ ǫ,δ ) and τ Z ( e V δ , e X, γ ǫ,δ )( θ ) . For the proofof Theorem 7.12, as before, we computeln τ ( e V δ , e X, γ ǫ,δ ) = − Z ∞ t − Z e V δ tr s ( N K ǫ,δ ( t, x, x ) P ⊥ ǫ,δ ( x, x )) dx dt − Z t − Z e V δ " tr s ( N K ǫ,δ ( t, x, x )) − X i =0 a ǫ,δi ( x ) t i − dx dt − X i =0 Z e V δ a ǫ,δi ( x ) i − dx + Γ ′ (1) Z e V δ h a ǫ,δ ( x ) − tr s ( N P ǫ,δ ( x, x )) i dx andln τ Z ( e V δ , e X, γ ǫ,δ ) = − Z ∞ t − Z e V δ tr s ( N K ǫ,δ ( t, x, θ ( x )) P ⊥ ǫ,δ ( x, θ ( x ))) dx dt − Z dtt "Z e V δ tr s ( N K ǫ,δ ( t, x, θ ( x ))) dx − X i =0 t i − Z E b i ( z ) dz + Z E b ( z ) dz + Γ ′ (1) (cid:20)Z E b ( z ) dz − Z e V δ tr s ( N P ǫ,δ ( x, θ ( x ))) dx (cid:21) . We study the behavior of each term in the right hand side as ǫ → δ →
0. Forthis, we set I ( ǫ, δ ; ρ ) := − Z dtt Z e V (3 δ ) " tr s ( N K ǫ,δ, ∞ ( t, x, x )) − X i =0 a ǫ,δi ( x ) t i − dx − X i =0 Z e V (3 δ ) a ǫ,δi ( x ) i − dx + Γ ′ (1) Z e V (3 δ ) a ǫ,δ ( x ) dx,J ( ǫ, δ ; ρ ) := − Z dtt "Z e V (3 δ ) tr s ( N K ǫ,δ, ∞ ( t, x, θ ( x ))) dx − X i =0 t i − Z E b ǫi ( z ) dz + Z E b ǫ ( z )2 dz + Γ ′ (1) Z E b ǫ ( z ) dz. Since K ǫ,δ, ∞ ( t, x, y ) = ⊕ q K qǫ,δ, ∞ ( t, x, y ) is the heat kernel of ( T ∗ P , κ ǫ,δ ), I ( ǫ, δ ; ρ )and J ( ǫ, δ ; ρ ) depend only on ǫ, δ ∈ (0 ,
1] with ǫδ − ≤ ǫ ( ρ ) and the cut-off function ρ . Since e V δ is a k -copies of e V (3 δ ), we haveln τ ( e V δ , e X, γ ǫ,δ ) = − Z ∞ dtt Z e V δ tr s ( N K ǫ,δ ( t, x, x ) P ⊥ ǫ,δ ( x, x )) dx − Z dtt Z e V δ tr s { N K ǫ,δ ( t, x, x ) − N K ǫ,δ, ∞ ( t, x, x ) } dx − Γ ′ (1) Z e V δ tr s ( N P ǫ,δ ( x, x )) dx + k · I ( ǫ, δ ; ρ )and similarlyln τ Z ( e V δ , e X, γ ǫ,δ ) = − Z ∞ dtt Z e V δ tr s ( N K ǫ,δ ( t, x, θ ( x )) P ⊥ ǫ,δ ( x, θ ( x ))) dx − Z dtt Z e V δ tr s { N K ǫ,δ ( t, x, θ ( x )) − N K ǫ,δ, ∞ ( t, x, θ ( x )) } dx − Γ ′ (1) Z e V δ tr s { N P ǫ,δ ( x, θ ( x )) } dx + k · J ( ǫ, δ ; ρ ) . Limit of the first integral.
Proposition 7.13.
The following equality holds: lim δ → lim ǫ → Z ∞ dtt Z e V δ tr s ( N K ǫ,δ ( t, x, x ) P ⊥ ǫ,δ ( x, x )) dx = 0 . The same is true for the first integral in the expression of ln τ Z ( e V δ , e X, γ ǫ,δ ) . NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 39
Proof.
Let ν > T = T ( ν ) > ν such that(7.17) Z ∞ T t − Z e V δ (cid:12)(cid:12) tr s ( N K ǫ,δ ( t, x, x ) P ⊥ ǫ,δ ( x, x )) (cid:12)(cid:12) dx dt < ν for all ǫ, δ ∈ (0 ,
1] with ǫ ≤ min { ǫ ( ρ ) δ , δ } , which will be assumed throughout theproof. By Theorem 7.7,(7.18) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T t − Z e V δ tr s [ N { K ǫ,δ ( t, x, x )) − K ǫ,δ, ∞ ( t, x, x ) } ] dx dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( T ) vol( e V δ ) , where C ( T ) is a constant depending only on T . By (7.14), we get(7.19) Z e V δ tr s [ N ( P ǫ,δ ( x, x )] dx = Z e V δ η ∧ ¯ η k η k L ≤ k η ∧ ¯ η/γ k L ∞ k η k L Vol( e V δ ) . By (7.13), (7.18), (7.19), we get(7.20) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T t − Z e V δ tr s [ N { K ǫ,δ ( t, x, x ) P ⊥ ǫ,δ ( x, x ) − K ǫ,δ, ∞ ( t, x, x ) } ] dx dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ { C ( T ) + 2 k η ∧ ¯ η/γ k L ∞ k η k L log T } vol( e V δ ) . By Proposition 6.9, there is a constant
A > Z T t − Z e V δ | K ǫ,δ, ∞ ( t, x, x ) | dx dt ≤ Ae C ( ǫδ − +1) T log T · vol( e V δ )for all ǫ, δ ∈ (0 ,
1] with ǫ/δ − ≤ ǫ ( ρ ). By (7.20), (7.21), we get(7.22) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T t − Z e V δ tr s [ N K ǫ,δ ( t, x, x ) P ⊥ ǫ,δ ( x, x )] dx dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ e C ( T ) vol( e V δ ) , where e C ( T ) = C ( T ) + (2 k η ∧ ¯ η/γ k L ∞ k η k L + Ae CT ) log T . Since ν > e V δ ) goes to zero as δ →
0, theresult follows from (7.17), (7.22). (cid:3)
Limit of the second integral.
Proposition 7.14.
The following equality holds: lim δ → lim ǫ → Z dtt Z e V δ tr s { N K ǫ,δ ( t, x, x ) − N K ǫ,δ, ∞ ( t, x, x ) } dx = 0 . The same is true for the second integral in the expression of ln τ Z ( e V δ , e X, γ ǫ,δ ) .Proof. The proof is the same as above, using the estimate of Theorem 7.7. Indeed,we have, for all ǫ, δ ∈ (0 ,
1] with ǫ ≤ min { ǫ ( ρ ) δ , δ } , there is a constant C > | tr s { N K ǫ,δ ( t, x, x ) − N K ǫ,δ, ∞ ( t, x, x ) }| ≤ C t for all ( x, t ) ∈ e V δ × (0 , (cid:12)(cid:12)(cid:12)(cid:12)Z dtt Z e V δ tr s { N K ǫ,δ ( t, x, x ) − N K ǫ,δ, ∞ ( t, x, x ) } dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C Vol( e V δ , γ ǫ,δ ) . By the fact that lim δ → lim ǫ → Vol( e V δ , γ ǫ,δ ) = lim δ → Vol( e V δ , γ ) = 0 , we get the result. (cid:3) Proof of Theorem 7.12.
By (7.14), we getlim δ → lim ǫ → Z e V δ tr s ( N P ǫ,δ ( x, x )) dx = lim δ → lim ǫ → Z e V δ tr s ( N P ǫ,δ ( x, θ ( x ))) dx = 0 . From Propositions 7.13 and 7.14, it follows thatlim δ → lim ǫ → ln τ ( e V δ , e X, γ ǫ,δ ) = k lim δ → lim ǫ → I ( ǫ, δ ; ρ ) , lim δ → lim ǫ → ln h ǫ k/ τ Z ( e V δ , e X, γ ǫ,δ ) i = k lim δ → lim ǫ → (cid:20) J ( ǫ, δ ; ρ ) + 13 ln ǫ (cid:21) . Since the right hand side depend only on the choice of ρ , we get the result by settingln C EH0 ( ρ ) := lim δ → lim ǫ → I ( ǫ, δ ; ρ ) , ln C EH1 ( ρ ) := lim δ → lim ǫ → (cid:20) J ( ǫ, δ ; ρ ) + 13 ln ǫ (cid:21) . This completes the proof, provided that these double limits exist. This will beaddressed in what follows. (cid:3)
Remark . C EH0 ( ρ ), respectively C EH1 ( ρ ), is renormalized (resp. equivariant)analytic torsion for the asymptotically conical space e V ( ∞ ) = ( T ∗ P , γ EH ).7.7. Proof of Theorem 7.1.
Sinceln τ ( e X, γ ǫ,δ ) = ln τ ( e X δ , e X, γ ǫ,δ ) + ln τ ( e V δ , e X, γ ǫ,δ )and ln τ Z ( e X, γ ǫ,δ ) = ln τ Z ( e X δ , e X, γ ǫ,δ ) + ln τ Z ( e V δ , e X, γ ǫ,δ )by the definition of partial (equivariant) analytic torsion, we get by Corollary 7.11and Theorem 7.12(7.24) lim δ → lim ǫ → ln τ ( e X, γ ǫ,δ ) = ln τ ( X, γ ) + k ln C EH0 ( ρ ) , (7.25) lim δ → lim ǫ → ln h ǫ k/ τ Z ( e X, γ ǫ,δ ) i = ln τ Z ( X, γ ) + k ln C EH1 ( ρ ) . As the double limits on the left hand side of (7.24), (7.25) exist by virtue of Corol-lary 5.3 and Proposition 7.2, so do the double limits in defining ln C EH0 ( ρ ) andln C EH1 ( ρ ).On the other hand, again by Corollary 5.3 and Proposition 7.2, the double limitslim δ → lim ǫ → ln τ ( e X, γ ǫ,δ ) and lim δ → lim ǫ → ln h ǫ k/ τ Z ( e X, γ ǫ,δ ) i are independentof the choice of ρ . Hence C EH0 ( ρ ) and C EH1 ( ρ ) in (7.24), (7.25) are in fact indepen-dent of ρ . This completes the proof of Theorem 7.1. (cid:3) A holomorphic torsion invariant of log-Enriques surfaces
In this section, we introduce a holomorphic torsion invariant of log-Enriquessurfaces and give its explicit formula as a function on the moduli space.
NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 41
A construction of invariant.Theorem 8.1.
There is a constant C ( k ) depending only on k = Y ) with τ M ( e X, θ ) = C ( k )Vol( Y, γ ) − k τ ( Y, γ ) × Y p ∈ Sing( X ) ( | f p (0) | Vol(
Y, γ ) k η k L ( Y ) ) × exp Z Y log ( η ∧ ¯ ηγ / · Vol(
Y, γ ) k η k L ( Y ) ) c ( Y, γ ) ! . Proof.
Since M ⊥ k ∼ = Λ k (2), we have − r ( H ( e X, Z ) + )4 = − k . By its independence ofthe choice of θ -invariant K¨ahler metric on e X , τ M ( e X, θ ) is given bylim δ → lim ǫ → τ ( e X, γ ǫ,δ ) τ Z ( e X, γ ǫ,δ )( θ ) Vol( e X, γ ǫ,δ ) − k Vol( e X θ , γ ǫ,δ | e X θ ) τ ( e X θ , γ ǫ,δ | e X θ ) × A M ( e X, θ, γ ǫ,δ ) exp Z e X log ( η ∧ ¯ ηγ ǫ,δ / · Vol( e X, γ ǫ,δ ) k η k L ( e X ) ) c ( e X, γ ǫ,δ ) ! = lim δ → lim ǫ → { ǫ k τ ( e X, γ ǫ,δ ) τ Z ( e X, γ ǫ,δ )( θ ) }× lim δ → lim ǫ → Y p ∈ Sing( X ) ǫ − Vol( E p , γ ǫ,δ | E p ) τ ( E p , γ ǫ,δ | E p ) × lim δ → lim ǫ → A M ( e X, θ, γ ǫ,δ ) × lim δ → lim ǫ → exp Z e X log ( η ∧ ¯ ηγ ǫ,δ / · Vol( e X, γ ǫ,δ ) k η k L ( e X ) ) c ( e X, γ ǫ,δ ) ! . By Propositions 5.2, 5.4, 5.5, Corollary 5.3 and Theorem 7.1, we get τ M ( e X, θ ) = ( C EH0 C EH1 ) k τ ( X, γ ) τ Z ( X, γ )( ι ) { Y, γ ) } − k × { Vol( P , ω FS ) τ ( P , ω FS ) } k × Y p ∈ Sing( X ) ( | f p (0) | Vol(
X, γ ) k η k L ( X ) ) + × exp " Z X log ( η ∧ ¯ ηγ / · Vol(
X, γ ) k η k L ( X ) ) c ( X, γ ) . Since τ ( Y, γ ) = τ ( X, γ ) τ Z ( X, γ )( ι ) , Vol(
X, γ ) / k η k L ( X ) = Vol( Y, γ ) / k η k L ( Y ) and since X is a double covering of Y , we get the result by setting(8.1) C ( k ) = 2 { − C EH0 C EH1
Vol( P , ω FS ) τ ( P , ω FS ) } k . This completes the proof. (cid:3)
Theorem 8.2.
Let γ be a K¨ahler form on Y in the sense of orbifolds. Then thefollowing equality holds: τ ( Y, γ )Vol(
Y, γ ) τ ( Y, ω η )Vol( Y, ω η ) = Y p ∈ Sing( Y ) ω η γ ! ( p ) − exp ( − Z Y log ω η γ ! c ( Y, γ ) ) . Proof.
Let p ∈ Sing( Y ) and let ( U p , ⊂ ( C ,
0) be an open subset which uni-formizes the germ ( Y, p ). We have an isomorphism ( Y, p ) ∼ = ( C / Γ p ,
0) of germs,where Γ p = Z / Z = h i i , such that ω η and γ lift to K¨ahler metrics on U p . FollowingMa [32], we define Y Σ as the union Y Σ := Y i ∐ Y i ∐ Y i , where Y i ν = { p i ν } p ∈ Sing( Y ) and the germ ( Y i ν , p i ν ) is equipped with orbifold structure ( Y i ν , p i ν ) ∼ = ( C / h i ν i , Σ ( T Y ) supported on the singular locus of Y appears in the Riemann-Roch theorem for orbifolds, for which we refer the readerto e.g. [32]. By the anomaly formula for Quillen metrics for orbifolds [32], we get(8.2)log (cid:18) τ ( Y, γ )Vol(
Y, γ ) τ ( Y, ω η )Vol( Y, ω η ) (cid:19) = 14 Z Y Σ f Td Σ ( T Y ; γ, ω η ) + 124 Z Y g c c ( T Y ; γ, ω η )= 14 X p ∈ Sing( Y ) 3 X ν =1 g Tde ! ν/ ( T U p ; γ, ω η ) (0 , ( p ) + 124 Z Y g c c ( T Y ; γ, ω η ) (2 , . Here, for θ ∈ R and a square matrix A , we define (cid:0) Tde (cid:1) θ ( A ) := det (cid:16) II − e πiθ A (cid:17) and( ] Td / e) θ is the Bott-Chern secondary class associated to (Td / e) θ ( A ) such that forany holomorphic vector bundle E and Hermitian metrics h , h ′ on E − dd c g Tde ! θ ( E ; h, h ′ ) = (cid:18) Tde (cid:19) θ (cid:18) − πi R ( E, h ) (cid:19) − (cid:18) Tde (cid:19) θ (cid:18) − πi R ( E, h ′ ) (cid:19) . Similarly, g c c is the Bott-Chern secondary class associated to the invariant poly-nomial c ( A ) c ( A ) such that for any holomorphic vector bundle E and Hermitianmetrics h , h ′ on E − dd c g c c ( E ; h, h ′ ) = c ( E, h ) c ( E, h ) − c ( E, h ′ ) c ( E, h ′ ) . For A = diag( λ , λ ), we have (cid:18) Tde (cid:19) ν ( A ) = 1(1 − i − ν ) { − i − ν − i − ν c ( A ) + O (2) } . Thus we get(8.3) X ν =1 g Tde ! ν ( T U p ; γ, ω η ) (0 , ( p ) = − X ν =1 i − ν (1 − i − ν ) e c ( T U p ; γ, ω η )( p )= 58 e c ( T U p ; γ, ω η )( p ) = −
58 log (cid:0) ω η /γ (cid:1) ( p ) . On the other hand, by the same computations as in (3.3), we get(8.4) g c c ( T Y ; γ, ω η ) (2 , = − log (cid:0) ω η /γ (cid:1) c ( T Y, γ ) . Substituting (8.3) and (8.4) into (8.2), we get the result. (cid:3)
Theorem 8.3.
For every Ricci-flat log-Enriques surface ( Y, ω ) , one has Vol(
Y, ω ) − k τ ( Y, ω ) = C ( k ) − τ M ( e X, θ ) , where C ( k ) is the same constant as in Theorem 8.1. NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 43
Proof.
We put γ = γ in Theorem 8.2. Then we get by Theorem 8.1 τ ( Y, ω η )Vol( Y, ω η ) = τ ( Y, γ )Vol( Y, γ ) − k Vol(
Y, γ ) k × { Y p ∈ Sing( X ) ω η γ ! ( p ) } exp " Z Y log ω η γ ! c ( Y, γ ) = C ( k ) − τ M ( e X, θ ) Vol(
Y, γ ) k × Y p ∈ Sing( X ) ( | f p (0) | Vol(
Y, γ ) k η k L ( Y ) ) − × exp " − Z Y log ( η ∧ ¯ ηγ / · Vol(
Y, γ ) k η k L ( Y ) ) c ( Y, γ ) × { Y p ∈ Sing( X ) ω η γ ! ( p ) } exp (cid:20) Z Y log (cid:18) η ∧ ¯ ηγ / (cid:19) c ( Y, γ ) (cid:21) . Since | f p (0) | = [ η ∧ ¯ η/ ( γ / p ) = [ ω η /γ ]( p ), we get τ ( Y, ω η )Vol( Y, ω η )= C ( k ) − τ M ( e X, θ ) Vol(
Y, γ ) k × Y p ∈ Sing( X ) (cid:18) | f p (0) | Vol(
Y, γ )Vol( Y, ω η ) (cid:19) − × { Y p ∈ Sing( X ) | f p (0) | } exp (cid:20) − Z Y log (cid:26) Vol(
Y, γ )Vol( Y, ω η ) (cid:27) c ( Y, γ ) (cid:21) = C ( k ) − τ M ( e X, θ ) Vol(
Y, γ ) k (cid:18) Vol(
Y, γ )Vol( Y, ω η ) (cid:19) − k exp (cid:20) − − k
32 log (cid:18)
Vol(
Y, γ )Vol( Y, ω η ) (cid:19)(cid:21) = C ( k ) − τ M ( e X, θ ) Vol(
Y, ω η ) k , where we used the second assertion of Proposition 5.1 to get the second equality.This proves the result. (cid:3) Theorem 8.4.
Let γ be a K¨ahler form on Y in the sense of orbifolds and let Ξ ∈ H ( Y, K ⊗ Y ) \ { } be a nowhere vanishing bicanonical form on Y . Then τ k ( Y ) := τ ( Y, γ )Vol(
Y, γ ) k Ξ k − k L ( Y ) Y p ∈ Sing( Y ) (cid:18) γ / | Ξ | (cid:19) ( p ) × exp (cid:20) Z Y log (cid:18) | Ξ | γ / (cid:19) c ( Y, γ ) (cid:21) is independent of the choices of γ and Ξ , where | Ξ | := p Ξ ⊗ Ξ is the Ricci-flatvolume form on Y induced by Ξ . In fact, τ k ( Y ) = C ( k ) − τ M ( e X, θ ) . Proof.
Let ω be a Ricci-flat K¨ahler form on Y in the sense of orbifolds such that ω /
2! = | Ξ | . Since Vol( Y, ω ) = k Ξ k L ( Y ) , we get by Theorem 8.3(8.5)Vol( Y, ω ) τ ( Y, ω ) = Vol(
Y, ω ) k Vol(
Y, ω ) − k τ ( Y, ω ) = C ( k ) − k Ξ k k L ( Y ) τ M ( e X, θ ) . Let ξ ∈ H ( e X, K e X ) be a nowhere vanishing holomorphic 2-form on e X such that( p ◦ π ) ∗ Ξ = ξ ⊗ . Since ω = ω ξ , i.e., ω /
2! = ξ ∧ ξ = | Ξ | , we get by Theorem 8.2(8.6) τ ( Y, γ )Vol(
Y, γ ) τ ( Y, ω )Vol(
Y, ω ) = Y p ∈ Sing( Y ) (cid:18) | Ξ | γ / (cid:19) ( p ) − exp (cid:20) − Z Y log (cid:18) | Ξ | γ / (cid:19) c ( Y, γ ) (cid:21) . Comparing (8.5) and (8.6), we get(8.7) τ ( Y, γ )Vol(
Y, γ ) = C ( k ) − τ M ( e X, θ ) k Ξ k k L ( Y ) Y p ∈ Sing( Y ) (cid:18) | Ξ | γ / (cid:19) ( p ) − × exp (cid:20) − Z Y log (cid:18) | Ξ | γ / (cid:19) c ( Y, γ ) (cid:21) . From (8.7), we get τ k ( Y ) = C ( k ) − τ M ( e X, θ ) / . Since the right hand side is inde-pendent of the choices of γ and Ξ , so is τ k ( Y ). This completes the proof. (cid:3) Del Pezzo surfaces and an explicit formula for the invariant τ k . Inthis subsection, we give an explicit formula for τ k as an automorphic function onthe K¨ahler moduli of Del Pezzo surfaces. Let 1 ≤ k ≤
9. We define the unimodularLorentzian lattices L k and U ( −
1) as L k := (cid:18) − I − k (cid:19) ( k = 2) , L := (cid:18) − (cid:19) or (cid:18) (cid:19) , U ( −
1) := (cid:18) − − (cid:19) . We fix an isometry of lattices Λ k ∼ = U ( − ⊕ L k and identify Λ k with U ( − ⊕ L k .Let V be a Del Pezzo surface of degree k , i.e., k = deg V = Z V c ( V ) . Then V = Bl − k ( P ) is the blowing-up of P at 9 − k points in general positionwhen k = 8. When k = 8, V ∼ = Σ or Σ , where Σ n = P ( O P ⊕ O P ( n )) is theHirzebruch surface. Notice that Σ = P × P and Σ = Bl ( P ). When V = Σ , H ( V, Z ) endowed with the cup product pairing is isometric to L k by identifying H, E , . . . , E − k with the standard basis of L k , where H ∈ H ( V, Z ) is the classobtained from the hyperplane class of H ( P , Z ) and E i ( i = 1 , . . . , − k ) are theclasses of exceptional divisors. Similarly, H ( V, Z ) endowed with the Mukai pairingis isometric to Λ k . In what follows, we identify L k (resp. Λ k ) with H ( V, Z ) (resp. H ( V, Z )) in this way.Recall that the type IV domain Ω k associated with Λ k was defined in Section 3.We identify Ω H ( V, Z ) with the tube domain H ( V, Z ) ⊗ R + i C H ( V, Z ) ⊂ H ( V, C )via the map(8.8) H ( V, Z ) ⊗ R + i C H ( V, Z ) ∋ y → [exp( y )] := (cid:2)(cid:0) , y, y / (cid:1)(cid:3) ∈ Ω H ( V, Z ) , where C H ( V, Z ) := { v ∈ H ( V, R ); v > } is the positive cone of H ( V, R ). NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 45
Let K V ⊂ C H ( V, Z ) be the K¨ahler cone of V , i.e., the cone of H ( V, R ) consistingof K¨ahler classes on V . Let Eff( V ) ⊂ H ( V, R ) be the effective cone of V , i.e., thedual cone of the K¨ahler cone K V . Definition 8.5.
Define the infinite product Φ V ( z ) on H ( V, Z ) ⊗ R + i K V byΦ V ( z ) := e πi h c ( V ) ,z i Y α ∈ Eff( V ) (1 − e πi h α,z i ) c (0) k ( α ) × Y β ∈ Eff( V ) , β/ ≡ c ( V ) / H ( V, Z ) (1 − e πi h β,z i ) c (1) k ( β / , where { c (0) k ( l ) } l ∈ Z , { c (1) k ( l ) } l ∈ Z + k/ are defined by the generating functions X l ∈ Z c (0) k ( l ) q l = η (2 τ ) θ A ( τ ) k η ( τ ) η (4 τ ) , X l ∈ k + Z c (1) k ( l ) q l = − η (4 τ ) θ A +1 / ( τ ) k η (2 τ ) . Here θ A + ǫ/ ( τ ) := P n ∈ Z q ( n + ǫ/ and η ( τ ) := q / Q n> (1 − q n ).Let C + H ( V, Z ) be the connected component of C H ( V, Z ) that contains K V and letΩ + H ( V, Z ) be the component of Ω H ( V, Z ) corresponding to H ( V, R )+ i C + H ( V, Z ) via theisomorphism (8.8). By Borcherds [8, Th. 13.3] (cf. [44]), Φ V ( z ) converges absolutelyfor those z ∈ H ( V, R ) + i K V with ℑ z ≫ + H ( V, Z ) for O + ( H ( V, Z )) of weight deg V + 4 with zero divisor div(Φ V ) = P d ∈ H ( V, Z ) , d = − d ⊥ under the identification H ( V, R ) + i C + H ( V, Z ) ∼ = Ω + H ( V, Z ) .Recently, an explicit Fourier series expansion of Φ V ( z ) is discovered by Gritsenko[22, Cor. 5.1]. It is also remarkable that Φ V is the denominator function of a gen-eralized Kac-Moody algebra, whose real and imaginary simple roots are explicitlygiven by the Fourier series expansion of Φ V [23, § V associated to Del Pezzo surfaces isquite analogous to the Borcherds Φ-function of rank 10.We define the Petersson norm of Φ V ( z ) by k Φ V ( z ) k := hℑ z, ℑ z i V | Φ V ( z ) | , where z ∈ H ( V, R ) + i C + H ( V, Z ) . Then k Φ V k is an O + ( H ( V, Z ))-invariant C ∞ function on Ω + H ( V, Z ) . Hence k Φ V k is identified with a C ∞ function on M deg V inthe sense of orbifolds. Theorem 8.6.
Let ≤ k ≤ . There exists a constant e C ( k ) > depending onlyon k such that for every -elementary K surface ( e X, θ ) of type M k := Λ k (2) ⊥ , τ M k ( e X, θ ) = e C ( k ) k Φ V ( ̟ ( e X, θ )) k − / , where k = deg V .Proof. See [44, Th. 4.2 (1)] and [45, Th. 0.1]. (cid:3)
Theorem 8.7.
Let ≤ k ≤ . Then there exists a constant C k > depending onlyon k such that for every good log-Enriques surface Y with Y ) = deg V , τ deg V ( Y ) = C deg V k Φ V ( ̟ ( Y )) k − / . Proof.
We set k = deg V . When k = 2, we define V = Σ when Y is of eventype and V = Σ when Y is of odd type. Let ( e X, θ ) be the 2-elementary K M k associated to Y . By the definition of the period of Y , we have ̟ ( Y ) = ̟ ( e X, θ ). Hence(8.9) k Φ V ( ̟ ( Y )) k = k Φ V ( ̟ ( e X, θ )) k . By Theorems 5.11 and 7.2 and (8.9), we get(8.10) τ k ( Y ) = C ( k ) − τ M k ( e X, θ ) / = C ( k ) − e C ( k ) k Φ V ( ̟ ( e X, θ )) k − / = C ( k ) − e C ( k ) k Φ V ( ̟ ( Y )) k − / . Setting C k := C ( k ) − e C ( k ) in (8.10), we get the result. (cid:3) The quasi-pullback of Φ V . Let π : e V := Bl p ( V ) → V be the blow-up of V at p and let E := π − ( p ) be the exceptional curve of π . Then we have a map ofcohomologies π ∗ : H ( V, Z ) → H ( e V , Z ), which induces the canonical identification H ( V, Z ) ∼ = π ∗ H ( V, Z ) = { [ x ] ∈ H ( e V , Z ); h [ E ] , x i = 0 } . Since [ E ] is a norm ( − H ( e V , Z ), this implies that KM ( V ) is identifiedwith a component of the Heegner divisor of norm ( − KM ( e V ). Since O ( H ( e V , Z )) acts transitively on the norm ( − H ( e V , Z ) except the casedeg e V = 7, i.e., H ( e V , Z ) ∼ = U ⊕ ⊕ h− i , KM ( V ) coincides with the Heegner divisorof norm ( − KM ( e V ) when deg V = 7. When deg V = 7, the Heegnerdivisor of norm ( − KM ( e V ) consists of two components; one is givenby KM ( Σ ) and the other is given by KM ( Σ ), where Σ n = P ( O P ⊕ O P ( n )) isthe Hirzebruch surface. In the following theorem, we use the convention that a DelPezzo surface of degree is an Enriques surface . Theorem 8.8. Φ V is the quasi-pullback of Φ e V to KM ( V ) = [ E ] ⊥ , up to a constant.Namely, in the infinite product expression in Definition 8.5, the following equalityholds Φ V = Const . Φ e V ( · ) h· , [ E ] i (cid:12)(cid:12)(cid:12)(cid:12) [ E ] ⊥ , where h z, [ E ] i is the linear form on H ( e V , C ) defined by the norm ( − -vector [ E ] .Proof. The result is a special case of [29, Th. 1.1]. See also [29, Example 3.17]. (cid:3)
This theorem can be summarized as the following diagrams: KM (Enr) ⊃ KM (dP ) ⊃ · · · ⊃ KM (dP ) ⊃ KM (Σ ) ⊃ KM ( P )Φ Enr → Φ dP → · · · → Φ dP → Φ Σ → Φ P η − − → η − − θ → · · · → η − − θ → η − − θ → η − − θ and KM (dP ) ⊃ KM (Σ )Φ dP → Φ Σ η − − θ → η − − θ where the inclusion implies the embedding as the discriminant divisor, the arrowin the second line implies the quasi-pullback (up to a constant), and the arrow inthe third line describe the change of elliptic modular form for Γ (4) correspondingto Φ V . We remark that there are no inclusions of KM ( P ) into KM (Σ ). NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 47 The invariant τ k and the BCOV invariant The BCOV invariant of log-Enriques surfaces.
In this subsection, weprove that the invariant τ k is viewed as the BCOV invariant of good log-Enriquessurfaces. Recall that for a compact connected K¨ahler orbifold ( V, γ ), the BCOVtorsion T BCOV ( V, γ ) is defined as T BCOV ( V, γ ) := exp( − X p,q ≥ ( − p + q pq ζ ′ p,q (0)) , where ζ p,q ( s ) is the spectral zeta function of the Laplacian (cid:3) p,q acting on ( p, q )-forms on V in the sense of orbifolds. As before, the analytic torsion of the trivialline bundle on V is denoted by τ ( V, γ ). Lemma 9.1. If dim V = 2 , then the following equality holds: T BCOV ( V, γ ) = τ ( V, γ ) − . Proof.
Since (cid:3) p,q and (cid:3) − q, − p are isospectral via the Hodge ∗ -operator, we have ζ p,q ( s ) = ζ − q, − p ( s ). Since (cid:3) p,q and (cid:3) q,p are isospectral via the complex conjuga-tion, we have ζ p,q ( s ) = ζ q,p ( s ). Using these relations, we have(9.1) − log T BCOV ( V, γ ) = 4 ζ ′ , (0) − ζ ′ , (0) + ζ ′ , (0) . Since ζ , ( s ) − ζ , ( s ) + ζ , ( s ) = 0 and ζ , ( s ) − ζ , ( s ) + ζ , ( s ) = 0, we have4 ζ ′ , (0) − ζ ′ , (0) = − ζ ′ , (0) and ζ ′ , (0) = ζ ′ , (0) + ζ ′ , (0) = ζ ′ , (0) + ζ ′ , (0) =2 ζ ′ , (0). Substituting these into (9.1), we get the result. (cid:3) Now we have the following:
Theorem 9.2.
Let Y be a good log-Enriques surface with k singular points. Let γ be a K¨ahler form on Y in the sense of orbifolds and let Ξ ∈ H ( Y, K ⊗ Y ) \ { } be anowhere vanishing bicanonical form on Y . Then τ BCOV ( Y ) := T BCOV ( Y, γ )Vol(
Y, γ ) − k Ξ k k L ( Y ) Y p ∈ Sing( Y ) (cid:18) γ / | Ξ | (cid:19) ( p ) − × exp (cid:20) − Z Y log (cid:18) | Ξ | γ / (cid:19) c ( Y, γ ) (cid:21) is independent of the choices of γ and Ξ . In fact, τ BCOV ( Y ) = τ k ( Y ) − = C − k k Φ V ( ̟ ( Y )) k , where C k is the same constant as in Theorem 8.7.Proof. Since τ BCOV ( Y ) = τ k ( Y ) − by Theorem 8.4 and Lemma 9.1, we get the firstclaim. The second claim follows from Theorem 8.7. (cid:3) We call τ BCOV ( Y ) the BCOV invariant of Y . When γ is Ricci-flat and | Ξ | = γ / τ BCOV ( Y ) = T BCOV ( Y, γ )Vol(
Y, γ ) k − . As in the case of Enriques surfaces, the BCOV invariant of good log-Enriquessurfaces is expressed by the Peterssion norm of a Borcherds product. In particular,the BCOV invariant of log-Enriques surfaces is not a birational invariant, for thebirational equivalence classes of log-Enriques surfaces consist of a single class.
Problem . For a good log-Enriques surface Y , there exists a log-Enriques surface Y ′ with a unique singular point admitting a birational morphism Y → Y ′ (cf.[47]). In general, the singularity of Y ′ is worse than those of Y . Can one constructa holomorphic torsion invariant of Y ′ using some ALE instanton instead of theEguchi-Hanson instanton? If this is the case, compare the holomorphic torsioninvariants between Y and Y ′ . Problem . Let Y be a good log-Enriques surface. Let p : e Y → Y be a resolutionsuch that p − (Sing Y ) is a disjoint union of smooth ( − Y and that of the pair ( e Y , p − (Sing Y )) defined by Zhang [49]. Problem . Can one construct a holomorphic torsion invariant of log-Enriquessurfaces with index ≥ The BCOV invariant of certain Borcea-Voisin type orbifolds.
Let Y be a good log-Enriques surface with k singular points and let X be the canonicaldouble covering of Y . Then X is a nodal K k nodes endowed withan anti-symplectic involution ι with fixed point set Sing X = { p , . . . , p k } . Let T be an elliptic curve. We define V = V ( X,ι,T ) := ( X × T ) / ( ι × ( − T ) . Then V is a Calabi-Yau orbifold of dimension 3. Let e V be the Borcra-Voisin orbifold e V = e V ( e X,θ,T ) := ( e X × T ) / ( θ × ( − T ) , where π : e X → X is the minimal resolution of X and θ is the involution on e X in-duced from ι . As before, we set E i := π − ( p i ) ∼ = P . The birational morphism from e V to V induced by π is denoted again by π . Then π : e V → V is a partial resolutionsuch that the k cyclic quotient singularities of type ( , , ) of V are replaced bythe milder cyclic quotient singularities of type ( , , e V and V .Let γ X (resp. γ e X ) be a Ricci-flat K¨ahler from on X (resp. e X ) and let γ T be theflat K¨ahler form with Vol( V, γ T ) = 1. Let π : V → Y = X/ι and π : V → T / ( − T be the projections. Similarly, let e π : e V → e X/θ and e π : e V → T / ( − T be theprojections. We define a Ricci-flat K¨ahler form γ (resp. e γ ) on V (resp. e V ) by γ := π ∗ γ X + π ∗ γ T , e γ := e π ∗ γ e X + e π ∗ γ T . Since Sing( X × T ) = ( { p } × T ) ∐ · · · ∐ ( { p k } × T ), we haveSing V = ( { p } × T / ( − T ) ∐ · · · ∐ ( { p k } × T / ( − T ) ∐ ( X ι × T [2])= ( { p } × T / ( − T ) ∐ · · · ∐ ( { p k } × T / ( − T ) ∐ (Sing X × T [2]) , where T [2] denotes the points of order 2 of T . Similarly,Sing e V = e X θ × T [2] = ( E × T [2]) ∐ · · · ∐ ( E k × T [2]) . Hence the 1-dimensional strata of Sing V (resp. Sing e V ) consist of k -copies of thequotient T / ( − T (resp. 4-copies of E , . . . , E k ), which are endowed with the flatorbifold K¨ahler form γ T (resp. K¨ahler form γ e X | E i induced from γ e X ). NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 49
Recall from [46, (6.12)] that the orbifold BCOV invariant of V is defined by(9.3) τ orbBCOV ( V ) = T BCOV ( V, γ )Vol(
V, γ ) − χ orb( V )12 Vol L ( H ( V, Z ) , γ ) − × k Y i =1 τ ( { p i } × ( T / ( − T ) , γ T ) − Vol(
T / ( − T , γ T ) − = T BCOV ( V, γ )Vol(
V, γ ) − χ orb( V )12 Vol L ( H ( V, Z ) , γ ) − k τ ( T, γ T ) − k , where we used the facts τ ( T / ( − T , γ T ) = τ ( T, γ T ) / and Vol( T / ( − T , γ T ) = 1 / L ( H ( V, Z ) , γ ) is the covolume of the lattice H ( V, Z ) fr := H ( V, Z ) / Torsion with respect to the L metric induced by γ . (Inwhat follows, for a finitely generated Z -module M , we set M fr := M/ Tors( M ).)For the definition of χ orb ( V ), see [46, (6.2)]. By [46, Prop. 6.2], χ orb ( V ) coincideswith the Euler characteristic of a crepant resolution of V . Similarly, we have τ orbBCOV ( e V ) = T BCOV ( e V , e γ )Vol( e V , e γ ) − χ orb( e V )12 Vol L ( H ( e V , Z ) , e γ ) − × { k Y i =1 τ ( E i , γ e X | E i )Vol( E i , γ e X | E i ) } − . Let q : X × T → V and e q : e X × T → e V be the quotient maps. Let H ( X × T, Z ) + (resp. H ( e X × T, Z ) + ) be the invariant subspace with respect to the ι × ( − T (resp. θ × ( − T )-action on X × T (resp. e X × T ). We define H ( X, Z ) + and H ( e X, Z ) + in the same way. Let r := rk Z H ( X, Z ) + and e r := rk Z H ( e X, Z ) + .Then e r = r + k = 10 + k . The maps of cohomologies q ∗ : H ( V, Z ) fr → H ( X × T, Z ) +fr = H ( X, Z ) +fr ⊕ H ( T, Z ) , e q ∗ : H ( e V , Z ) fr → H ( e X × T, Z ) + = H ( e X, Z ) + ⊕ H ( T, Z ) , have finite cokernel. Let disc( H ( X, Z ) +fr ) be the discriminant of the lattice H ( X, Z ) +fr with respect to the intersection pairing h· , ·i on H ( X, Z ) fr ⊂ H ( X, Q ). Namely,if { e , . . . , e r } is a basis of H ( X, Z ) fr , then disc( H ( X, Z ) +fr ) := det( h e i , e j i ). Ob-viously, | Coker q ∗ | , | Coker e q ∗ | , disc( H ( X, Z ) +fr ), disc( H ( e X, Z ) + ) depend only on k . Recall that the constant C ( k ) was defined in (8.1), which is the k -th power ofthe product of the normalized analytic torsion of the Eguchi-Hanson instanton andthe analytic torsion of P endowed with the Fubini-Study metric, up to a universalconstant. Theorem 9.6.
The following equality holds: τ orbBCOV ( V ) τ orbBCOV ( e V ) = 2 − k − C ( k ) (cid:18) | Coker q ∗ || Coker e q ∗ | (cid:19) − | disc( H ( X, Z ) +fr ) || disc( H ( e X, Z ) + ) | ! − . Proof.
We express T BCOV ( V, γ ) in terms of τ Z ( X, γ X )( ι ) and τ ( T, γ T ). As is easilyverified, Lemmas 8.3-8.7 of [46] hold true for V without any change. Since h , ( X ) =20 − k , the coefficient 21 of ζ T, + ( s ) in [46, Lemma 8.8] should be replaced by 21 − k .Hence, for V , the equality corresponding to [46, Eq. (8.28)] becomes X p,q ( − p + q pq ζ p,q ( s ) = (24 − k ) ζ T, + ( s ) + 8 { ζ X, + ( s ) − ζ X, − ( s ) } . As a result, we get the following equality as in the first equality of [46, p. 358](9.4) T BCOV ( V, γ ) = τ Z ( X, γ X )( ι ) − τ ( T, γ T ) − (12 − k ) . By [46, l.2-3], we have(9.5) T BCOV ( e V , e γ ) = τ Z ( e X, γ e X )( θ ) − τ ( T, γ T ) − . Since e X θ consists of k copies of mutually disjoint P , we get χ orb ( V ) = χ orb ( e V ) = χ ( e X × T ) + χ ( e X θ × T [2]) = 12 k by [46, Prop. 6.1 and (6.3)]. Hence(9.6) Vol( V, γ ) − χ orb( V )12 = Vol( V, γ ) − k = 2 − k Vol(
X, γ X ) − k , where we used the fact Vol( T, γ T ) = 1 and Vol( V, γ ) = Vol(
X, γ X )Vol( T, γ T ) / e V , e γ ) − χ orb( e V )12 = 2 − k Vol( e X, γ e X ) − k . Let { f , . . . , f r +1 } be a basis of H ( V, Z ) fr . By definition, we haveVol L ( H ( V, Z ) , γ ) = | det( h f i , f j i L ) | , where h· , ·i L denotes the L inner product on H ( V, R ) induced from γ . SinceVol L ( H ( T, Z ) , γ T ) = 1, the same calculations as in [19, Lemma 13.4] yield that(9.8) Vol L ( H ( V, Z ) , γ ) = | Coker q ∗ | Vol L ( H ( X, Z ) +fr ⊕ H ( T, Z ) , γ X ⊕ γ T )= | Coker q ∗ | Vol L ( H ( X, Z ) +fr , γ X ) Vol( X, γ X ) /
2= 2 − ( r +1) | Coker q ∗ | | disc( H ( X, Z ) +fr ) | Vol(
X, γ X ) . Similarly, we have(9.9) Vol L ( H ( e V , Z ) , e γ ) = 2 − ( e r +1) | Coker e q ∗ | | disc( H ( e X, Z ) + ) | Vol( e X, γ e X ) . Substituting (9.4), (9.6), (9.8) into (9.3) and using (3.7), we get(9.10) τ orbBCOV ( V ) = 2 r +4 | Coker q ∗ | − | disc( H ( X, Z ) +fr ) | − × τ Z ( X, γ X )( ι ) − Vol(
X, γ X ) − k τ ( T, γ T ) − = 2 r +4 | Coker q ∗ | − | disc( H ( X, Z ) +fr ) | − × τ ( X, γ X ) − τ Z ( X, γ X )( ι ) − Vol(
X, γ X ) − k τ ( T, γ T ) − = 2 r − k | Coker q ∗ | − | disc( H ( X, Z ) +fr ) | − τ ( Y, γ Y ) − Vol(
Y, γ Y ) − k τ ( T, γ T ) − = 2 r − k | Coker q ∗ | − | disc( H ( X, Z ) +fr ) | − C ( k ) τ M ( e X, θ ) − τ ( T, γ T ) − , where we used the equality τ ( Y, γ Y ) = τ ( X, γ X ) τ Z ( X, γ X )( ι ) to get the thirdequality and Theorem 8.3 to get the last equality. Similarly, substituting (9.5), NALYTIC TORSION FOR LOG-ENRIQUES SURFACES AND BORCHERDS PRODUCT 51 (9.7), (9.9) into [46, (6.12)], we get(9.11) τ orbBCOV ( e V ) = τ Z ( e X, γ e X )( θ ) − τ ( T, γ T ) − { k Y i =1 τ ( E i , γ e X | E i )Vol( E i , γ e X | E i ) } − × − k Vol( e X, γ e X ) − k e r +1 | Coker e q ∗ | − | disc( H ( e X, Z ) + ) | − Vol( e X, γ e X ) − = 2 e r +4 − k | Coker e q ∗ | − | disc( H ( e X, Z ) + ) | − τ ( T, γ T ) − × τ Z ( e X, γ e X )( θ ) − Vol( e X, γ e X ) − k { k Y i =1 τ ( E i , γ e X | E i )Vol( E i , γ e X | E i ) } − = 2 r +4 | Coker e q ∗ | − | disc( H ( e X, Z ) + ) | − τ M ( e X, θ ) − τ ( T, γ T ) − . Comparing (9.10) and (9.11), we get the result. (cid:3)
We define the BCOV invariant of elliptic curve T as τ BCOV ( T ) := Vol( T, ω ) − τ BCOV ( T, ω ) exp (cid:20) − Z T log (cid:18) i ξ ∧ ξω (cid:19) c ( T, ω ) (cid:21) , where ω is an arbitrary K¨ahler from on T . By [46, Th. 8.1], τ BCOV ( T ) is independentof the choice of ω and is expressed by the Petersson norm of the Dedekind η -function. By definition, we have τ BCOV ( T ) = τ ( T, γ T ) − . By (9.10), we have thefollowing factorization of the orbifold BCOV invariant of V . Corollary 9.7.
The following equality of BCOV invariants holds: τ orbBCOV ( V ) = 2 r − k | Coker q ∗ | − | disc( H ( X, Z ) +fr ) | − τ BCOV ( Y ) τ BCOV ( T ) . Proof.
The result follows from (9.2) and the third equality of (9.10). (cid:3)
Remark . In [46, p.357 l.7], it seems that the equality H ( X, Z ) = H ( S × T, Z ) + does not hold in general. As the difference of these two quantities, | Coker e q ∗ | shouldappear in the formula for τ orbBCOV ( e V ) as in (9.11). Remark . In this subsection, for the sake of simplicity of notation, we adopt thedefinitions Vol(
V, γ ) = R V γ /
3! and h α, β i L = R V ( H α ) ∧ ∗ ( H β ) etc., where H ( · )denotes the harmonic projection. If we follow the tradition in Arakelov geometry, itis more natural to define the L -inner product by Vol( V, γ ) = (2 π ) − dim V R V γ / h α, β i L = (2 π ) − dim V R V ( H α ) ∧ ∗ ( H β ) etc. Notice that in [46], this latterdefinition is adopted. Problem . Is the orbifold BCOV invariant [46] a birational invariant of Calabi-Yau orbifolds? (To our knowledge, it is still open that the BCOV invariant ofKLT Calabi-Yau varieties [20] coincides with the orbifold BCOV invariant [46].)If the answer is affirmative, then it follows from Theorem 9.6 that the normalizedanalytic torsion of the Eguchi-Hanson instanton will essentially be given by theanalytic torsion of P with respect to the Fubini-Study metric. Once a comparisonformula for the BCOV invariants for birational Calabi-Yau orbifolds is obtained,then one will get a formula for the normalized analytic torsion of the Eguchi-Hansoninstanton through Theorem 9.6. References [1] Barth, W., Peters, C., Van de Ven, A.
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