Bergman kernel and period map for curves
aa r X i v : . [ m a t h . AG ] F e b BERGMAN KERNEL AND PERIOD MAP FORCURVES
ALESSANDRO GHIGI, CAROLINA TAMBORINI
Abstract.
As for any symmetric space the tangent space to Siegelupper-half space is endowed with an operation coming from the Liebracket on the Lie algebra. We consider the pull-back of this oper-ation to the moduli space of curves via the Torelli map. We charac-terize it in terms of the geometry of the curve, using the Bergmankernel form associated to the curve. It is known that the secondfundamental form of the Torelli map outside the hyperelliptic lo-cus can be seen as the multiplication by a certain meromorphicform. Our second result says that the Bergman kernel form is theharmonic representative - in a suitable sense - of this meromorphicform.
Contents
1. Introduction 12. Bergman kernel 33. Lie bracket 54. The form ˆ η Introduction X be a Riemannian symmetric space. For a fixed point x ∈ X we have X = G/K , where G is a Lie group (independent of x ) and K = G x is the stabilizer of x . Moreover there is a Cartan decomposition g = k ⊕ p such that [ p , p ] ⊂ k and [ k , p ] ⊂ p . Since p ∼ = T x X , the Liebracket on g gives rise to a kind of operation B x : T x X × T x X ∼ = p × p → k = g x . Let S = G × K k be the homogeneous bundle over X corresponding to the adjoint representation of K . Then S x = g x forany x , so B is a section of Λ T ∗ X ⊗ S . Since the differential geometryof X can be studied by means of Lie theory, the tensor B , which reflectsLie bracket, is of central importance. Mathematics Subject Classification. B is invariant by the action of G it makes sensealso on any locally symmetric space X . In this paper we consider thecase where X is A g , the moduli space of principally polarized abelianvarieties of dimension g over C , which is a locally symmetric spaceobtained as a quotient of the Siegel upper half-space S g . Denote by M g the moduli space of curves of genus g . We are interested in the Torellimap j : M g → A g , which associates to [ C ] ∈ M g its Jacobian variety[ J C ] ∈ A g . Our motivation comes from the study of totally geodesicsubvarieties of A g that are generically contained in j ( M g ) [6, 2, 4]. Thisis also connected to the Coleman-Oort conjecture [8]. The tensor B controls the local geometry of A g and its pull-back B = j ∗ B to M g should give important information on the extrinsic geometry of theinclusion j ( M g ) ⊂ A g . For example we expect that the study of B willgive constraints on the existence of Lie triples tangent to M g .1.3. The first step in this direction is the computation of B at a modulipoint [ C ] ∈ M g in terms of the geometry of the curve C . This is thefirst main result of this note. Let ¯ C denote the conjugate curve, i.e.with the opposite complex structure. We first show that the dual mapof B can be seen as map B ∗ : H ( K C ) ⊗ H ( K C ) → H (2 K C ) ⊗ H (2 K C ) . Secondly, we consider the algebraic surface Z = C × C . By K¨unnethformula H ( Z, K Z ) ≃ H ( K C ) ⊗ H ( K C ) and H (2 K Z ) ≃ H (2 K C ) ⊗ H (2 K C ). With these identifications we prove the following. Theorem A.
The map B ∗ : H ( Z, K Z ) −→ H ( Z, K Z ) coincides with the multiplication by − i K , where K ∈ H ( Z, K Z ) is theBergman kernel of the curve C . (See Theorem 3.7. See 2.1 for the definition of Bergman kernel in thesense we need.) In other words the Bergman kernel K governs therestriction of the Lie bracket to dj ( T M g ).1.4. Another approach to the extrinsic geometry of inclusion j ( M g ) ⊂ A g outside the hyperelliptic locus uses the second fundamental form.If C is non-hyperelliptic, the second fundamental form at [ C ] has beeninterpreted as the multiplication by a holomorphic section ˆ η of theline bundle K S (2∆), where S = C × C and ∆ ⊂ S is the diagonal(see [2, 3]). This leads back the study of the behavior of the secondfundamental form to the study of the 2 − form ˆ η ∈ H ( S, K S (2∆)).1.5. The form ˆ η has been further studied in [1] in relation with pro-jective structures on compact Riemann surfaces. Section 5 of [1] isdedicated to the study of the cohomology class of the form ˆ η and con-tains a characterization of ˆ η as the unique element (up to multiples) ERGMAN KERNEL AND PERIOD MAP FOR CURVES 3 of H ( S, K S (2∆)) with cohomology class in H ( S − ∆) of pure type(1 , η : Theorem B.
The Bergman kernel is the (1 , − harmonic representa-tive of the cohomology class of ˆ η ∈ H ( S, K S (2∆)) in H ( S − ∆ , Z ) .More precisely, there exists α ∈ H ( S, A , (∆)) such that ˆ η − π K = dα, that is ∂α = ˆ η and ¯ ∂α = − π K . (See Theorem 4.1.) It is quite hard to control the behaviour of ˆ η outside of the diagonal. Only along ∆ its behaviour admits an algebraicdescription, via the second Gaussian map µ , see [3]. We expect theabove result to allow some better understanding of ˆ η and the secondfundamental form. Acknowledgements . The idea to study the Lie bracket restricted tomoduli space arose from discussions with Gian Pietro Pirola. The ideaof using Bergman kernel in the study of the map B ∗ was suggested byIndranil Biswas. We heartily thank both of them. We also thank PaolaFrediani for several interesting discussions related to the subject of thispaper. 2. Bergman kernel C be a smooth complex projective curve of genus g ≥ S := C × C and let p, q : S → C be the projections p ( x, y ) = x , q ( x, y ) = y .Let ¯ C denote the conjugate variety and set Z := C × ¯ C . Z coincideswith S as a real manifold, but has a different complex structure. Theprojections p : Z → C , q : Z → ¯ C are holomorphic.Denote by h the Hodge Hermitian product on H ( C, K C ), definedby h ( α, β ) =: i Z α ∧ β. Definition 2.1.
Let ω , ..., ω g be a unitary basis for H ( C, K C ) . Then K := g X j =1 p ∗ ω j ∧ q ∗ ω j is a well-defined (1 , − form on S independent of the choice of theunitary basis. It is called the Bergman kernel form of the algebraiccurve C . ALESSANDRO GHIGI, CAROLINA TAMBORINI
This is the definition of the Bergman kernel form on an arbitrary com-plex manifold due to Kobayashi [7]. It generalizes the classical Bergmankernel on open domains in C n . K can also be seen as a holomorphic2-form on Z . In particular it is a harmonic form with respect to anyK¨ahler metric on Z . If we consider it as a (1,1)-form on S , it is harmonicfor any K¨ahler metric on S which is K¨ahler also on Z . In particular itis harmonic for any product metric.If x, y ∈ C , T ( x,y ) S = T x C ⊕ T y C . Thus elements of T ( x,y ) S are pairs( u, v ) with u ∈ T x C and v ∈ T y C . Since K is a (1,1)-form, its behaviouris controlled by the values K (( u, , (0 , v )) for u ∈ T , x C , v ∈ T , y C .2.2. Although not needed in the following, it is interesting to pointout the following relation between the Bergman kernel and the periodmatrix associated to the algebraic curve C (cf. [10, eq. (2.4)]). Let Q denote the intersection form on H ( C, Z ) and let { a i , b i } be a symplec-tic basis for ( H ( C, Z ) , Q ). Consider a basis ω , ..., ω g of H ( C, K C )normalized with respect to { a i , b i } and the period matrix Z = ( z ij )with z ij = R b j ω i . Then, with respect this basis, the Bergman kernelhas the form K = 12 X i,j (Im Z ) ij p ∗ ω j ∧ q ∗ ω j (2.1)where (Im Z ) ij denote the coefficients of (Im Z ) − .To check this observe first that h ( ω i , ω j ) = 2 Im z ij . Indeed let B = { a ∗ i , b ∗ j } be the dual basis. If D : H ( C ) → H ( C ) is Poincar´e duality,then Da ∗ i = b i and Db ∗ i = − a i , so B is symplectic for Q ∗ ( α, β ) = R C α ∪ β . Since ω i = a ∗ i + P nk =1 z ik b ∗ k , the result follows.Now (2.1) is a consequence of the following general fact: if α , ..., α g is a basis of H ( C, K C ) and A is the matrix with entries a ij := h ( α i , α j ),then K = P i,j a ij p ∗ α i ∧ q ∗ α j .2.3. Next we show how to recover the Bergman kernel using the so-called elementary potentials. Let ( U, z ) be a chart centered at x ∈ C and set u = ∂∂z ( x ) . Classical results ensure the existence of a harmonicfunction f u ∈ C ∞ ( C − { x } ) such that f u = − z + g ( z ) on U − { x } forsome g ∈ C ∞ ( U ). The function f u is unique up to an additive constantand it is called elementary potential (see [2, § ∂f u is smooth on C and that Z C ω ∧ ( − ¯ ∂f u ) = 2 πiω ( u ) , for ω ∈ H ( C, K C ) . (2.2)(See [2, Section 3] for more details.) This shows that elementary poten-tials are related to evaluation and the canonical map. Therefore theyare clearly related to Bergman kernel as we show now. ERGMAN KERNEL AND PERIOD MAP FOR CURVES 5 x ∈ C and u ∈ T x C , let ev u : H ( C, K C ) → C be theevaluation map and let k u ∈ H ( K C ) be such thatev u = h ( · , k u ) . (2.3) Lemma 2.2.
For u ∈ T , x C , v ∈ T , y C , with x, y ∈ C , we have K (( u, , (0 , v )) = h ( k v , k u ) = k v ( u ) . Proof.
Let ω , ..., ω g be a unitary basis for H ( K C ). Then k v = P j λ j ω j , with λ j = h ( k v , ω j ) = ω j ( v ). Thus h ( k v , k u ) = k v ( u ) = P j ω j ( v ) ω j ( u ) = K (( u, , (0 , v )) . (cid:3) Lemma 2.3.
Let x, y ∈ C , and u ∈ T , x C , v ∈ T , y C . If f u is anelementary potential, then ¯ ∂f u = 2 πk u . In particular K (( u, , (0 , v )) = 12 π ¯ ∂f u ( v ) . Proof.
From (2.2) and (2.3) it follows that R C ω ∧ ( ¯ ∂f u ) = − πiω ( u ) = − πi h ( ω, k u ) = 2 π R C ω ∧ k u . So ¯ ∂f u and 2 πk u have the same co-homology class. Since both are harmonic they coincide. Next by theprevious Lemma K (( u, , (0 , v )) = h ( k v , k u ) = h ( k u , k v ) = k u ( v ) = k u ( v ) = π ¯ ∂f u ( v ). (cid:3) Lie bracket
In this section we study the Lie bracket i.e. the tensor B on Siegelspace, introduced in 1.1, and prove Theorem A. We start by recall-ing something about Siegel upper half-space. Next we go throughseveral identications of the tangent space to S g and write down B in terms of them (Proposition 3.4). Given a curve, we apply this toits Jacobian, i.e. we consider B as in 1.2 (Proposition 3.6). We re-call further identifications using the conjugate curve and the surface Z = C × ¯ C . This allows to understand the dual map B ∗ as a map H ( Z, K Z ) → H ( Z, K Z ). Finally using the elementary potentials weprove our main result Theorem 3.7.Let ( V, Q ) be a real symplectic vector space. If J ∈ End V satisfies J = − I V and J ∗ Q = Q , the bilinear form g J ( v, v ′ ) := Q ( v, J v ′ ) issymmetric. Siegel upper half-space is defined as S := S ( V, Q ) := { J ∈ End V : J = − I V , J ∗ Q = Q, g J ≫ } . For every J we denote V − , ( J ) and V , − ( J ) the ± i -eigenspaces of J on V C . We also set H , J := Ann V , − H , J := Ann V − , ( J ) . We usually drop J in the notation. When V = R g and Q is thestandard form, we write S g . The symplectic group Sp := Sp( V, Q ) actson S by conjugation. This action is transitive and S is a Hermitiansymmetric space. For X ∈ End V set Q X := Q ( · , X · ). Then sp = ALESSANDRO GHIGI, CAROLINA TAMBORINI sp ( V, Q ) = { X ∈ End V : Q X is symmetric } . If sp = sp J ⊕ p is theCartan decomposition at J ∈ S , then p = { X ∈ sp : XJ + J X = 0 } , sp J = { X ∈ sp : [ J, X ] = 0 } . We endow p ∼ = T J S with the complex structure ˆ I := (1 / J . Then p C = { X ∈ sp C : X ( V − , ) ⊂ V , − and X ( V , − ) ⊂ V − , } p , = { X ∈ Hom( V , − , V − , ) : Q X is symmetric } , (3.1)3.1. We have an isomorphism ϕ Q : V C −→ V ∗ C , ϕ Q ( v ) := Q ( · , v ) . Its inverse is denoted by ψ Q := ϕ − Q . For any Lagrangian subspace L ⊂ V C the isomorphism ϕ Q maps L onto Ann( L ). Therefore ϕ Q givesan isomorphism V , − ∼ = H , .As mentioned in 1.1 we are interested in the Lie bracket which canbe seen as a section of a bundle over the symmetric space S . We wishto compute B J ∈ Λ T ∗ J S ⊗ sp J . As usual it is useful to look at B thourgh its complexification B J : ( T J S ) C × ( T J S ) C −→ ( sp J ) C . Recall that ( T J S ) C = p C = p , ⊕ p , , where p , is given by (3.1) and( sp J ) C = { X ∈ End V C : X ( V − , ) ⊂ V − , ,X ( V , − ) ⊂ V , − , Q X is simmetric } . (3.2) Lemma 3.1.
The map B J is of type (1 , , i.e. it vanishes on vectorsof the same type.Proof. Since B J is real, it is enough to show that it vanishes on pairsof (1 , X, Y ∈ p , , then X ( V C ) ⊂ V − , and Y | V − , = 0,thus Y X = 0. For the same reason also XY = 0. Thus B J ( X, Y ) = XY − Y X = 0. (cid:3) If X ∈ End V C , let X ∗ ∈ End V ∗ C denote the transpose. The trans-position map X X ∗ is a canonical isomorphism End V C ∼ = End V ∗ C .It is useful to reinterpret everything in terms of End V ∗ C rather thanEnd V C . Set Q ∗ := ψ ∗ Q Q. (Notation as in 3.1.) Then Q ∗ is a symplectic form on V ∗ C . Lemma 3.2. If X ∈ End V C , then Q X is symmetric iff Q ∗ X ∗ is sym-metric.Proof. Define ˜ X ∈ End V C by Q ( Xa, b ) = Q ( a, ˜ Xb ). Then X ∗ ϕ Q a = X ∗ Q ( · , a ) = Q ( X · , a ) = Q ( · , ˜ Xa ) = ϕ Q ˜ Xa.
ERGMAN KERNEL AND PERIOD MAP FOR CURVES 7
Given α, β ∈ V ∗ C let a = ψ Q α, b = ψ Q β . Then − Q ∗ X ∗ ( α, β ) = Q ∗ ( X ∗ α, β ) = Q ( ψ Q X ∗ ϕ Q a, ψ Q ϕ Q b ) == Q ( ˜ Xa, b ) = Q ( a, Xb ) = Q X ( a, b ) . So − Q ∗ X ∗ ( α, β ) = Q X ( a, b ). The statement follows. (cid:3) Lemma 3.3. If X ∈ End V C , then X | V − , = 0 = ⇒ Im X ∗ ⊂ H , Im X ⊂ V − , = ⇒ X ∗ | H , = 0 . Proof.
Recall that for any linear map L : E → F of vector spacesAnn Im L = ker L ∗ . Now X | V − , = 0 = ⇒ V − , ⊂ ker X = ⇒ H , =Ann V − , ⊃ Ann ker X = Im X ∗ . And Im X ⊂ V − , = ⇒ ker X ∗ =Ann Im X ⊃ Ann V − , = H , . (cid:3) Proposition 3.4.
There are canonical isomorphisms p , ∼ = { t ∈ Hom( H , , H , ) : Q ∗ t is symmetric } , (3.3) ( sp J ) C ∼ = End H , . Using these isomorphism B J gets identified with the map B J : p , × p , −→ End H , ( s, ¯ t ) ¯ ts. Proof.
The first isomorphism is simply the restriction to p , of the map X X ∗ . Lemmata 3.2 and 3.3 show that indeed the image of p , is theset of X ∗ ∈ End V ∗ C that vanish on H , , have image in H , and suchthat Q ∗ X ∗ is symmetric. To describe the second isomorphism start from(3.2). Again the Lemmata show that the map X X ∗ sends ( sp J ) C to the set of X ∗ ∈ End V ∗ C that preserve each H p,q and such that Q ∗ X ∗ is symmetric. The latter means that Q ∗ ( u, X ∗ v ) = Q ∗ ( v, X ∗ u ). Thisidentity is trivial if u and v have the same type, since in that case bothterm vanish. Hence X ∗ | H , is an arbitrary endomorphism of H , . Onthe contrary the identity shows that for any v ∈ H , the value X ∗ v isdetermined by X ∗ | H , . Hence( sp J ) C −→ End H , , X X ∗ | H , (3.4)is the desired isomorphism. Now let X, Y ∈ p , and set s := X ∗ , t := Y ∗ . Then B ( X, Y ) ∗ = ¯ ts − s ¯ t . Since ¯ t | H , = 0, in the isomorphism(3.4) B ( X, Y ) ∈ ( sp J ) C corresponds to B ( X, Y ) ∗ | H , = ¯ ts . (cid:3) Since H , = Ann V , − , we have Ann H , = V , − . So there is acanonical isomorphism ( H , ) ∗ ∼ = V C / Ann H , = V C /V , − ∼ = V − , .We treat this isomorphism as an identity. By 3.1 ϕ Q maps V − , iso-morphically onto H , . Thus ψ Q = ϕ − Q restricts to an isomorphism ψ Q : H , ∼ = −→ ( H , ) ∗ . (3.5) Lemma 3.5.
For ¯ ω ∈ H , , we have ψ Q (¯ ω ) = Q ∗ (¯ ω, · ) . ALESSANDRO GHIGI, CAROLINA TAMBORINI
Proof.
First we claim that ϕ Q ∗ ϕ Q = − id V C i.e. ψ Q = − ϕ Q ∗ . Indeedfix v ∈ V C . That ϕ Q ∗ ϕ Q ( v ) = − v means that Q ∗ ( · , ϕ Q ( v )) = − v , i.e.that Q ∗ ( λ, ϕ Q ( v )) = − λ ( v ) for any λ ∈ V ∗ C . Assume λ = ϕ Q ( w ) for w ∈ V C . Then λ ( v ) = Q ( v, w ) and Q ∗ ( λ, ϕ Q ( v )) = Q ∗ ( ϕ Q ( w ) , ϕ Q ( v )) = Q ( w, v ). (cid:3) Now consider the period map j : M g −→ A g . Let x ∈ M g bethe moduli point of a curve C : x = [ C ]. If we fix a symplecticbasis of H ( C, Z ) we get a sympletic isomorphism of H ( C, R ) withthe intersection form onto ( R g , Q ). Thus the Hodge decomposition H ( C, C ) = H ( C ) ⊕ H , ( C ) gives a complex structure on H ( C, C ),hence a point J ∈ S g .In the following we use T x M g to denote the real tangent space i.e.the tangent space of M g as a differentiable manifold (and similarly for A g ). Thus ( T x M g ) C = T , x M g ⊕ T , x M g and T , x M g = H ( C, T C ), while( T x A g ) C = ( T J S g ) C = p , ⊕ p , . By a theorem of Griffiths the map dj x : H ( C, T C ) −→ p , using the interpretation (3.3) is given by dj x ( ξ ) = ξ ∪ · : H , −→ H , , dj x ( ξ )( ω ) = ξ ∪ ω. (See e.g. [9, pp. 234ff].) Now T , x M g = H ( C, T C ) is the conju-gate vector space, i.e. it has the same underlying real vector space as H ( C, T C ) but multiplication by i is replaced with multiplication by − i . Since j is holomorphic, its differential is a direct sum of the map dj x : H ( C, T C ) → p , and its conjugate. Hence for ¯ η ∈ H ( C, T C ) and ω ∈ H , ( C ) we have dj x (¯ η ) = ¯ η ∪ · : H , −→ H , , dj x (¯ η )(¯ ω ) = η ∪ ω. As mentioned in the Introduction our goal is to study the map B x := dj ∗ x B J : H ( C, T C ) × H ( C, T C ) −→ End H , ( C ) . The following is a consequence of Proposition 3.4.
Proposition 3.6.
For ξ, η ∈ H ( C, T C ) and ω ∈ H , ( C ) , we have B ( ξ, ¯ η )( ω ) = ¯ η ∪ ( ξ ∪ ω ) . Once again it is useful to dualize. This time we dualize the map B itself. Using (3.5) we can describe the domain of B ∗ as follows:(End H , ) ∗ = ( H , ∗ ⊗ H , ) ∗ = H , ⊗ H , ∗ ∼ = H , ⊗ H , . More explicitely, let ω, ω ∈ H , and t ∈ End H , . Then ω ⊗ ¯ ω ′ ∈ H , ⊗ H , . Recalling Lemma 3.5 one easily verifies that the correspondingelement of (End H , ) ∗ is the linear functional mapping t to Q ∗ (¯ ω ′ , tω ).The dual of H ( C, T C ) is H ( C, K C ). Thus the dual of B is definedon H ( C, K C ) ⊗ H ( C, K C ) and maps to H , ( C ) ⊗ H , ( C ).Denoting by ¯ C the conjugate variety we have H ( C, K C ) = H ( ¯ C, K ¯ C ) , H , ( C ) = H , ( ¯ C ) . ERGMAN KERNEL AND PERIOD MAP FOR CURVES 9
Thus B ∗ is a map from H , ( C ) ⊗ H , ( ¯ C ) to H ( C, K C ) ⊗ H ( ¯ C, K ¯ C ).We further reinterpret domain and target of B ∗ as spaces of sectionsof appropriate bundles on Z = C × ¯ C . Denoting by p : Z → C and q : Z → ¯ C the projections and given bundles L → C and M → ¯ C , set L ⊠ M := p ∗ L ⊗ q ∗ M . The map H ( C, L ) ⊗ H ( ¯ C, M ) −→ H ( Z, L ⊠ M ) , s ⊗ t p ∗ s ⊗ q ∗ t, is an isomorphism. For any positive integer n there is a canonicalisomorphism K nZ ∼ = K nC ⊠ K n ¯ C , (3.6)obtained as follows: if α ∈ K C,x and β ∈ K ¯ C,y , denote by α n ∈ ( K C,x ) ⊗ n and β n ∈ ( K ¯ C,y ) ⊗ n the tensor powers. Then α n ⊗ β n ∈ ( K nC ⊠ K n ¯ C ) ( x,y ) ,while ( p ∗ α ∧ q ∗ β ) n ∈ K Z, ( x,y ) . The isomorphism (3.6) maps α n ⊗ β n to( p ∗ α ∧ q ∗ β ) n .We now prove Theorem A. Theorem 3.7.
The map B ∗ : H ( Z, K Z ) −→ H ( Z, K Z ) coincides with the multiplication by − i K .Proof. Fix a point x ∈ C and a chart ( U, z ) centered in x . Set u = ∂∂z ( x )and consider the Schiffer variation ξ u at x ∈ C . We recall that ξ u ∪ = − π ev u ⊗ k u . (3.7)Indeed fix a Dolbeault representative ϕ = ¯ ∂bz ∂∂z , where b ∈ C ∞ isa bump function which is equal to 1 in a neighbourhood of x . For ω ∈ H ( K C ), with local expression ω = h ( z ) dz on U , it holds that dj x ( ξ u )( ω ) = ξ u ∪ ω = [ ϕ · ω ] = (cid:20) ¯ ∂ ( bh ) z (cid:21) . If f u is an elementary potential, the functions bz + f u and b · h − h (0) z are smooth on C . Hence¯ ∂ (cid:18) bhz (cid:19) = ¯ ∂ (cid:18) b · h − h (0) z (cid:19) + h (0) · ¯ ∂ (cid:18) bz + f u (cid:19) − h (0) ¯ ∂f u . Thus dj x ( ξ u )( ω ) = h (0) · [ − ¯ ∂f u ] . As usual we identify H , ( C ) with thespace of antiholomorphic forms. Thus since h (0) = ω ( u ), using Lemma2.3 and the fact that k u is antiholomorphic, we get (3.7).We also recall (see [2, Lemma 2.3]) that for β ∈ H ( C, K C ) = H ( C, T C ) ∗ , we have β ( ξ u ) = 2 πiβ ( u ). It follows that for Φ ∈ H ( Z, K Z )Φ(( u, , (0 , ¯ v )) = − π Φ( ξ u ⊗ ξ ¯ v ) . Now we can prove the statement. Without loss of generality we canassume that Ω = p ∗ ω ∧ q ∗ ω ′ , with ω, ω ′ ∈ H ( K C ). Then( B ∗ Ω) ( x,y ) (( u, , (0 , v )) = − π B ∗ ( p ∗ ω ∧ q ∗ ω ′ )( ξ u ⊗ ξ v ) == − π ( p ∗ ω ∧ q ∗ ω ′ )( B ( ξ u ⊗ ξ v )) . It follows from (3.7) that ξ v ∪ ( ξ u ∪ ω ) = − πω ( u ) · ξ v ∪ k u = − πω ( u ) · ξ v ∪ k u == 4 π ω ( u ) · k u ( v ) · k v . Using this and Lemma 3.5 we get( B ∗ Ω) ( x,y ) (( u, , (0 , v )) = − π Q ∗ ( ω ′ , B ( ξ u ⊗ ξ v ) ω ) == − π Q ∗ ( ω ′ , ξ v ∪ ( ξ u ∪ ω )) = − ω ( u ) · k u ( v ) · Q ∗ ( ω ′ , k v ) . Since Q ∗ ( ω ′ , k v ) = i · ω ′ ( v ) and using Lemma2.2 we finally get( B ∗ Ω) ( x,y ) (( u, , (0 , v )) = − iω ( u ) ω ′ ( v ) · k u ( v ) == − i · (Ω · K ) ( x,y ) (( u, , (0 , v )) (cid:3) The form ˆ η Fix a smooth complex projective curve C of genus g > ⊂ S = C × C be the diagonal. In this section we recall the definitionof the meromorphic form ˆ η ∈ H ( C, K S (2∆)) constructed in [3, 2],which governs the second fundamental form of the Torelli map withrespect to the Siegel metric. Next we recall from [1] the analysis ofits cohomology class. Finally we prove our second main result, i.e.Theorem B.4.1. The construction of the form ˆ η goes as follows. For x ∈ C , let j x : H ( C, K C (2 x )) ֒ → H ( C − { x } , C ) = H ( C, C )be the map that associates to ω ∈ H ( C, K C (2 x )) its de Rham coho-mology class. This map is an injection since C = P . As H , ( C ) ⊂ j x ( H ( C, K C (2 x ))) and h ( C, K C (2 x )) = g +1, the preimage j − x ( H , ( C ))is a line. Thus, fixed a chart ( U, z ) centered at x ∈ C , there exists aunique element ϕ in this line such that on U − { x } ϕ = (cid:18) z + h ( z ) (cid:19) dz with h ∈ O C ( U ). Set u = ∂∂z ( x ) , and define the map η x : T , x C → H ( C, K C (2 x )) , λu η x ( λu ) := λϕ. ERGMAN KERNEL AND PERIOD MAP FOR CURVES 11
It is easy to see that η x does not depend on the choice of the localcoordinate. In the following we will also use the fact that if f u is anelementary potential, then ∂f u = η u , see [2, Lemma 3.1].Next consider the line bundle L := K S (2∆) on S and set V := p ∗ ( q ∗ K C (2∆)) , E := p ∗ L. By the projection formula E = K C ⊗ V . Also, since q ∗ K C (2∆) | { x }× C = q ∗ K C (2 x ), we have that H ( p − ( x ) , q ∗ K C (2∆)) ≃ H ( C, K C (2 x )) andthe fiber of the holomorphic vector bundle V → C on x ∈ C is iso-morphic to H ( C, K C (2 x )). Thus η x ∈ E x . More precisely, the map x η x is a holomorphic section of E ([2, Proposition 3.4]).Finally, since E = p ∗ L , there is an isomorphism between H ( C, E )and H ( S, L ) that associates to α ∈ H ( C, E ) the section ˆ α of L suchthat α x = ˆ α | { x }× C ∈ E x . The form ˆ η ∈ H ( S, K S (2∆)) is defined asthe holomorphic section of L corresponding to η ∈ H ( C, E ). Notethat, in particular, for u ∈ T , x C and v ∈ T , x C with x = y , it holds η x ( u )( v ) = ˆ η ( u, v ) . (4.1)4.2. The form ˆ η also appears in an unpublished book of Gunning [5]under the name of intrinsic double differential of the second kind .4.3. The importance of ˆ η comes from the fact that the second funda-mental form of the Torelli map outside the hyperelliptic locus coincideswith the multiplication by ˆ η [3, 2]. The form ˆ η has been further stud-ied in [1] in relation with projective structures on compact Riemannsurfaces. Moreover Section 5 in [1] contains an analysis of the coho-mology class of the form ˆ η . Denoting j : S − ∆ ֒ → S the inclusionmap, it follows from the exact sequence of homology groups for thepair ( S, ∆) and Poincar ˜A © and Lefschetz dualities, that the homo-morphism j ∗ : H ( S, Z ) → H ( S − ∆ , Z ) is surjective and its kernelis generated by the (pure) class of the diagonal. Consequently, theHodge decomposition of H ( S ) induces a decomposition of H ( S − ∆).In particular, for any ζ ∈ H ( C, K S (2∆)) there is [ γ ] ∈ H ( S ) suchthat [ ζ ] = j ∗ [ γ ] ∈ H ( S − ∆). Moreover the (0 ,
2) part of [ γ ] vanishes.So [ γ ] = γ , + γ , where γ , is holomorphic and γ , is harmonic of type (1 , η is that[ˆ η ] = j ∗ [ γ , ] ∈ H ( S − ∆) , where γ , is a harmonic (1 , − form on S . That is, ˆ η has cohomologyclass in H ( S − ∆) of pure type (1 , η as the unique element (up to multiples) of H ( S, K S (2∆)) with cohomology class in H ( S − ∆) of pure type (1 , γ , of ˆ η .4.4. Denote by A p,q the sheaf of smooth differential forms of type( p, q ) on S . Denote by A p,q ( n ∆) be the sheaf of ( p, q ) − forms havinga pole of order at most n on ∆, i.e. those forms ω such that x n ω issmooth of type ( p, q ), where x = 0 is a local equation of ∆.For u ∈ ( T x C ) C , denote by u , the (1 , − component of u and set f u := f u , . For u ∈ ( T x C ) C and v ∈ ( T y C ) C , set α ( u, v ) := 2 f v ( p ) + f u ( q )It is clear that α ∈ H ( S, A , (∆)).We can now prove Theorem B. Theorem 4.1.
The Bergman kernel is the (1 , − harmonic represen-tative of the cohomology class of ˆ η ∈ H ( S, K S (2∆)) in H ( S − ∆) .More precisely, ˆ η − π K = dα, that is ∂α = ˆ η and ¯ ∂α = − π K .Proof. We first observe that for u ∈ ( T x C ) C and v ∈ ( T y C ) C , denotedby U, V two vector fields on C such that U x = u and V y = v , since[( U, , (0 , V )] = 0, we have that dα (( u, , (0 , v )) = ( U, α (0 , v )) − (0 , V )( α ( u, . Now assume u ∈ T , x C , v ∈ T , y C and that U and V are (1,0). For x = y , ∂f u = η x ( u ) and we get ∂α (( u, , (0 , v )) = 2( U, f v − (0 , V ) f u == 2 ∂f v ( u ) − ∂f u ( v ) = 2 η y ( v )( u ) − η x ( u )( v ) = ˆ η (( u, , (0 , v )) , where for the last equality we used (4.1) and the symmetry of ˆ η (see[2, Lemma 3.5]). Similarly using Lemma 2.3¯ ∂α (( u, , (0 , v )) = − (0 , V ) f u = − ¯ ∂f u ( v ) = − π K (( u, , (0 , v )) . (cid:3) References [1] I. Biswas, E. Colombo, P. Frediani, and G.P. Pirola,
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