In this section, we prove our main theorem, characterising projective special Kähler manifolds in terms of the deviance. We start by deriving necessary conditions on the deviance, reflecting the curvature conditions of Proposition 3.2.
Proposition 7.1
For a projective special Kähler manifold ((pi :{widetilde{M}}rightarrow M,nabla )) with (({widetilde{M}},{widetilde{g}},{widetilde{I}},{widetilde{omega }},nabla ,xi )), and a local section (s:Urightarrow S), then the corresponding deviance (eta) satisfies
$$begin{aligned} d^{LC}eta =2itau wedge eta end{aligned}$$
where (tau =s^*varphi in Omega ^1(U)).
Proof
Thanks to Proposition 6.3, we know that there exists (z=re^{ivartheta }) and (eta in T_{1,0}Uotimes T^{0,1}Uotimes T_{1,0}U) such that on (pi ^{-1}(U)) we have ({widetilde{eta }}={mathfrak {R}}(z^2pi ^*eta )).
Now we would like to describe ({widetilde{d}}^{LC}{widetilde{eta }}) in terms of (d^{LC}eta). Notice that
$$begin{aligned} begin{aligned} {widetilde{d}}^{LC}{widetilde{eta }}&={widetilde{d}}^{LC}{mathfrak {R}}(z^2pi ^*eta ) ={mathfrak {R}}({widetilde{d}}^{LC}(z^2pi ^*eta )) ={mathfrak {R}}(2zdzwedge pi ^*eta +z^2{widetilde{d}}^{LC}pi ^*eta )\&={mathfrak {R}}left( z^2left( 2(frac{1}{r}dr+idvartheta ) wedge pi ^*eta +{widetilde{d}}^{LC}pi ^*eta right) right) . end{aligned} end{aligned}$$
(9)
The next step is to compute ({widetilde{d}}^{LC}pi ^*eta), but since we are using the Levi-Civita connection, it is equivalent to compute (sharp _2({widetilde{d}}^{LC}pi ^*sigma )), where (sigma =flat _2 eta in S_{3,0}U). Let us consider a local coframe (theta) in M and the corresponding lifting ({widetilde{theta }}) as in (4), so that we can denote explicitly (sigma =sigma _{k,j,h}theta ^kotimes theta ^{j}otimes theta ^{h}). We have
$$begin{aligned} {widetilde{nabla }}^{LC}pi ^*theta ^k&={widetilde{nabla }}^{LC}frac{{widetilde{theta }}^k}{r} =-frac{dr}{r^2}otimes {widetilde{theta }}^k-frac{1}{r}left( ({widetilde{omega }}^{LC})^{k}_{j}otimes {widetilde{theta }}^{j}right) \&=-frac{dr}{r}otimes pi ^*theta ^k-frac{1}{r}left( sum _{j=1}^{n}pi ^*(omega ^{LC})^{k}_{j}otimes {widetilde{theta }}^{j}+i{widetilde{varphi }}otimes {widetilde{theta }}^j+pi ^*theta ^kotimes theta ^{n+1}right) \&=-frac{dr}{r}otimes pi ^*theta ^k-pi ^*left( (omega ^{LC})^{k}_{j}otimes theta ^{j}right) -i{widetilde{varphi }}otimes pi ^*theta ^j-pi ^*theta ^kotimes frac{1}{r}theta ^{n+1}\&=pi ^*left( nabla ^{LC} theta ^{k}right) – frac{1}{r}theta ^{n+1}otimes pi ^*theta ^k-pi ^*theta ^kotimes frac{1}{r}theta ^{n+1}. end{aligned}$$
We can now compute the following for (Xin {mathfrak {X}}{left( {{pi ^{-1}(U)}} right) }):
$$begin{aligned} {widetilde{nabla }}^{LC}_X&pi ^*sigma ={widetilde{nabla }}^{LC}_Xpi ^*(sigma _{k,j,h}theta ^kotimes theta ^{j}otimes theta ^{h}) ={widetilde{nabla }}^{LC}_X(pi ^*sigma _{k,j,h}pi ^*theta ^kotimes pi ^*theta ^{j}otimes pi ^*theta ^{h})\&=dpi ^*sigma _{k,j,h}(X)theta ^kotimes theta ^{j}otimes theta ^{h} +pi ^*sigma _{k,j,h}left( {widetilde{nabla }}^{LC}_Xpi ^*theta ^kotimes pi ^*theta ^{j}otimes pi ^*theta ^{h}right. \&quad left. +pi ^*theta ^kotimes {widetilde{nabla }}^{LC}_Xpi ^*theta ^{j}otimes pi ^*theta ^{h} +pi ^*theta ^kotimes pi ^*theta ^{j}otimes {widetilde{nabla }}^{LC}_Xpi ^*theta ^{h}right) \&=pi ^*dsigma _{k,j,h}(X)theta ^kotimes theta ^{j}otimes theta ^{h} +pi ^*sigma _{k,j,h}pi ^*left( nabla ^{LC} theta ^{k}right) _X otimes pi ^*theta ^{j}otimes pi ^*theta ^{h}\&quad +pi ^*sigma _{k,j,h}pi ^*theta ^kotimes pi ^*left( nabla ^{LC} theta ^{j}right) _Xotimes pi ^*theta ^{h}\&quad +pi ^*sigma _{k,j,h}pi ^*theta ^kotimes pi ^*theta ^{j}otimes pi ^*left( nabla ^{LC} theta ^{j}right) _X -frac{3}{r}{widetilde{theta }}^{n+1}(X)pi ^*sigma \&quad -frac{1}{r}left( pi ^*sigma _{k,j,h}pi ^*theta ^k(X){widetilde{theta }}^{n+1}otimes pi ^*theta ^{j}otimes pi ^*theta ^{h}right. \&quad left. +pi ^*sigma _{k,j,h}pi ^*theta ^kotimes pi ^*theta ^{j}(X){widetilde{theta }}^{n+1}otimes pi ^*theta ^{h}right. \&quad left. +pi ^*sigma _{k,j,h}pi ^*theta ^kotimes pi ^*theta ^{j}otimes pi ^*theta ^{h}(X){widetilde{theta }}^{n+1}right) \&=pi ^*left( nabla ^{LC}sigma right) _X -frac{2}{r}{widetilde{theta }}^{n+1}(X)pi ^*sigma -frac{1}{r}{widetilde{theta }}^{n+1}(X)pi ^*sigma -frac{1}{r}{widetilde{theta }}^{n+1}otimes pi ^*sigma (X,cdot ,cdot )\&quad -frac{1}{r}pi ^*sigma (cdot ,Xotimes {widetilde{theta }}^{n+1},cdot ) -frac{1}{r}pi ^*sigma (cdot ,cdot ,Xotimes {widetilde{theta }}^{n+1}). end{aligned}$$
In general then, if (sigma =theta ^kotimes sigma _k), where (sigma _k=sigma _{k,j,h}theta ^jotimes theta ^hin S_{2,0}U), we have by symmetry
$$begin{aligned} {widetilde{nabla }}^{LC}pi ^*sigma&=pi ^*left( nabla ^{LC}sigma right) -frac{2}{r}{widetilde{theta }}^{n+1}otimes pi ^*sigma -frac{2}{r}(({widetilde{theta }}^{n+1})(pi ^*theta ^k))otimes pi ^*(sigma _{k,j,h}theta ^jotimes theta ^h)\&quad -frac{2}{r}left( pi ^*(sigma _{k,j,h}theta ^kotimes theta ^j)otimes (({widetilde{theta }}^{n+1})(pi ^*theta ^h))right) . end{aligned}$$
Notice in particular that the last two rows are symmetric in the first two indices.
In order to compute ({widetilde{d}}^{LC}pi ^*sigma) we need to antisymmetrise ({widetilde{nabla }}^{LC}pi ^*sigma) in the first two indices and multiply by two, so only the first row survives and we get
$$begin{aligned} {widetilde{d}}^{LC}pi ^*sigma =pi ^*(d^{LC}sigma )-frac{2}{r}{widetilde{theta }}^{n+1}wedge pi ^*sigma , end{aligned}$$
and therefore
$$begin{aligned} {widetilde{d}}^{LC}pi ^*eta =pi ^*(d^{LC}eta )-frac{2}{r}{widetilde{theta }}^{n+1}wedge pi ^*eta . end{aligned}$$
Substituting this value in (9), we obtain
$$begin{aligned} {widetilde{d}}^{LC}{widetilde{eta }}&={mathfrak {R}}left( z^2left( 2(frac{1}{r}dr+idvartheta ) wedge pi ^*eta +pi ^*(d^{LC}eta )-frac{2}{r}{widetilde{theta }}^{n+1}wedge pi ^*eta right) right) \&={mathfrak {R}}left( z^2left( pi ^*d^{LC}eta -2i({widetilde{varphi }}-dvartheta ) wedge pi ^*eta right) right) . end{aligned}$$
As observed in Remark 6.8, ({widetilde{varphi }}-dvartheta =pi ^*tau), so we have
$$begin{aligned} {widetilde{d}}^{LC}{widetilde{eta }}&={mathfrak {R}}left( z^2pi ^*left( d^{LC}eta -2itau wedge eta right) right) . end{aligned}$$
From Proposition 3.2, we know that ({widetilde{d}}^{LC}{widetilde{eta }}=0), and since (eta in Omega ^1(U,T_{0,1}otimes T^{1,0})), (eta) and ({overline{eta }}) are linearly independent, so this quantity vanishes if and only if (z^2 pi ^*left( d^{LC}eta -2itau wedge eta right)) does. Therefore,
$$begin{aligned} d^{LC}eta -2itau wedge eta =0, end{aligned}$$
ending the proof. (square)
Let us now look at the final ingredient of the curvature tensor, that is (frac{1}{2}[{widetilde{eta }}wedge {widetilde{eta }}]). In the setting of Proposition 6.3, given a section (s:Urightarrow S), and the induced deviance (eta), then
$$begin{aligned} frac{1}{2}[{widetilde{eta }}wedge {widetilde{eta }}]&=frac{1}{2}[{mathfrak {R}}(z^2pi ^*eta )wedge {mathfrak {R}}(z^2pi ^*eta )] =frac{1}{2}[z^2pi ^*eta +{overline{z}}^2pi ^*{overline{eta }}wedge z^2pi ^*eta +{overline{z}}^2pi ^*{overline{eta }}]\&=frac{1}{2}{mathfrak {R}}left( z^4[pi ^*eta wedge pi ^*eta ]right) +|z|^4[pi ^*eta wedge pi ^*{overline{eta }}]. end{aligned}$$
We can compute this tensor for a local coframe (theta) on M. Since we have
$$begin{aligned} pi ^*theta ^kcirc pi ^*theta _h =frac{1}{r}{widetilde{theta }}^k(frac{1}{r}{widetilde{theta }}_h) =frac{1}{r^2}{widetilde{theta }}^k({widetilde{theta }}_h) =frac{1}{r^2}delta ^k_h =frac{1}{r^2}pi ^*(theta ^kcirc theta _h) end{aligned}$$
and (pi ^*theta ^kcirc pi ^*overline{theta _h}=pi ^*overline{theta ^k}circ pi ^*theta _h=0), then
$$begin{aligned}{}[pi ^*eta wedge pi ^*eta ]&=[pi ^*eta ^{j}_{k,h}pi ^*theta ^{k}otimes pi ^*overline{theta _{j}}otimes pi ^*theta ^{h}wedge pi ^*eta ^{j’}_{k’,h’}pi ^*theta ^{k’}otimes pi ^*overline{theta _{j’}}otimes pi ^*theta ^{h’}]\&=pi ^*eta ^{j}_{k,h}pi ^*theta ^{k}wedge pi ^*eta ^{j’}_{k’,h’}pi ^*theta ^{k’}otimes [pi ^*overline{theta _{j}}otimes pi ^*theta ^{h},pi ^*overline{theta _{j’}}otimes pi ^*theta ^{h’}] =0 end{aligned}$$
and
$$begin{aligned}{}[pi ^*eta wedgepi ^*{overline{eta }}] &=[pi ^*eta ^{j}_{k,h}pi ^*theta ^{k}otimes pi ^*overline{theta _{j}}otimes pi ^*theta ^{h}wedge pi ^*overline{eta ^{j’}_{k’,h’}}pi ^*overline{theta ^{k’}}otimes pi ^*theta _{j’}otimes pi ^*overline{theta ^{h’}}]\&=pi ^*eta ^{j}_{k,h}pi ^*theta ^{k}wedge pi ^*overline{eta ^{j’}_{k’,h’}}pi ^*overline{theta ^{k’}}otimes [pi ^*overline{theta _{j}}otimes pi ^*theta ^{h},pi ^*theta _{j’}otimes pi ^*overline{theta ^{h’}}]\&=pi ^*(eta ^{j}_{k,h}theta ^{k}wedge overline{eta ^{j’}_{k’,h’}}overline{theta ^{k’}})otimes frac{1}{r^2}pi ^*(overline{theta _{j}}otimes theta ^{h}(theta _{j’})otimes overline{theta ^{h’}}-theta _{j’}otimes overline{theta ^{h’}}(overline{theta _{j}})otimes theta ^{h})\&=frac{1}{r^2}pi ^*[eta wedge {overline{eta }}]. end{aligned}$$
Therefore
$$begin{aligned} frac{1}{2}[{widetilde{eta }}wedge {widetilde{eta }}] =frac{|z|^4}{r^2}pi ^*[eta wedge {overline{eta }}] =r^2pi ^*[eta wedge {overline{eta }}]. end{aligned}$$
(10)
Remark 7.2
Note that ([eta wedge {overline{eta }}]) is independent on the local coframe, and if we consider another section such that (s’=sa) on the intersection of their domains, with a taking values in (S^1), if (eta ‘) is the deviance corresponding to (s’), then ([eta ‘wedge overline{eta ‘}]=[eta awedge {overline{eta }}{overline{a}}]=|a|^2[eta wedge {overline{eta }}]=[eta wedge {overline{eta }}]). So, there is a globally defined section (Mrightarrow S^2(mathfrak {u}(n))) mapping p to ([eta _pwedge overline{eta _p}]).
For a projective special Kähler manifold ((pi :{widetilde{M}}rightarrow M,nabla )) of real dimension 2n, Proposition 3.2, interpreted in the light of the last observations and the ones made in Section 5 (see Remark 5.4), says that (0=r^2pi ^*(Omega ^{LC}+Omega _{{mathbb {P}}_{{mathbb {C}}}^n}+[eta wedge {overline{eta }}])); thus, we have the following equation:
$$begin{aligned} Omega ^{LC}+Omega _{{mathbb {P}}_{{mathbb {C}}}^n}+[eta wedge {overline{eta }}]=0. end{aligned}$$
(11)
This is a curvature tensor, so we can compute its Ricci and scalar component.
Proposition 7.3
Let ((pi :{widetilde{M}}rightarrow M,nabla )) be a projective special Kähler manifold of dimension 2n, then
$$begin{aligned} mathrm {Ric}_M(X,Y)+2(n+1)g(X,Y)-{mathfrak {R}}(h(overline{eta _X},eta _Y))=0; end{aligned}$$
(12)
$$begin{aligned} mathrm {scal}_M+2(n+1)-frac{2}{n}leftVert {eta }rightVert _h^2=0. end{aligned}$$
(13)
Proof
The first summand in (11) gives the Ricci tensor of M, the second gives the Ricci tensor of the projective space (6). In order to compute the last term, consider a unitary frame (theta); from previous computations,
$$begin{aligned}{}[eta wedge {overline{eta }}]&=(eta ^{j}_{k,h}theta ^{k}wedge overline{eta ^{j’}_{k’,h’}}overline{theta ^{k’}})otimes (delta ^{h}_{j’}overline{theta _{j}}otimes overline{theta ^{h’}}-delta ^{h’}_{j}theta _{j’}otimes theta ^{h})\&={mathfrak {R}}left( eta ^{j}_{k,h}overline{eta ^{h}_{k’,h’}}theta ^{k}wedge overline{theta ^{k’}}otimes overline{theta _{j}}otimes overline{theta ^{h’}}right) end{aligned}$$
then the Ricci component (mathrm {Ric}([eta wedge {overline{eta }}])) evaluated on (X={mathfrak {R}}(X^ktheta _k)) and (Y={mathfrak {R}}(Y^ktheta _k)) is the trace of ([eta wedge {overline{eta }}](cdot ,Y)X), which is
$$begin{aligned}{}[eta wedge{overline{eta }}](cdot ,Y)X&=eta ^{j}_{k,h}overline{eta ^{h}_{u,v}}(theta ^{k} overline{Y^{u}}-Y^k overline{theta ^{u}})otimes overline{theta _{j}}otimes overline{X^{v}}+ overline{eta ^{j}_{k,h}}eta ^{h}_{u,v}(overline{theta ^{k}} Y^{u}-overline{Y^{u}}theta ^{k})otimes theta _{j}otimes X^{v}\&={mathfrak {R}}left( eta ^{j}_{k,h}overline{eta ^{h}_{u,v}}(theta ^{k} overline{Y^{u}}-Y^k overline{theta ^{u}})otimes overline{theta _{j}}otimes overline{X^{v}}right) . end{aligned}$$
Its trace is therefore
$$begin{aligned} -{mathfrak {R}}left( eta ^{j}_{k,h}overline{eta ^{h}_{j,v}}Y^k overline{X^{v}}right) =-{mathfrak {R}}left( eta ^{j}_{k,h}overline{eta ^{h}_{u,j}}Y^k overline{X^{u}}right) =-{mathfrak {R}}(h(overline{eta _X},eta _Y)), end{aligned}$$
or equivalently, (mathrm {Ric}([eta wedge {overline{eta }}])=-{mathfrak {R}}left( overline{eta ^{h}_{u,j}}eta ^{j}_{k,h}overline{theta ^{u}}theta ^k right)). Thus, we obtain (12).
From this tensor, we can now obtain (13) by computing the scalar component, that is by taking the trace, raising the indices with g and then dividing it by the dimension of M. Thus, the first summand gives (mathrm {scal}_M), the second gives (2(n+1)) and the third
$$begin{aligned} frac{1}{2n}mathrm {tr}left( -{mathfrak {R}}left( overline{eta ^{h}_{u,j}}eta ^{j}_{k,h}(overline{theta ^{u}})_sharp theta ^k right) right)&=-frac{1}{2n}mathrm {tr}left( {mathfrak {R}}left( overline{eta ^{h}_{u,j}}eta ^{j}_{k,h}(2theta _{u})theta ^k right) right) \&=-frac{1}{n}sum _{j,h,k} {mathfrak {R}}left( eta ^{j}_{k,h}overline{eta ^{h}_{k,j}}right) =-frac{2}{n}leftVert {eta }rightVert _h^2. end{aligned}$$
(square)
In particular, since the norm of (eta) is non-negative, we obtain a lower bound for the scalar curvature:
Corollary 7.4
Let ((pi :{widetilde{M}}rightarrow M,nabla )) be a projective special Kähler manifold, then
$$begin{aligned} mathrm {scal}_Mge -2(n+1). end{aligned}$$
Equality holds at a point if and only if the deviance vanishes at that point.
Remark 7.5
The lower bound is reached by projective special Kähler manifolds with zero deviance; we will see that this condition characterises the complex hyperbolic space (Proposition 9.5).
We can now state the main result:
Theorem 7.6
On a 2n-dimensional Kähler manifold ((M,g,I,omega )), to give a projective special Kähler structure is equivalent to give an (S^1)-bundle (pi _S:Srightarrow M) endowed with a connection form (varphi) and a bundle map (gamma :Srightarrow sharp _2 S_{3,0} M) such that:
- 1.
(dvarphi =-2pi _S^*omega);
- 2.
(gamma (u a)=a^2gamma (u)) for all (ain S^1);
- 3.
for a certain choice of an open covering ({U_alpha |alpha in {mathcal {A}}}) of M and a family ({s_alpha :U_alpha rightarrow S}_{alpha in {mathcal {A}}}) of sections, denoting by (eta _alpha) the local 1-form taking values in (T^{0,1}Motimes T_{1,0}M) determined by (gamma circ s_{alpha }), for all (alpha in {mathcal {A}}):
(bf {D1} qquad {Omega ^{LC}+Omega _{{mathbb {P}}^n_{{mathbb {C}}}}+[eta_alpha wedge overline{eta _alpha }]=0});
({bf {D2} qquad d^{LC}eta _alpha =2i s_{alpha }^*varphi wedge eta _alpha}).
In this case, 3 is satisfied by every such family of sections.
Proof
Given a projective special Kähler manifold, we define (S:=r^{-1}(1)subset {widetilde{M}}) and (varphi :=-iota _xi omega |_{S}). The principal action on S is generated by (Ixi) which is tangent to S since (T_uS=ker (dr)) and (dr(Ixi )=-frac{1}{r}xi ^flat (Ixi )=-frac{{widetilde{g}}(xi ,Ixi )}{r}). The curvature is then (dvarphi =-2pi _S^*omega) as shown in Remark 4.4, so the first point is satisfied. The second condition holds thanks to Proposition 6.5. For the third point, we get D1 from the arguments leading to equation (11) and D2 from Proposition 7.1.
In order to prove the other direction, define ({widetilde{M}}:=Stimes {mathbb {R}}^+), (pi :=pi _Scirc pi _1:{widetilde{M}}rightarrow M), and (t:=pi _2in {mathcal {C}}^{infty }{left( {{widetilde{M}},{mathbb {R}}^+} right) }), where (pi _1:Stimes {mathbb {R}}^+rightarrow S) and (pi _2:Stimes {mathbb {R}}^+rightarrow {mathbb {R}}^+) are the projections. Let ({widetilde{varphi }}:=pi _1^*varphi), in particular (d{widetilde{varphi }}=pi _1^*dvarphi =-2pi ^*omega) as expected. Define now
$$begin{aligned} {widetilde{g}}:=t^2pi ^*g-t^2{widetilde{varphi }}^2-dt^2 end{aligned}$$
which is non-degenerate, since (r{widetilde{varphi }}) and dt are linearly independent and transverse to (pi), so we can form a basis for the 1-forms according to which we can see that ({widetilde{g}}) has signature (2n, 2). Extend now I to ({widetilde{I}}) so that ({widetilde{I}}cdot (pi ^*alpha )=pi ^*Ialpha) for all (alpha in T^*M) and ({widetilde{I}}cdot (dt)=t{widetilde{varphi }}).
The metric ({widetilde{g}}) is compatible with ({widetilde{I}}) since
$${{widetilde{I}}cdot {widetilde{g}}=t^2{widetilde{I}}cdot pi ^*g-({widetilde{I}}cdot t{widetilde{varphi }})^2-({widetilde{I}}cdot dt)^2=t^2pi ^*(Icdot g)-(-dt)^2-(t{widetilde{varphi }})^2=t^2pi ^*(Icdot g)-dt^2-t^2{widetilde{varphi }}^2={widetilde{g}}}.$$
We thus have a Kähler manifold (({widetilde{M}},{widetilde{g}},{widetilde{I}},{widetilde{omega }})), where
$$begin{aligned} {widetilde{omega }}:=t^2pi ^*omega +t{widetilde{varphi }}wedge dt. end{aligned}$$
Let (xi :=tpartial _t) where (partial _t) is the vector field corresponding to the coordinate derivation on ({mathbb {R}}^+). Notice that the function (r=sqrt{-{widetilde{g}}(xi ,xi )}) coincides with t, as (sqrt{-{widetilde{g}}(tpartial _t,tpartial _t)}=sqrt{-t^2 {widetilde{g}}(partial _t,partial _t)}=t). In particular, ({widetilde{g}}(xi ,xi )=-t^2ne 0) and ({widetilde{g}}({widetilde{I}}xi ,{widetilde{I}}xi )={widetilde{g}}(xi ,xi )<0), so ({widetilde{g}}) is negative definite on (langle xi ,Ixi rangle) and hence positive definite on the orthogonal complement.
Let now (theta) be a unitary coframe on an open subset (Usubseteq M), then we can lift it to a complex coframe ({widetilde{theta }}) on (pi ^{-1}(U)) defined as in (4). It is straightforward to check that ({widetilde{theta }}) is adapted to the pseudo-Kähler structure of ({widetilde{M}}). Notice that the proof of Proposition 5.2 is still valid in this situation even though we do not know whether ({widetilde{M}}rightarrow M) has a structure of projective special Kähler manifold; this gives us a description of the Levi-Civita connection form on ({widetilde{M}}) with respect to ({widetilde{theta }}). Notice that ({widetilde{theta }}^k(xi )=0) for (kle n) and ({widetilde{theta }}^{n+1}(xi )=dt(tpartial _t)+i{widetilde{varphi }}(tpartial _t)=t) so (xi ={mathfrak {R}}(t{widetilde{theta }}_{n+1})). We can thus compute
$$begin{aligned} {widetilde{nabla }}^{LC}xi&=dtotimes {mathfrak {R}}({widetilde{theta }}_{n+1})+t{widetilde{nabla }}^{LC}{mathfrak {R}}({widetilde{theta }}_{n+1})\&={mathfrak {R}}(dtotimes {widetilde{theta }}_{n+1})+frac{t}{r}{mathfrak {R}}left( sum _{k=1}^n {widetilde{theta }}^kotimes {widetilde{theta }}_k+ i{text {Im}}({widetilde{theta }}^{n+1})otimes {widetilde{theta }}_{n+1}right) \&={mathfrak {R}}left( sum _{k=1}^{n+1} {widetilde{theta }}^kotimes {widetilde{theta }}_kright) =mathrm {id}. end{aligned}$$
Each section (s_alpha) corresponds to the trivialisation ((pi |_{pi ^{-1}(U)},z_{alpha }):pi ^{-1}Urightarrow Utimes {mathbb {C}}^*) in the sense that (s(pi (u))cdot z_{alpha }(u)=u) for all (uin pi ^{-1}(U_alpha )). For all (alpha) on (pi ^{-1}(U_{alpha })), define the tensor ({widetilde{eta }}_alpha :={mathfrak {R}}(z_alpha ^2pi ^*eta _alpha )). The family ({{widetilde{eta }}_alpha }_{alpha in {mathcal {A}}}) is compatible on intersections (U_1cap U_2), in fact if (s_1=cs_2) for (cin mathrm {U}(1)), then (z_2=cz_1) and (eta _1=gamma circ s_1=gamma circ cs_2=c^2gamma circ s_2=c^2eta _2), so
$$begin{aligned} widetilde{eta _1} ={mathfrak {R}}(z_1^2pi ^*eta _1) ={mathfrak {R}}(z_1^2c^2pi ^*eta _2) ={mathfrak {R}}(z_2^2pi ^*eta _2) =widetilde{eta _2}. end{aligned}$$
Therefore, this family glues to form a tensor ({widetilde{eta }}in sharp _2 S^3{widetilde{M}}).
We can build another connection (nabla :={widetilde{nabla }}^{LC}+{widetilde{eta }}). Notice that (nabla xi ={widetilde{nabla }}^{LC}xi +{widetilde{eta }}(xi )=mathrm {id}+{mathfrak {R}}(z_{alpha }^2pi ^*eta _{alpha })(xi )=mathrm {id}) because locally (eta _alpha) is horizontal for all (alpha).
In order to prove that (nabla) is symplectic, since the Levi-Civita connection is symplectic, it is enough to prove that ({widetilde{omega }}({widetilde{eta }},cdot )+{widetilde{omega }}(cdot ,{widetilde{eta }})=0). Locally, ({widetilde{omega }}=frac{1}{2i}sum _{k=1}^{n+1}overline{{widetilde{theta }}^k}wedge {widetilde{theta }}^k) and in fact, for all (X={mathfrak {R}}(X^k{widetilde{theta }}_k)), (Y={mathfrak {R}}(Y^k{widetilde{theta }}_k)), (Z={mathfrak {R}}(Z^k{widetilde{theta }}_k)) vector fields on ({widetilde{M}}):
$$begin{aligned} 2i({widetilde{omega }}({widetilde{eta }}_X Y,Z)+{widetilde{omega }}(Y,{widetilde{eta }}_X Z))&=sum _{k=1}^{n+1}left( overline{{widetilde{theta }}^k}({widetilde{eta }}_X Y){widetilde{theta }}^k(Z) -{widetilde{theta }}^k({widetilde{eta }}_X Y)overline{{widetilde{theta }}^k}(Z)right. \&quad left. +overline{{widetilde{theta }}^k}(Y)wedge {widetilde{theta }}^k({widetilde{eta }}_X Z) -{widetilde{theta }}^k(Y)wedge overline{{widetilde{theta }}^k}({widetilde{eta }}_X Z)right) \&=sum _{k=1}^{n+1}left( z pi ^*eta ^k_{u,v}X^u Y^v Z^k -overline{Z^k}{overline{z}}^2overline{pi ^*eta }^k_{u,v}overline{X^u} overline{Y^v} right. \&quad +left. {overline{Y}}^k{overline{z}}^2overline{pi ^*eta }^k_{u,v}overline{X^u} overline{Z^v} -z^2pi ^*eta ^k_{u,v}X^u Z^v Y^k right) \&=sum _{k=1}^{n+1}{mathfrak {R}}left( z^2 pi ^*eta ^k_{u,v}X^u Y^v Z^k -z^2pi ^*eta ^k_{u,v}X^u Z^v Y^k right) \&=sum _{k=1}^{n+1}{mathfrak {R}}left( z^2 pi ^*(eta ^k_{u,v}-eta ^v_{u,k})X^u Y^v Z^k right) . end{aligned}$$
By the symmetry of (eta), this quantity vanishes.
Proving that (d^{nabla }{widetilde{I}}=0), is equivalent to proving that (nabla {widetilde{I}}) is symmetric in the two covariant indices, and thus, (nabla {widetilde{I}}={widetilde{nabla }}^{LC}{widetilde{I}}+[{widetilde{eta }},{widetilde{I}}]=[{widetilde{eta }},{widetilde{I}}]). Since ({widetilde{I}}={mathfrak {R}}(i{widetilde{theta }}_k{widetilde{theta }}^k)), we have
$$begin{aligned}{}[{widetilde{eta }},{widetilde{I}}]&=iz^2pi ^*eta ^u_{v,w}{widetilde{theta }}^votimes overline{{widetilde{theta }}_u}otimes {widetilde{theta }}^w -ioverline{z^2pi ^*eta ^u_{v,w}{widetilde{theta }}^votimes overline{{widetilde{theta }}_u}otimes {widetilde{theta }}^w}\&quad +iz^2pi ^*eta ^u_{v,w}{widetilde{theta }}^votimes overline{{widetilde{theta }}_u}otimes {widetilde{theta }}^w -ioverline{z^2pi ^*eta ^u_{v,w}{widetilde{theta }}^votimes overline{{widetilde{theta }}_u}otimes {widetilde{theta }}^w} =2i{widetilde{eta }}=-2{widetilde{I}}{widetilde{eta }}, end{aligned}$$
which is symmetric, proving (d^{nabla }I=0).
For the flatness of (nabla), we compute the curvature locally
$$begin{aligned} Omega ^nabla =domega ^{nabla }+frac{1}{2}[omega ^{nabla }wedge omega ^{nabla }] ={widetilde{Omega }}^{LC}+{widetilde{d}}^{LC}{widetilde{eta }}+frac{1}{2}[{widetilde{eta }}wedge {widetilde{eta }}]. end{aligned}$$
By Proposition 5.2, ({widetilde{Omega }}^{LC}=r^2pi ^*(Omega ^{LC}+Omega _{{mathbb {P}}^n_{{mathbb {C}}}})). For the same reasoning exposed in the proof of Proposition 7.1, ({widetilde{d}}^{LC}{widetilde{eta }}=0) if and only if (d^{LC}eta -2is^*varphi wedge eta =0), which is granted by D2.
Finally, the computations leading to equation (10) still apply, and thus, we can deduce that
$$begin{aligned} Omega ^{nabla }=rpi ^*(Omega ^{LC}+Omega _{{mathbb {P}}^n_{mathbb {C}}}+[eta wedge {overline{eta }}])=0, end{aligned}$$
making the connection (nabla) flat.
Notice that (pi :{widetilde{M}}rightarrow M) is a principal ({mathbb {C}}^*)-bundle, where for all (l e^{itheta }in {mathbb {C}}^*) and ((u,t)in {widetilde{M}}):
$$begin{aligned} (u,t)l e^{itheta }:=(u e^{itheta },tl). end{aligned}$$
The infinitesimal vector field corresponding to 1 at ((u,t_0)) is (xi _{(u,t_0)}) and the one corresponding to i is (X:=frac{d}{dt}((u,t_0)exp (it))|_{t=0}=frac{d}{dt}(ue^{it},t_0)|_{t=0}), which is vertical and such that ({widetilde{varphi }}(X)=varphi (p_*X)=varphi (frac{d}{dt}(ue^{it})|_{t=0})=1) and (dr(X)=0). This means that (X=Ixi) since ({widetilde{g}}(X,cdot )=-r^2{widetilde{varphi }}=-rIdr=Ixi ^{flat }).
We are only left to prove that M is the Kähler quotient or ({widetilde{M}}) with respect to the (mathrm {U}(1))-action and in order to do so, notice that ({widetilde{omega }}(Ixi ,cdot )=-{widetilde{g}}(xi ,cdot )=rdr=dleft( frac{r^2}{2}right)), so (mu :=frac{r^2}{2}) is a moment map for (Ixi). Notice that (mu ^{-1}(frac{1}{2})=Stimes {1}) and S is a principal bundle so, by definition of ({widetilde{g}}) and ({widetilde{omega }}), (S/mathrm {U}(1)) is isometric to M and this ends the proof. (square)
Remark 7.7
Starting from the family ({eta _alpha }_{alpha }), we can build a bundle map (gamma :Srightarrow M) as long as the (eta _alpha)’s are linked by the relation (eta _alpha =g_{alpha ,beta }^2eta _beta) where (g_{alpha ,beta }) is a cocycle defining S.
Remark 7.8
Let (M, g, I) be a Kähler manifold, then if (H^2(M,{mathbb {Z}})=0), in particular, every complex line bundle and every circle bundle are trivial. Moreover, by de Rham’s theorem, (H^2_{dR}(M)=H^2(M,{mathbb {R}})=H^2(M,{mathbb {Z}})otimes {mathbb {R}}=0), so in particular (omega =dlambda) for some (lambda in Omega ^1(M)).
Corollary 7.9
A Kähler 2n-manifold ((M,g,I,omega )) such that (H^2(M,{mathbb {Z}})=0), has a projective special Kähler structure if and only if there exists a section (eta :Mrightarrow sharp _2 S_{3,0}M) such that
- D1(^{*}):
(qquad Omega ^{LC}+Omega _{{mathbb {P}}^n_{{mathbb {C}}}}+[eta wedge {overline{eta }}]=0;)
- D2(^{*}):
(qquad d^{LC}eta =-4ilambda wedge eta);
for some (lambda in Omega ^{1}(M)) such that (dlambda =omega).
Proof
If M has a projective special Kähler structure, then from Theorem 7.6 we obtain an (S^1)-bundle (p:Srightarrow M) and the map (gamma :Srightarrow sharp _2 S_{3,0}M). Consider the corresponding line bundle (L=Stimes _{mathrm {U}(1)}{mathbb {C}}). As noted in Remark 7.8, we can assume (L=Mtimes {mathbb {C}}) and (S=Mtimes S^1). In particular, there is a global section (s:Mrightarrow S) and if we call (eta =gamma circ s:Mrightarrow sharp _2S_{3,0}M), it is a global section satisfying the curvature equation thanks to Theorem 7.6. Defining (lambda :=-frac{1}{2}s^*varphi), we have (dlambda =-frac{1}{2}s^*(-2pi _S^*omega )=(pi _S s)^* omega =omega), and thus, also the differential condition is satisfied by Theorem 7.6.
Conversely, by de Rham’s Theorem, we have (lambda in Omega ^1(M)) such that (dlambda =omega). We define (pi _S=pi _1:S=Mtimes S^1rightarrow M) and choose as connection the form (varphi =pi _2^*dvartheta -2pi _S^*lambda), where (dvartheta) is the fundamental 1-form on (S^1=mathrm {U}(1)). Then, (dvarphi =0-2pi _S^*dlambda =-2pi _S^*omega), so (Srightarrow M) has the desired curvature. Moreover, it is trivial, so we have a global section (s:Mrightarrow S) mapping p to (p, 1).
Given (eta :Mrightarrow sharp _2S_{3,0}M) as in the statement, we define (gamma :Srightarrow sharp _2 S_{3,0}M) such that (gamma (p,a):=a^2eta (p)) for all (pin M) and (ain mathrm {U}(1)). Notice that (gamma circ s=gamma (cdot ,1)=eta), so the curvature equation of this corollary gives the curvature equation in Theorem 7.6 and the same is true for the differential condition, since (s^*varphi =s^*pi _2^*dvartheta -2s^*pi _S^*lambda =0-2lambda). By Theorem 7.6, M is thus projective special Kähler. (square)
Remark 7.10
Instead of requiring a section (eta) as in Corollary 7.9, we could use a section (sigma) of (S_{3,0}M) such that (sharp _2sigma =eta).
