GEODESIC RAYS IN THE DONALDSONUHLENBECKYAU THEOREM MATTIAS JONSSON NICHOLAS MCCLEEREY AND SANAL SHIVAPRASAD

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GEODESIC RAYS IN THE DONALDSON–UHLENBECK–YAU

THEOREM

MATTIAS JONSSON, NICHOLAS MCCLEEREY, AND SANAL SHIVAPRASAD

Abstract. We give new proofs of two implications in the Donaldson–Uhlenbeck–Yau

theorem. Our proofs are based on geodesic rays of Hermitian metrics, inspired by

recent work on the Yau–Tian–Donaldson conjecture.

1. Introduction

Let (Xn, ω) be a compact K¨ahler manifold, and Ea holomorphic vector bundle of

rank rover X. A Hermite-Einstein metric on Eis a Hermitian metric hwhich satisﬁes

Θ(h)∧ωn−1=γ ωnIdE,

where Θ(h) is the curvature of h, IdEis the identity endomorphism, and γis a cohomo-

logical constant.

The celebrated Donaldson–Uhlenbeck–Yau theorem states that Eadmits a Hermite-

Einstein metric if and only if Eis slope stable [Don87; UY86]. We will consider the

following version:

Theorem 1.1. Suppose that Eis a holomorphic vector bundle over a compact K¨ahler

manifold (Xn, ω). Then the following conditions are equivalent:

(1) Eis slope stable;

(2) The Donaldson functional Mis proper on the space of Hermitian metrics;

(3) Eadmits a unique Hermite-Einstein metric.

Here Eis slope stable if and only if µE< µEfor any nontrivial holomorphic torsion

free subsheaf E ⊂ E, where µFdenotes the slope of a holomorphic sheaf Fover Xwith

respect to ω. The Donaldson functional Mis a functional on the space of Hermitian

metrics whose minimizers are exactly the Hermite–Einstein metrics. See §2 for details.

The equivalence (1)⇔(3) in Theorem 1.1 was the ﬁrst result to link solvability of a

geometric PDE to a stability condition in the sense of GIT, and has proven to be deeply

inﬂuential in shaping subsequent results and conjectures, e.g. the Yau–Tian–Donaldson

conjecture, as discussed shortly.

We have chosen the above formulation of Theorem 1.1 in order to emphasize the

similarities with the variational approach to the Yau–Tian–Donaldson (YTD) conjecture

on the existence of (unique) cscK metrics on polarized complex manifolds (X, L). Despite

much recent progress, this conjecture is still open in general, but it is settled for Fano

manifolds, when L=−KX, see [Ber16; CDS15a; CDS15b; CDS15c; Tia15; DS16;

CSW18; BBJ21], and even for (possibly singular) log Fano pairs [Li22].

2020 Mathematics Subject Classiﬁcation. Primary: 53C07, Secondary: 14J60, 32Q15, 58E15.

arXiv:2210.09246v2 [math.DG] 19 Nov 2022

2 MATTIAS JONSSON, NICHOLAS MCCLEEREY, AND SANAL SHIVAPRASAD

In the general YTD conjecture, the Donaldson functional is replaced by the Mabuchi

K-energy functional, and slope stability by a suitable version of K-stability, a condition

on the space of test conﬁgurations for (X, L) [Tia97; Don02].

The analogue of the equivalence (2)⇔(3) is known in full generality [BBEGZ19; DR17;

CC21a; CC21b], and versions of (3)⇒(1) and (1)⇒(2) have been shown in [BHJ19]

and [Li21], respectively (albeit with two diﬀerent, conjecturally equivalent [Li21; BJ22],

deﬁnitions of stability). In each of these proofs, the notion of a geodesic ray in the space

of singular semipositive metrics on Lplays a crucial role.

Going back to the Hermite–Einstein problem, there is also a natural notion of geodesic

rays in the space of Hermitian metrics on E, and the goal of the present paper is to give

new proofs of the implications (3)⇒(1) and (1)⇒(2) by utilizing geodesic rays, paralleling

the recent work on the YTD conjecture.

Our ﬁrst result constructs a geodesic ray from any ﬁltration. As in the YTD con-

jecture, we will actually construct rays of singular metrics. Let H1,p be the space of

trace-free endomorphisms on Ewith coeﬃcients in the Sobolev space W1,p, 1 6p6∞.

Theorem A. Let h0be a hermitian metric on Eand

0 =: Em+1 ⊂ Em⊂. . . ⊂ E1:= E

a ﬁltration of Eby holomorphic subsheaves. Let Fi:= Ei/Ei+1,16i6m. Then there

exists w∈ H1,∞, such that the geodesic ray of singular hermitian metrics ht:= etwh0,

t>0,satisﬁes:

(1.1) lim

t→∞ Mω(ht, h0)

k=1

2π(m−k+ 1) rk(Fk) (µFk−µE),

where rk(Fk)is the rank of Fk, and µFkis its slope (with respect to ω).

Theorem A, which can be viewed as an analogue of Theorem A in [BHJ19], easily gives

the implication (3) ⇒(1) in Theorem 1.1, using the fact that Hermite–Einstein metric

are exactly the minimizers of the Donaldson functional, which is furthermore convex

along any geodesic ray. Indeed, when m= 2, the sign of the right-hand side of (1.1)

is evidently related to the slope stability of E, since for any E ⊂ E, rk(E)(µE−µE) =

rk(E/E)(µE/E−µE).

We construct the desired ray in Theorem A by ﬁrst passing to a smooth resolution

π:e

X→Xof the ﬁltration (see e.g. [Jac14; Sib15]). We then explicitly describe a

smooth endomorphism won π∗Ewhich acts on π∗h0by scaling the induced metrics on

the quotient bundles π∗Fkby k; pushing wforward produces the desired ray downstairs.

As we see from Theorem A, it is useful to consider geodesics of singular metrics of the

form etwh0, for some w∈ H1,p – note that etwh0will generally be much more singular

than w. With this setup, the largest natural space of metrics to consider is:

S:= {w∈ H1,1| M(ewh0, h0)<∞}.

Proposition 4.1 ([Don87]) shows that actually S ⊂ H1,pmax , where pmax := 2n

2n−1, and it

seems natural to interpret H1,p as an analogue of the space E1of metrics (or potentials)

of ﬁnite energy in the study of cscK metrics (although it is not obvious to us that the

GEODESIC RAYS 3

constant pmax is optimal, c.f. Theorem A). Our next result can be viewed as an analogue

of [BBJ21, Theorem 2.16].

Theorem B. If the Donaldson functional Mis non-proper on Swith respect to the

W1,p-norm, for any 1< p < pmax, then there exists a geodesic ray in Salong which M

is bounded above.

Note that Proposition 4.1 actually implies that Mwill be proper with respect to the

W1,pmax -norm if Eadmits an HE metric, so that Theorem B can likely be improved.

There are two main ingredients in the proof of Theorem B. The ﬁrst is the lower semi-

continuity of Min the weak W1,p-topology, a fact for which we give a new, elementary

proof, see Proposition 4.2. The analogous fact in the cscK case is that the Mabuchi

functional is lsc with respect to the strong topology on the space of metrics of ﬁnite

energy, see [BBEGZ19]. The second is a compactness statement, which here boils down

to the Banach–Alaoglu theorem. The analogue in the cscK case is that sets of bounded

entropy are strongly compact, as proved in [BBEGZ19].

Finally we go from geodesic rays to ﬁltrations:

Theorem C. Suppose that Eadmits a geodesic ray in Salong which the Donaldson

functional is bounded from above. Then there exists a nontrivial ﬁltration of Eby holo-

morphic subsheaves {Ek}m+1

k=1 such that µEk>µEfor at least one k. In particular, Eis

not slope stable.

Theorem C follows from a formula for M(ht, h0) in terms of the eigenvalues of

log(h0h−1

t), due to Donaldson [Don87]. Using this, we show that M(ht) can only be

bounded from above under very restrictive circumstances: essentially, the geodesic ray

(ht) must have come from a construction similar to Theorem A. The weakly holomorphic

W1,2-projection theorem of Uhlenbeck-Yau [UY86; UY89] can then be used to produce

the desired ﬁltration; applying (1.1) shows that at least one of the subsheafs in the

ﬁltration has slope larger than µE.

The role of Theorem C in the cscK case (X, L) is played by Theorem 6.4 in [BBJ21],

which to any geodesic ray (of linear growth) of metrics of ﬁnite energy associates a psh

metric of ﬁnite energy on the Berkovich analytiﬁcation of the line bundle with respect

to the (non-Archimedean) trivial absolute value on C. As proved by C. Li in [Li21], the

slope at inﬁnity of the Mabuchi functional along the ray is bounded below by the Mabuchi

functional evaluated at the non-Archimedean metric. In the setting of Theorem C, the

limiting object is simpler, given by a ﬁltration of Eby holomorphic subsheaves.

The combination of Theorems B and C evidently give us the implication (1)⇒(2) in

Theorem 1.1. As already mentioned, (3)⇒(1) follows from Theorem A. The remaining

implication (2)⇒(3) can be shown by an easy application of the Hermitian-Yang-Mills

ﬂow [Don85]. In the K¨ahler-Einstein case, any minimizer of the Mabuchi functional is a

K¨ahler–Einstein metric [BBGZ13], and the corresponding result in the cscK case holds

as well [CC21a; CC21b]. It is reasonable to believe that a minimizer in H1,p of the

Donaldson functional must in fact be a (smooth) Hermitian metric.

Comparison with previous works: In terms of history, L¨ubke [L¨ub83] and Kobayashi

[Kob87] ﬁrst proved the implication (3)⇒(1) in Theorem 1.1 by using vanishing theo-

rems for E. The more diﬃcult implication (1)⇒(3) was proved by Donaldson [Don83]

4 MATTIAS JONSSON, NICHOLAS MCCLEEREY, AND SANAL SHIVAPRASAD

for projective surfaces and by Uhlenbeck and Yau [UY86] in general. Donaldson gave

another proof in [Don87] for projective manifolds, using induction on dimension and a

theorem of Mehta–Ramanathan [MR84], and the implications (1)⇒(2) and (2)⇒(3) can

be extracted from results in that paper.

Subsequent work of Simpson simultaneously uniﬁed and generalized the approaches in

[UY86] and [Don87], establishing a version of Theorem 1.1 for Higgs bundles over certain

non-compact K¨ahler manifolds. His usage of a blow-up argument along a non-proper

ray to extract a limiting endomorphism and subsequent ﬁltration is similar in spirit to

our proof of Theorem B, but with several diﬀerences; ﬁrstly, his notion of properness is

less general than ours, only holding for a special subclass of metrics with L1-curvature.

Several simpliﬁcations follow from this – for instance, it is easy to show the lower semi-

continuity of Mon this smaller space, and he has no need to work with regularizations

of subsheaves.

Quite recently, Hashimoto and Keller [HK19; HK21] have given a new proof of the

implication (3)⇒(1), and a conditional new proof of (3)⇒(1), both in the polarized case

when ω∈c1(L), for an ample line bundle L. Like ours, their approach is variational in

nature, but uses geodesics in the space of Hermitian norms on global sections of E⊗Lk

for k0.

There has also been a great deal of work on singular versions of Theorem 1.1. In

[BS94], Bando and Siu introduced the notion of an admissible Hermite-Einstein metric

on a torsion-free sheaf, and showed that a reﬂexive sheaf on a K¨ahler manifold admits an

admissible Hermite-Einstein metric if and only if it is polystable. Subsequent work has

focused on similar results on singular varieties, see e.g. Chen and Wentworth [CW21].

[BS94] also introduced a regularization procedure for holomorphic subsheaves of E,

which was elaborated upon by Jacobs [Jac14] and Sibley [Sib15] (see also [Buc99]).

This procedure is an important tool in our proofs (c.f. Theorem 3.7). It was used by

Jacobs [Jac14] to generalize Theorem 1.1 to semi-stable bundles, motivated by work of

Kobayashi [Kob87].

Other generalizations of Theorem 1.1 include generalizations to compact Hermitian

manifolds ([Buc99; LY87]), and very recent work of Feng-Liu-Wan [FLW18], which ex-

panded Theorem 1.1 to include the existence of Finsler-Einstein metrics.

Organization: In Section 2 we set some deﬁnitions and collect several background

results. In Section 3 we prove Theorem A (c.f. Theorem 3.7). In Section 4 we prove

Proposition 4.1, which can be seen as a reverse Sobolev inequality for w∈ S, and show

the lower semicontinuity of Mon H1,p. We then show Theorems B and C in Section 5.

Acknowledgements. We would like to thank Yoshinori Hashimoto for pointing out

an inaccuracy in an earlier draft. The ﬁrst author was supported by NSF grants DMS-

1900025 and DMS-2154380.

2. Background Material

2.1. Slope stability and Sobolev Endomorphisms. For any holomorphic, torsion-

free sheaf Eon X, write:

µE:= RXc1(E)∧ωn−1

rk(E),

GEODESIC RAYS 5

for the slope of E, with respect to ω. We say Eis slope stable if:

µE< µE

for all proper, saturated torsion-free subsheaves E ⊂ E.Eis said to be slope semi-stable

if µE6µEfor all such E, and slope unstable otherwise. A subsheaf satisfying µE> µE

is said to be destabilizing.

Fix a Hermitian metric h0on E. We can identify Herm(E), the space of all smooth

Hermitian metrics on E, with e

H, the space of h0-self adjoint endomorphisms of Eby:

h∈Herm(E)7→ log(hh−1

0);

note that geodesics in Herm(E) map to straight line segements in e

H, and visa versa.

We have e

H=H ⊕ R, where His the (geodesically complete) subspace of trace-free

endomorphims.

Write H1,p := H ⊗C∞W1,p, for any p>1. We refer to straight line segments in H1,p

as weak geodesics, sometimes without the adjective if the lack of regularity is clear from

context. For any w∈ H1,p, we deﬁne kwkp

Lto be the Lp-norm of the operator norm of

wi.e.

kwkp

Lp=ZX

tr(ww∗)p/2ωn.

The duality pairing between H1,p and H1,q,q:= p

p−1, is deﬁned to be:

hw, ui:= ZX

tr(wu)ωn+√−1ZX

tr(D0w∧∂u)∧ωn−1,

where D=D0+∂is the Chern connection of h0; it follows that H1,q, is linearly dual to

H1,p. Standard functional analysis implies H1,p is reﬂexive for p > 1, so by the Banach–

Alaoglu Theorem, any bounded subset of H1,p is weakly compact, i.e. if 

∂wi

Lp6C,

then there exists a subsequence of the isuch that, after relabeling:

wi w ∈ H1,p,

in the sense that:

hwi, ui→hw, ui,

for every u∈ H1,q.

For the purposes of this paper, it will be convenient to ﬁx 1 < p 6pmax := 2n

2n−1.

Then p < 2, and the Sobolev conjugate of pis p0:= 2np

n−pand:

p0>p

2−p=: p∗

Note that p0

max =p∗

max, The Gagliardo-Nirenberg-Sobolev inequality can be given as:

(2.1) kwkLp06CSob kwkW1,p for any w∈ H1,p,

and by the Sobolev embedding theorem, the inclusion W1,p →Lp∗is compact – this

will be the only reason we need to restrict to p<pmax in the proof of Theorem B.

Given w∈ H, write λ1>. . . >λrfor the eigenvalues of w; these will be Lipschitz

functions on Xsuch that Pr

i=1 λi= 0. When w∈ H1,p, the λimay only be in Lp0.

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GEODESICRAYSINTHEDONALDSON{UHLENBECK{YAUTHEOREMMATTIASJONSSON,NICHOLASMCCLEEREY,ANDSANALSHIVAPRASADAbstract.WegivenewproofsoftwoimplicationsintheDonaldson{Uhlenbeck{Yautheorem.OurproofsarebasedongeodesicraysofHermitianmetrics,inspiredbyrecentworkontheYau{Tian{Donaldsonconjecture.1.IntroductionLet(Xn...

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