A note on Demailly's approach towards a conjecture of Griffiths
aa r X i v : . [ m a t h . DG ] F e b A NOTE ON DEMAILLY’S APPROACH TOWARDS ACONJECTURE OF GRIFFITHS
VAMSI PRITHAM PINGALI
Abstract.
We prove that a “cushioned” Hermitian-Einstein-type equationproposed by Demailly in an approach towards a conjecture of Griffiths on theexistence of a Griffiths positively curved metric on a Hartshorne ample vectorbundle, has an essentially unique solution when the bundle is stable. This resultindicates that the proposed approach must be modified in order to attack theaforementioned conjecture of Griffiths. Introduction
The notion of ampleness/positivity is paramount in algebraic geometry. For aholomorphic line bundle, there is only one notion of differentio-geometric positivity,i.e., there is a smooth Hermitian metric whose curvature form is a K¨ahler form. Bythe Kodaira embedding theorem, it coincides with algebro-geometric ampleness.A holomorphic vector bundle E is said to be Hartshorne ample if O E ∗ (1) is anample line bundle over P ( E ∗ ). There is no unique differentio-geometric notion ofpositivity of curvature Θ of a smooth Hermitian metric h . There are several compet-ing inequivalent notions. The most natural of these notions are Griffiths positivity ( h v, √− v i is a K¨ahler form for all v = 0), Nakano positivity (the bilinear formdefined by √−
1Θ on T , M ⊗ E is positive-definite), and dual-Nakano positivity (the Hermitian holomorphic bundle ( E ∗ , h ∗ ) is Nakano negative ). Nakano positivityand dual-Nakano positivity imply Griffiths positivity and all three of them implyHartshorne ampleness. A famous conjecture of Griffiths [4] asks whether Hartshorneample vector bundles admit Griffiths positively curved metrics. This conjecture isstill open. However, a considerable amount of work has been done to provide evi-dence in its favour [1, 2, 3, 5, 7, 8, 9, 12].Relatively recently, Demailly [3] proposed a programme to prove the aformen-tioned conjecture of Griffiths for a holomorphic rank − r vector bundle E on a com-pact K¨ahler manifold ( X, ω ). In fact, if Demailly’s method works, it will end upproving a stronger conjecture : Do Hartshorne-ample bundles admit dual-Nakanopositively curved metrics ? Demailly’s approach involves solving a family (depend-ing on a parameter 0 ≤ t ≤
1) of vector bundle Monge-Amp`ere equations (dis-tinct from the one introduced in [10]) in conjunction with “cushioned” Hermitian-Einstein-type equations (Theorem 2.17 in [3]):det
T X ⊗ E ∗ (Θ h t + (1 − t ) αω ⊗ I E ∗ ) /r = f t (det h ) µ (det h t ) λ ω n , (1.1) (cid:18) √− F h t − √− r tr F h t (cid:19) ω n − = − ǫ (det h ) µ (det h ) µ ln (cid:18) hh − det( hh − ) /r (cid:19) ω n , (1.2) where h is a smooth background Hermitian metric, µ, λ ≥ α > h + αω is dual-Nakano positively curved,and f t > Theorem 1.1.
Let E be an ω -stable rank − r holomorphic bundle on X . Let H bea Hermitian-Einstein metric on E with respect to ω , that is, √− F H ω n − = λω n .Let h be a smooth metric on E solving the following cushioned Hermitian-Einsteinequation for given parameters ǫ ≥ , µ ≥ . (cid:18) √− F h − √− r tr F h (cid:19) ω n − = − ǫ (det H ) µ (det h ) µ ln (cid:18) hH − det( hH − ) /r (cid:19) ω n , (1.3) where h, H are matrices (any holomorphic trivialisation will do). Then h = H e − f for some smooth function f . As a result, if we consider the system of the vector bundle Monge-Amp`ere equa-tion and the cushioned Hermitian-Einstein-type equation on an ω -stable ample E ,and if solutions exist all the way till t = 1, the final t = 1 solution, by virtue ofthe fact that it satisfies the cushioned Hermitian-Einstein-type equation, has to beof the form H e f . This condition might be a strong restriction (which is unlikelyto be met owing to [6, 11] without a restriction on the second Chern character).On the other hand, if we replace ω by say (1 − t ) ω + t √− tr ( F h t ) (or the choicein Section 2.19 in [3] for instance), the above argument will not be applicable andthere might be some hope for the approach to yield an affirmative solution to theGriffiths Conjecture. Acknowledgements.
This work is partially supported by grant F.510/25/CAS-II/2018(SAP-I) from UGC (Govt. of India), and a MATRICS grant MTR/2020/000100from SERB (Govt. of India). The author thanks Jean-Pierre Demailly for fruitfuldiscussions and for encouraging me to write up this note.2.
Proof of uniqueness
In a holomorphic trivialisation, our conventions are : h v, w i H = v T H ¯ w , if g is anendomorphism then g.s = [ g ] T ~s , ∇ s = ds + A T s , A = ∂HH − , F = dA − A ∧ A =¯ ∂A , and ∇ g = dg + [ g, A ].The proof is motivated by a similar one by Donaldson for Riemann surfaces. Ingeneral, h = qH where q is some smooth H -Hermitian positive-definite endomor-phism of E . We decompose q further as q = e − f g where det( g ) = 1 and f is asmooth function. Thus, F h = F H + ∂ ¯ ∂f + ¯ ∂ ( ∂ gg − ). The trace-free part of thecurvature is F ◦ h = F ◦ + ¯ ∂ ( ∂ gg − ). Substituting these expressions in 1.3 and usingthe fact that H is Hermitian-Einstein with respect to ω , we get √− ∂ ( ∂ gg − ) ω n − = − ǫe rµf ln gω n . (2.1)Now we compute12 √− ∂∂tr ( g ) ω n − = √− ∂tr ( g∂ g ) ω n − = √− tr ( ¯ ∂g∂ g ) ω n − + √− tr ( g ¯ ∂∂ g ) ω n − = √− tr ( ¯ ∂g∂ g ) ω n − − ǫe rµf tr ( g lng ) ω n − √− tr ( g∂ gg − ¯ ∂g ) ω n − ≤ − ǫe rµf tr ( g lng ) ω n . (2.2) NOTE ON DEMAILLY’S APPROACH TOWARDS A CONJECTURE OF GRIFFITHS 3
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