On the optimal volume upper bound for Kähler manifolds with positive Ricci curvature (with an appendix by Yuchen Liu)
OON THE OPTIMAL VOLUME UPPER BOUND FOR KÄHLERMANIFOLDS WITH POSITIVE RICCI CURVATURE
KEWEI ZHANG(With an Appendix by Yuchen Liu)
Abstract.
Using δ -invariants and Newton–Okounkov bodies, we derive theoptimal volume upper bound for Kähler manifolds with positive Ricci cur-vature, from which we get a new characterization of the complex projectivespace. Introduction
Let ( X, g ) be an m -dimensional Riemannian manifold such thatRic ( g ) ≥ ( m − g. Then the well-known Bishop–Gromov volume comparison says that
Vol(
X, g ) ≤ Vol( S m , g Sm ) , and the equality holds if and only if ( X, g ) is isometric to the standard m -sphere S m . However, suppose in addition that X has a complex structure J such that ( X, g, J ) is Kähler, then G. Liu [24] shows that this volume upper bound is neversharp (unless X = P ), in the sense that there exists a dimensional gap (cid:15) ( n ) > such that Vol(
X, g ) ≤ Vol( S m , g Sm ) − ε ( n ) . This distinguishes the Kähler geometry from the Riemannian case. So it is naturalto ask what the optimal volume upper bound in the Kähler setting should be.A folklore conjecture predicts that, in the Kähler setting, the complex projectivespace equipped with the Fubini–Study metric should attain the maximal volume.Regarding this problem, a significant progress was made by K. Fujita [15], whogave an affirmative answer in the case of Kähler–Einstein manifolds, and whoseapproach is, interestingly enough, purely algebraic. K. Fujita’s breakthrough wasalso mentioned by S.K. Donaldson in his 2018 ICM talk (cf. [12]). However, ananswer for general Kähler manifolds with positive Ricci curvature is still missing.The purpose of this paper is to completely solve this problem by using some keyinput from algebraic geometry and convex geometry. Our main result is stated asfollows.
Theorem 1.1.
Let ( X, ω ) be an n -dimensional Kähler manifold with Ric ( ω ) ≥ ( n + 1) ω. Then one has
Vol(
X, ω ) ≤ Vol( P n , ω F S ) , and the equality holds if andonly if ( X, ω ) is biholomorphically isometric to ( P n , ω F S ) . Here ω F S denotes theFubini–Study metric so that (cid:82) P n ω nF S = (2 π ) n . This result gives a new characterization of the complex projective space in termsof Ricci and volume, and extends the previous works of Berman–Berndtsson [3], F. a r X i v : . [ m a t h . DG ] S e p KEWEI ZHANG
Wang [33] and K. Fujita [15] (see also Y. Liu [25]) to general Kähler classes. Aswe shall see, while the statement of Theorem 1.1 is differential geometric, its proofturns out to be rather algebraic.Indeed, Kähler manifolds with positive Ricci curvature are automatically Fano.These are simply connected projective manifolds with many additional algebraicproperties. So in what follows, unless otherwise specified, we will always assumethat X is an n -dimensional Fano manifold. Note that the Picard group Pic( X ) ∼ = H ( X, Z ) is a finitely generated torsion free Abelian group, hence a lattice. So theisomorphism classes of line bundles on X are in one-to-one correspondence with thelattice points of H ( X, Z ) . Also note that the Kähler cone K ( X ) of X coincideswith its ample cone , as H ( X, R ) = H , ( X, R ) by Kodaira vanishing and Hodgedecomposition. So any Kähler class in K ( X ) can be approximated by a sequenceof rational classes (corresponding to ample Q -line bundles).Based on this, instead of using traditional Riemannian geometry, we will proveTheorem 1.1 from an algebraic viewpoint, which requires several tools that havebeen developed in the K-stability theory. First of all, we will reformulate the cur-vature condition in terms of Tian’s greatest Ricci lower bound , which in turn canbe related to the algebraic δ -invariant (see (2.4)) thanks to the recent result ofthe author [10, Appendix] and independently [4, Theorem C]. Then by modifyingthe argument of K. Fujita [15], we get the desired volume upper bound for Käh-ler classes. The essential difficulty of Theorem 1.1 lies in the characterization ofthe equality. For this, we will compute the anti-canonical Seshadri constant as in[15]. However, when dealing with general Kähler classes, the irrationality causessome subtle issue. To overcome this, we need some key observations from convexgeometry. In particular, Newton–Okounkov bodies and the positivity criterion ofKüronya–Lozovanu [20] will be crucially used to treat the equality case of Theorem1.1.Apart from the optimal volume upper bound, quantitative volume rigidity is alsoan important property in geometry. Recall that, the classical sphere gap theoremin Riemannian geometry says the following: Theorem 1.2. [9, Theorem A.1.10] . There exists ε ( m ) > , such that if ( M, g ) isan m -dimensional Riemannian manifold with Ric ( g ) ≥ ( m − g and Vol(
M, g ) ≥ Vol( S m ) − ε ( m ) , then M is diffeomorphic to S m . So it is satisfactory to have the following Kählerian analogue, which was statedas a conjecture in an earlier version of this paper. The proof is due to Yuchen Liu(in the toric setting, this was also obtained by F. Wang [33] using combinatoricmethods).
Theorem 1.3.
There exists ε ( n ) > , such that if ( X, ω ) is an n -dimensionalKähler manifold with Ric ( ω ) ≥ ( n + 1) ω and Vol(
X, ω ) ≥ Vol( P n , ω F S ) − ε ( n ) ,then X is biholomorphic to P n . The rest of this paper is organized as follows. In Section 2 we review somenecessary notions and tools from the literature. In Section 3 we prove Theorem1.1 by assuming that [ ω ] is (a multiple of) a rational class. In Section 4, we proveTheorem 1.1 in full generality. In the appendix provided by Y. Liu, Theorem 1.3is proved. PTIMAL VOLUME UPPER BOUND FOR KÄHLER MANIFOLDS 3
Remark 1.4.
After completing the first draft of this paper, the author was kindlyinformed by Feng Wang that he had also independently obtained the inequality inTheorem 1.1 by adopting the argument of [15] to a sequence of conic KE metrics.
Acknowledgments.
The author would like to thank Kento Fujita, Feng Wangand Chuyu Zhou for many helpful discussions. He is also grateful to Yuchen Liu forproviding the proof of Theorem 1.3. Thanks also go to Xiaohua Zhu and Yanir Ru-binstein for valuable comments. Special thanks go to the anonymous referees whosecomments helped improve and clarify this manuscript. The author is supported bythe China post-doctoral grant BX20190014.2.
Preliminaries
In this section X is assumed to be an n -dimensional Fano manifold.2.1. The volume function on the Néron–Severi space.
Note that, H , ( X, R ) can be identified with the Néron–Severi space N ( X ) R , whichconsists of numerical equivalence classes of R -divisors on X . One can define a continuous volume function Vol( · ) on N ( X ) R . When restricted to the Kähler cone K ( X ) (i.e., the ample cone), Vol( · ) is the usual volume for Kähler classes (whichwill be treated as ample R -divisors in what follows).Also recall that, a class ξ ∈ N ( X ) R is called nef if for every curve C on Xξ · C ≥ . For nef classes ξ , Vol( ξ ) is simply equal to the top self-intersection number ξ n .A class ξ ∈ N ( X ) R is called big if Vol( ξ ) > . For more details on this subject, we refer the reader to the standard reference [21].2.2.
The greatest Ricci lower bound.
Let K ( X ) denote the Kähler cone of X . For any Kähler class ξ ∈ K ( X ) , one cannaturally define its greatest Ricci lower bound β ( X, ξ ) to be (2.1) β ( X, ξ ) := sup { µ > | ∃ Kähler form ω ∈ πξ s.t. Ric ( ω ) ≥ µω } . Note that, by the Calabi–Yau theorem, given any Kähler form α ∈ πc ( X ) , onecan always find ω ∈ πξ such that Ric ( ω ) = α > . By compactness of X we seeRic ( ω ) ≥ (cid:15)ω for some (cid:15) > . So β ( X, ξ ) is always a positive number. On the otherhand, β ( X, ξ ) is naturally bounded from above by the Seshadri constant(2.2) (cid:15) ( X, ξ ) := sup { µ > | c ( X ) − µξ is nef } . Thus we always have(2.3) < β ( X, ξ ) ≤ (cid:15) ( X, ξ ) . When ξ = c ( L ) for some ample Q -line bundle L , we will write β ( X, L ) := β ( X, c ( L )) for ease of notation. We put a factor π in the definition for convenience. KEWEI ZHANG
Remark 2.1.
When ξ = c ( X ) , the greatest Ricci lower bound was first studied byTian [32] , although it was not explicitly defined there. It was first explicitly definedby Rubinstein in [28, 29] , and was later further studied by Székelyhidi [31] , Li [23] ,Song–Wang [30] , Cable [7] , et al. The δ -invariant. Let L be an ample Q -line bundle on X . Following [16, 5], the δ -invariant of L isdefined by(2.4) δ ( X, L ) := inf E A X ( E ) S L ( E ) . Here E runs through all the prime divisors over X (i.e., E is a divisor contained insome birational model Y π −→ X over X ). Moreover, A X ( E ) := 1 + ord E ( K Y − π ∗ K X ) , denotes the log discrepancy, and S L ( E ) := 1Vol( L ) (cid:90) ∞ Vol( π ∗ L − xE ) dx denotes the expected vanishing order of L along E . Note that δ -invariant is alsocalled stability threshold in the literature, which plays important roles in the studyof K-stability and has attracted intensive research attentions. When L = − K X ,it was proved by the author in the appendix of his joint work with Cheltsov andRubinstein [10] that β ( X, − K X ) = min { , δ ( X, − K X ) } . For arbitrary ample Q -line bundles, we have the following independent result byBerman–Boucksom–Jonsson [4], giving a geometric interpretation of δ -invariantson Fano manifolds. Theorem 2.2.
Let L be an ample Q -line bundle on a Fano manifold X . Then onehas β ( X, L ) = min { (cid:15) ( X, L ) , δ ( X, L ) } . Newton–Okounkov bodies and positivity of R -line bundles. We briefly recall the definition of Newton–Okounkov bodies; for more details werefer the reader to [22]. Let Y be an n -dimensional projective manifold. Choose aflag of subvarieties Y • : Y = Y ⊃ Y ⊃ ... ⊃ Y n − ⊃ Y n = { pt. } , such that each Y i is an irreducible subvariety of codimension i and smooth at thepoint Y n . Such a flag is called admissible . Then any big class ξ ∈ N ( Y ) R canbe associated with a convex body ∆ Y • ( ξ ) in ( R ≥ ) n , which is called the Newton–Okounkov body of ξ with respect to the flag Y • . This generalizes the classicalpolytope construction for divisors on toric varieties. A crucial fact is that(2.5) Vol( ξ ) = n !Vol R n (∆ Y • ( ξ )) . In this way one can study the volume function
Vol( · ) on N ( Y ) R using convexgeometry. For more details of this construction we refer the reader to [22]. PTIMAL VOLUME UPPER BOUND FOR KÄHLER MANIFOLDS 5
It turns out that Newton–Okounkov bodies can also help us visualize the posi-tivity of R -line bundles. More precisely, for any big R -divisor ξ , one can define its restricted base loci by B − ( ξ ) := (cid:91) A B ( ξ + A ) , where the union is over all ample Q -divisors A on Y and B ( · ) denotes the stablebase loci (cf. [14]). Then it is easy to see that(2.6) ξ is nef if and only if B − ( ξ ) = ∅ . More precisely, B − ( ξ ) captures the non-nef locus of ξ (see [14, Example 1.18]).Indeed, suppose that there exists some curve C intersecting negatively with ξ , thenby adding a small amount of ample Q -divisor A, one still has ( ξ + A ) · C < , which implies that C ⊂ B ( ξ + A ) and hence C ⊂ B − ( ξ ) . The result of Küronya–Lozovanu says that one can characterize the restricted baseloci using Newton–Okounkov bodies.
Theorem 2.3. [20, Theorem A]
Let ξ be a big R -divisor. Then the following areequivalent. (1) q / ∈ B − ( ξ ) . (2) There exists an admissible flag Y • with Y n = { q } such that the origin ∈ ∆ Y • ( ξ ) ⊂ R n . (3) For any admissible flag Y • with Y n = { q } , one has ∈ ∆ Y • ( ξ ) ⊂ R n . Let us also record the following useful translation property of Newton–Okounkovbodies.
Proposition 2.4. [20, Proposition 1.6] . Let ξ be a big R -divisor and Y • an admis-sible flag on Y . Then for any t ∈ [0 , τ ( ξ, Y )) we have ∆ Y • ( ξ ) ν ≥ t = ∆ Y • ( ξ − tY ) + te , where τ ( ξ, Y ) := sup { µ > | ξ − µY is big } denotes the pseudo-effective threshold, ν denotes the first coordinate of R n and e = (1 , , ..., ∈ R n . Rational classes
In this section, we will verify Theorem 1.1 for rational classes. More precisely,we prove the following
Theorem 3.1.
Let L be an ample Q -line bundle on a Fano manifold X . Then onehas β ( X, L ) n Vol( L ) ≤ ( n + 1) n , with equality if and only if X is biholomorphic to P n .Proof. We follow the argument of K. Fujita [15]. Firstly, Theorem 2.2 implies that δ ( X, L ) ≥ β ( X, L ) and (cid:15) ( X, L ) ≥ β ( X, L ) . Pick any point p ∈ X and let ˆ X σ −→ X be the blow-up at p . Let E be the exceptionaldivisor of σ . Then one has A X ( E ) ≥ β ( X, L ) S L ( E ) , KEWEI ZHANG and hence, n = A X ( E ) ≥ β ( X, L ) S L ( E )= β ( X, L )Vol( L ) (cid:90) ∞ Vol( σ ∗ L − xE ) dx ≥ β ( X, L )Vol( L ) (cid:90) n √ Vol( L )0 (Vol( L ) − x n ) dx = nβ ( X, L ) n + 1 n (cid:112) Vol( L ) . Here we used [15, Theorem 2.3(1)]. Thus
Vol( L ) ≤ ( n + 1) n β ( X, L ) n , so the desired inequality is established. Now suppose that Vol( L ) = ( n +1) n β ( X,L ) n . Thenwe see that the equality Vol( σ ∗ L − xE ) = Vol( L ) − x n has to hold true for any x ∈ [0 , n +1 β ( X,L ) ] (as Vol( σ ∗ L − xE ) is a continuous functionin x ). So [15, Theorem 2.3(2)] implies that σ ∗ L − n + 1 β ( X, L ) E is nef.Now using (cid:15) ( X, L ) ≥ β ( X, L ) , we find that σ ∗ ( − K X ) − ( n + 1) E = σ ∗ (cid:18) − K X − β ( X, L ) L (cid:19) + β ( X, L ) (cid:18) σ ∗ L − n + 1 β ( X, L ) E (cid:19) is nef as well. Since p ∈ X can be chosen arbitrarily, we conclude that X ∼ = P n by[11, 17]. (cid:3) In the above proof, we actually obtained the following general inequality (cf.Blum–Jonsson [5, Theorem D])(3.1) δ ( X, L ) n Vol( L ) ≤ ( n + 1) n . This inequality reveals the deep relationship between singularities and volumes oflinear systems.
Remark 3.2.
In the toric case when L = − K X , the inequality in Theorem 3.1 wasfirst obtained by Berman–Berndtsson [3] using analytic methods, and the equalitycase was characterized by F. Wang [33] . General Kähler classes
Let us attend to general Kähler classes. The main result of this section is thefollowing.
Theorem 4.1.
Let ξ be a Kähler class of a Fano manifold X . Then one has β ( X, ξ ) n Vol( ξ ) ≤ ( n + 1) n , with equality if and only if X is biholomorphic to P n . PTIMAL VOLUME UPPER BOUND FOR KÄHLER MANIFOLDS 7
Remark 4.2.
Using Bishop–Gromov, one can quickly derive β ( X, ξ ) n Vol( ξ ) ≤ n +1 ( n !) (2 n − n (2 n )! . However this bound is much worse than ( n + 1) n (especially when n is large). The proof of Theorem 4.1 will be divided into several steps. We first showthe inequality by approximation and then characterize the equality using Newton–Okounkov bodies. We begin with a simple observation.
Lemma 4.3.
The greatest Ricci lower bound β ( X, · ) is a lower semi-continuous function on K ( X ) .Proof. Let { e , ..., e ρ } be a basis of H , ( X, R ) , then any Kähler class ξ ∈ K ( X ) can be written as ξ = ρ (cid:88) i =1 a i e i for some a i ∈ R . For each i ∈ { , ..., ρ } choose a smooth real (1 , -form η i ∈ e i .Now assume that there exists ω ∈ πξ such thatRic ( ω ) > µω for some µ > . For any (cid:126)(cid:15) = ( (cid:15) , ..., (cid:15) ρ ) ∈ R ρ with || (cid:126)(cid:15) || (cid:28) , we put ω (cid:126)(cid:15) := ω + ρ (cid:88) i =1 (cid:15) i η i . Then for || (cid:126)(cid:15) || (cid:28) one also has Ric ( ω (cid:126)(cid:15) ) > µω (cid:126)(cid:15) . So the lower semi-continuity of β ( X, · ) follows. (cid:3) As a consequence we get the following volume upper bound for general Kählerclasses in terms of its greatest Ricci lower bound.
Proposition 4.4.
Let ξ be a Kähler class on an n -dimensional Fano manifold X .Then one has (4.1) β ( X, ξ ) n Vol( ξ ) ≤ ( n + 1) n . Proof.
Choose a sequence of ample Q -line bundles L i such that L i → ξ in N ( X ) R . By Lemma 4.3 we have β ( X, ξ ) ≤ lim inf i β ( X, L i ) . So for any (cid:15) > and i (cid:29) , one has β ( X, L i ) ≥ β ( X, ξ ) − (cid:15). Thus Theorem 3.1 implies that ( β ( X, ξ ) − (cid:15) ) n Vol( L i ) ≤ ( n + 1) n . In fact it is proved in the author’s recent work [34] that β ( X, · ) is even continuous. KEWEI ZHANG
Using the continuity
Vol( L i ) → Vol( ξ ) and sending (cid:15) → , we get β ( X, ξ ) n Vol( ξ ) ≤ ( n + 1) n . (cid:3) Therefore, to finish the proof of Theorem 4.1, it remains to show that the equalityof (4.1) is exactly obtained by P n . Let us prepare the following lemma. Lemma 4.5.
Let X be a projective manifold. Pick any point p ∈ X and let ˆ X σ −→ X be the blow-up at p . Let E be the exceptional divisor of σ . Let ξ ∈ N ( X ) R be anef and big R -line bundle. (1) For any x ∈ R ≥ , one has Vol( σ ∗ ξ − xE ) ≥ Vol( ξ ) − x n . (2) Suppose in addition that X is Fano and that ξ is ample satisfying β ( X, ξ ) n Vol( ξ ) = ( n + 1) n , then for any x ∈ [0 , Vol( ξ ) /n ] , Vol( σ ∗ ξ − xE ) = Vol( ξ ) − x n . Proof.
This first part follows from [15, Theorem 2.3(1)] by approximation. Indeed,let L i be a sequence of ample Q -line bundles such that L i → ξ in N ( X ) R . Thenfor any x ∈ R ≥ , [15, Theorem 2.3(1)] says that Vol( σ ∗ ( L i ) − xE ) ≥ Vol( L i ) − x n , so the assertion follows by the continuity of Vol( · ) .For the second part, we rescale ξ such that β ( X, ξ ) = 1 and
Vol( ξ ) = ( n + 1) n . Let L i be a sequence of ample Q -line bundles such that L i → ξ in N ( X ) R . Forany (cid:15) > and i (cid:29) , Theorem 2.2 and Lemma 4.3 implies that δ ( X, L i ) ≥ β ( X, L i ) ≥ − (cid:15). Thus we get n = A X ( E ) ≥ (1 − (cid:15) ) S L i ( E )= 1 − (cid:15) Vol( L i ) (cid:90) ∞ Vol( σ ∗ L i − xE ) dx. Letting i → ∞ , by dominated convergence theorem and by sending (cid:15) → , we get n ≥ ξ ) (cid:90) ∞ Vol( σ ∗ ξ − xE ) dx, so that (recall Vol( ξ ) = ( n + 1) n ) n ≥ ξ ) (cid:90) Vol( ξ ) /n (Vol( ξ ) − x n ) dx = nn + 1 Vol( ξ ) /n = n. This gives
Vol( σ ∗ ξ − xE ) = Vol( ξ ) − x n for x ∈ [0 , Vol( ξ ) /n ] as claimed. (cid:3) PTIMAL VOLUME UPPER BOUND FOR KÄHLER MANIFOLDS 9
As we have seen in the proof of Theorem 3.1, K. Fujita [15, Theorem 2.3(2)] saysthat, for ample Q -line bundles, the condition Vol( σ ∗ L − xE ) = Vol( L ) − x n , x ∈ [0 , a ] implies that σ ∗ L − xE is nef for x ∈ [0 , a ] , whose proof however heavily relies on therationality of L and the ampleness criterion of [13]. To prove the same assertionfor general ample R -line bundles, there is some subtlety involved if we follow Fu-jita’s original argument. To overcome this, we take an alternative approach, usingNewton–Okounkov bodies.Before stating the key result, we recall the notion of local Seshadri constant .Given a nef R -divisor ξ and a point p ∈ X , let (cid:15) p := inf C ξ · C mult p ( C ) denote the Seshadri constant of ξ at p (where the inf is over all the curves passingthrough p ). Assume further that ξ is big and nef, then (cid:15) p > for any general point p ∈ X . Indeed, as ξ is big and nef, it can be written as ξ = A + F , where A is anample R -divisor and F is an effective R -divisor (see [21, Proposition 2.2.22]). Then ξ has positive Seshadri constant at any point p ∈ X − supp( F ) . Proposition 4.6.
Let X be a projective manifold. Let ξ be a big and nef R -divisor.Pick a point p ∈ X such that ξ has positive Seshadri constant at p . Let ˆ X σ −→ X bethe blow-up at p . Let E be the exceptional divisor of σ . Suppose that there exists a ∈ (0 , n (cid:112) Vol( ξ )] such that Vol( σ ∗ ξ − xE ) = Vol( ξ ) − x n for any x ∈ [0 , a ] . Then σ ∗ ξ − xE is nef for any x ∈ [0 , a ] .Proof. Let (cid:15) p > denote the Seshadri constant of ξ at p . Then σ ∗ ξ − xE is nef for any x ∈ [0 , (cid:15) p ] . Assume that (cid:15) p < a , otherwise we are done.We argue by contradiction. Suppose that σ ∗ ξ − x E is not nef for some x ∈ ( (cid:15) p , a ) . Then there exists some curve C intersecting σ ∗ ξ − x E negatively. Notethat such C necessarily intersects E by the projection formula. Thus the restrictedbase loci B − ( σ ∗ ξ − x E ) intersects E as well. So we can pick a point q ∈ B − ( σ ∗ ξ − x E ) ∩ E. Moreover, we build an admissible flag Y • on ˆ X such that Y := E and Y n := { q } . This is doable because E ∼ = P n − and Y ⊃ ... ⊃ Y n = { q } can be chosen to be aflag of linear subspaces in P n − . Then we get a Newton–Okounkov body ∆ Y • ( σ ∗ ξ ) .As σ ∗ ξ is nef, B − ( σ ∗ ξ ) = ∅ . So Theorem 2.3 implies that ∈ ∆ Y • ( σ ∗ ξ ) . Also note that, by our assumption,
Vol( σ ∗ ξ − xE ) = Vol( ξ ) − x n > for all x ∈ [0 , a ) . So σ ∗ ξ − xE is big for all x ∈ [0 , a ) and hence the pseudo-effective threshold τ ( ξ, E ) satisfies τ ( ξ, E ) ≥ a . Now by Proposition 2.4, for any x ∈ [0 , a ) , the Newton–Okounkov body ∆ Y • ( σ ∗ ξ − xE ) can be obtained from ∆ Y • ( σ ∗ ξ ) by truncating andtranslating in the first coordinate ν of R n . Therefore (recall (2.5)), for x ∈ (0 , a ) , Vol R n (cid:18) ∆ Y • ( σ ∗ ξ ) ∩ { ≤ ν ≤ x } (cid:19) = 1 n ! (cid:18) Vol( ξ ) − Vol( σ ∗ ξ − xE ) (cid:19) = x n n ! . For r ∈ (0 , a ) , consider the slice S r := ∆ Y • ( σ ∗ ξ ) ∩ { ν = r } , and put A ( r ) := Vol R n − ( S r ) . Then (cid:90) x A ( r ) dr = Vol R n (cid:18) ∆ Y • ( σ ∗ ξ ) ∩ { ≤ ν ≤ x } (cid:19) = x n n ! , x ∈ (0 , a ) , which implies that A ( r ) = r n − ( n − for r ∈ (0 , a ) . Note that the Brunn–Minkowski inequality in convex geometry says that A ( r ) n − is concave in its support. However in our case, A ( r ) n − is in fact linear , so we arein the equality case of the Brunn–Minkowski inequality. This means that all theslices S r are homothetic . We claim that, this forces ∆ Y • ( σ ∗ ξ ) to be a convex coneover the ( n − -dimensional convex set Σ := ∆ Y • ( σ ∗ ξ ) ∩ { v = a } . Indeed, for r ∈ (0 , a ) , consider the cone over S r : C ( S r ) := { λv | λ ∈ [0 , , v ∈ S r } . Then C ( S r ) ⊆ ∆ Y • ( σ ∗ ξ ) ∩ { ≤ ν ≤ r } by convexity. On the other hand, Vol R n ( C ( S r )) = (cid:90) r ( sr ) n − A ( r ) ds = r n n ! = Vol R n (cid:18) ∆ Y • ( σ ∗ ξ ) ∩ { ≤ ν ≤ r } (cid:19) . This implies that ∆ Y • ( σ ∗ ξ ) ∩ { ≤ ν ≤ r } = C ( S r ) , for any r ∈ (0 , a ) . Sending r → a , we find that ∆ Y • ( σ ∗ ξ ) ∩ { ≤ ν ≤ a } is a cone over Σ , as claimed.Now recall that σ ∗ ξ − xE is nef for x ∈ [0 , (cid:15) p ] , so B − ( σ ∗ ξ − xE ) = ∅ and hence byTheorem 2.3, ∈ ∆ Y • ( σ ∗ ξ − xE ) , for all x ∈ [0 , (cid:15) p ] . Then by Proposition 2.4, ∆ Y • ( σ ∗ ξ ) ∩ { ≤ ν ≤ a } contains the line segment { ( x, , ..., | x ∈ [0 , (cid:15) p ] } , so the cone property forces it to contain the whole linesegment { ( x, , ..., | x ∈ [0 , a ] } . Intuitively, one has the following picture.
PTIMAL VOLUME UPPER BOUND FOR KÄHLER MANIFOLDS 11
Figure 1. ∆ Y • ( σ ∗ ξ ) ∩ { ≤ ν ≤ a } Then by Proposition 2.4 again, ∈ ∆ Y • ( σ ∗ ξ − x E ) and hence q / ∈ B − ( σ ∗ ξ − x E ) by Theorem 2.3, which contradicts our choice of q . Thus σ ∗ ξ − xE is nef for any x ∈ [0 , a ] . (cid:3) Remark 4.7.
We are kindly informed by a referee that, the above result also followsfrom [27, Theorem 1.3] .Finishing the proof of Theorem 4.1.
It remains to consider the equality case: β ( X, ξ ) n Vol( ξ ) =( n + 1) n . Applying Lemma 4.5(2) and Proposition 4.6, σ ∗ ξ − n + 1 β ( X, ξ ) E is nef . On the other hand, by (2.3), − K X − β ( X, ξ ) ξ is nef . So we find that σ ∗ ( − K X ) − ( n + 1) E = σ ∗ (cid:18) − K X − β ( X, ξ ) ξ (cid:19) + β ( X, ξ ) (cid:18) σ ∗ ξ − n + 1 β ( X, ξ ) E (cid:19) is nef as well. Since p ∈ X can be chosen arbitrarily, we conclude that X ∼ = P n by[11, 17]. (cid:3) Finally, we are able to prove the main result of this paper.
Theorem 4.8 (=Theorem 1.1) . Let ( X, ω ) be an n -dimensional Kähler manifoldwith Ric ( ω ) ≥ ( n + 1) ω. Then one has (cid:90) X ω n ≤ (2 π ) n , and the equality holds if and only if ( X, ω ) is biholomorphically isometric to ( P n , ω F S ) .Proof. Consider the Kähler class ξ := π [ ω ] . Then β ( X, ξ ) ≥ ( n +1) . So Vol( ξ ) ≤ by Proposition 4.4. In other words, (cid:90) X ω n = (2 π ) n Vol( ξ ) ≤ (2 π ) n . And the equality holds if and only if X ∼ = P n by Theorem 4.1, in which case, theequality (cid:82) P n ω n = (2 π ) n implies that [ ω ] = 2 πc ( O P n (1)) . So ∂ ¯ ∂ -lemma gives some f ∈ C ∞ ( P n , R ) such that √− ∂ ¯ ∂f = Ric ( ω ) − ( n + 1) ω ≥ , which forces f to be a constant. Thus ω satisfies the Kähler–Einstein equationRic ( ω ) = ( n + 1) ω. Now by the uniqueness of KE metrics [1], we obtain ω = ω F S up to an automor-phism. (cid:3)
PTIMAL VOLUME UPPER BOUND FOR KÄHLER MANIFOLDS 13
Appendix A. Volume gap for Kähler manifolds with positive Riccicurvature by Yuchen LiuDepartment of Mathematics, Yale University,New Haven, CT 06511, USAEmail: [email protected]
In this appendix, we will prove Theorem 1.3, which is a direct consequence ofthe following theorem.
Theorem A.1.
Let X be an n -dimensional Fano manifold. Let ξ ∈ N ( X ) R be anample R -line bundle. Then there exists ε = ε ( n ) > such that if β ( X, ξ ) n · Vol(
X, ξ ) ≥ ( n + 1) n − ε, then X is biholomorphic to P n . Lemma A.2.
Let X be a Fano manifold. Let { L i } be a sequence of ample Q -line bundles on X . If (cid:15) ( X, L i ) has a positive lower bound, then { L i } is a boundedsequence in N ( X ) R .Proof. Choose a basis { [ C ] , [ C ] , · · · , [ C ρ ] } of N ( X ) R where ρ is the Picard rankof X and each C j is an irreducible curve on X . Let || · || be the norm on N ( X ) R asthe maximum norm with respect to the basis { [ C j ] } . We also denote its dual normon N ( X ) R by || · || as abuse of notation. Choose a > such that (cid:15) ( X, L i ) ≥ a > for any i . Then − K X − aL i is nef which implies < ( L i · C j ) ≤ a − ( − K X · C j ) for any i and j . Hence we know that || L i || = sup max j | c j | =1 | ( L i · ρ (cid:88) j =1 c j C j ) | ≤ ρ (cid:88) j =1 | ( L i · C j ) | ≤ a − ρ (cid:88) j =1 ( − K X · C j ) . Thus the proof is finished. (cid:3)
Proof of Theorem A.1.
Assume to the contrary that ( X i , ξ i ) is a sequence of Fanomanifolds not biholomorphic to P n with ample R -line bundles, such that lim i →∞ β ( X i , ξ i ) n · Vol( X i , ξ i ) = ( n + 1) n . Since β ( X, · ) is a lower semi-continuous function, after perturbing ξ i to ample Q -linebundles L i , we may assume that(A.1) lim i →∞ β ( X i , L i ) n · Vol( X i , L i ) = ( n + 1) n . By boundedness of Fano manifolds [8, 19], after passing to a subsequence we may as-sume that there exists a smooth Fano family π : X → T over an irreducible smoothbase T and a sequence of closed points { t i } ⊂ T such that X i ∼ = X t i . By Lefschetz (1 , theorem and Kodaira vanishing theorem we know that Pic( X ) ∼ = H ( X, Z ) for a Fano manifold X . Hence after replacing T by an irreducible component ofits étale cover, we may assume that R π ∗ Z is a trivial local system on T . Then L i extends to a Q -line bundle L i on X that is generically π -ample. Since the vol-ume of an ample Q -line bundle is the top self intersection number, we know that Vol( X i , L i ) = Vol( X t , L i,t ) for a general t ∈ T . Let t ∈ T be a very general closed point. By the lower semi-continuity of δ -invariants [6] (with respect to the Zariski topology) and the genericity of ampleness,we know that δ ( X t , L i,t ) ≥ δ ( X i , L i ) and (cid:15) ( X t , L i,t ) ≥ (cid:15) ( X i , L i ) . Therefore, by Theorem 2.2 we have β ( X t , L i,t ) ≥ β ( X i , L i ) . This together with(A.1) implies that(A.2) lim inf i →∞ β ( X t , L i,t ) n · Vol( X t , L i,t ) ≥ ( n + 1) n . Let us choose a sequence of positive rational numbers { b i } i ∈ Z > such that lim i →∞ b i · Vol( X t , L i,t ) n = n +1 . Denote by L (cid:48) i := b i L i the rescaled Q -line bundle of L i . Thenwe have(A.3) lim i →∞ Vol( X t , L (cid:48) i,t ) = lim i →∞ b ni · Vol( X t , L i,t ) = ( n + 1) n . Since β ( X t , L (cid:48) i,t ) = b − i · β ( X t , L i,t ) , (A.2) and (A.3) imply that(A.4) lim inf i →∞ β ( X t , L (cid:48) i,t ) ≥ . On the other hand, (A.3) together with (3.1) implies that(A.5) lim sup i →∞ δ ( X t , L (cid:48) i,t ) ≤ . By Theorem 2.2, we have β ( X t , L (cid:48) i,t ) = min { δ ( X t , L (cid:48) i,t ) , (cid:15) ( X t , L (cid:48) i,t ) } . Hence (A.4)and (A.5) imply that(A.6) lim i →∞ δ ( X t , L (cid:48) i,t ) = 1 and lim inf i →∞ (cid:15) ( X t , L (cid:48) i,t ) ≥ . Hence by Lemma A.2 we know that {L (cid:48) i,t } is a bounded sequence of N ( X t ) R ,which (after passing to a subsequence) converges to ξ ∞ ∈ N ( X t ) R . Then ξ ∞ is anef and big R -divisor with Vol( X t , ξ ∞ ) = ( n + 1) n . Pick a point p ∈ X t such that ξ ∞ has positive Seshadri constant at p . Let σ : ˆ X t → X t be the blow up of X t at p with exceptional divisor E . Let ε > be a positive constant. Then we have δ ( X t , L (cid:48) i,t ) ≥ − ε for i (cid:29) . Hence we have n = A X t ( E ) ≥ (1 − ε ) S L (cid:48) i,t ( E ) = 1 − ε Vol( X t , L (cid:48) i,t ) (cid:90) ∞ Vol( σ ∗ L (cid:48) i,t − xE ) dx. Then by the dominated convergence theorem similar to the proof of Lemma 4.5,we have n ≥ X t , ξ ∞ ) (cid:90) ∞ Vol( σ ∗ ξ ∞ − xE ) dx. Since
Vol( X t , ξ ∞ ) = ( n + 1) n , the proof of Lemma 4.5 proceeds to showing that Vol( σ ∗ ξ ∞ − xE ) = Vol( ξ ∞ ) − x n for any x ∈ [0 , Vol( ξ ∞ ) /n ] . Thus Proposition 4.6implies that σ ∗ ξ ∞ − ( n + 1) E is nef. By (A.6) we know that − K X t − ξ ∞ is nef,hence σ ∗ ( − K X t ) − ( n + 1) E is nef. In other words, the Seshadri constant of − K X t at p is at least n + 1 . Hence we conclude that X t ∼ = P n by [2, 26]. Then X i ∼ = P n by rigidity of P n under smooth deformation [18, Exercise V.1.11.12.2], which is acontradiction. (cid:3) PTIMAL VOLUME UPPER BOUND FOR KÄHLER MANIFOLDS 15
References [1] S. Bando, T. Mabuchi, Uniqueness of Einstein Kähler metrics modulo connected group ac-tions. Algebraic geometry, Sendai, 1985, 11-40, Adv. Stud. Pure Math., , North-Holland,Amsterdam, 1987.[2] T. Bauer, T. Szemberg, Seshadri constants and the generation of jets. J. Pure Appl. Algebra213, (2009), 2134—40.[3] R. Berman, B. Berndtsson, The volume of Kähler–Einstein Fano varieties and convex bodies.J. Reine Angew. Math. (2017), 127-152.[4] R. J. Berman, S. Boucksom, M. Jonsson. A variational approach to the Yau-Tian-Donaldsonconjecture, arXiv:1509.04561v2 (2018).[5] H. Blum, M. Jonsson, Thresholds, valuations, and K -stability, arXiv:1706.04548 (2017).[6] H. Blum and Y. Liu, Openness of uniform K-stability in families of Q-Fano varietiesarXiv:1808.09070 (2018).[7] J. Cable, Greatest Lower Bounds on Ricci Curvature for Fano T-manifolds of Complexity 1,Bull. Lond. Math. Soc. (2019), 34-42.[8] F. Campana, Connexité rationnelle des variétés de Fano. (French) Ann. Sci. École Norm. Sup.(4) (1992), 539-545.[9] J. Cheeger, T.H. Colding, On the structure of spaces with Ricci curvature bounded below. I.J. Differential Geom. (1997), 406-480.[10] I. Cheltsov, Y.A. Rubinstein, K. Zhang, Basis log canonical thresholds, local intersectionestimates, and asymptotically log del Pezzo surfaces, Selecta Math. (N.S.) 25 (2019), Art. 34,36 pp.[11] K. Cho, Y. Miyaoka and N. I. Shepherd-Barron, Characterizations of projective spaces andapplications to complex symplectic manifolds, Higher dimensional birational geometry (Kyoto,1997), 1-88, Adv. Stud. Pure Math. , Math. Soc. Japan, Tokyo, 2002.[12] S.K. Donaldson, Some recent developments in Kähler geometry and exceptional holonomy.Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. I.Plenary lectures, 425-451, World Sci. Publ., Hackensack, NJ, 2018.[13] T. de Fernex, A. Küronya and R. Lazarsfeld, Higher cohomology of divisors on a projectivevariety, Math. Ann. (2007), 443-455.[14] L. Ein, R. Lazarsfeld, M. Mustaţă, M. Nakamaye, M. Popa, Asymptotic invariants of baseloci. Ann. Inst. Fourier (Grenoble) (2006), 1701-1734.[15] K. Fujita, Optimal bounds for the volumes of Kähler–Einstein Fano manifolds. Amer. J.Math. (2018), 391-414.[16] K. Fujita, Y. Odaka, On the K-stability of Fano varieties and anticanonical divisors. TohokuMath. J. (2) (2018), 511-521.[17] S. Kebekus, Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron,Complex geometry (Göttingen, 2000), 147-155, Springer, Berlin, 2002.[18] J. Kollár. Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Gren-zgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, . Springer-Verlag, Berlin,1996. viii+320 pp[19] J. Kollár, Y. Miyaoka, and S. Mori, Rational connectedness and boundedness of Fano mani-folds. J. Differential Geom. (1992), 765-779.[20] A. Küronya, V. Lozovanu, Positivity of line bundles and Newton–Okounkov bodies. Doc.Math. (2017), 1285-1302.[21] R. Lazarsfeld, Positivity in Algebraic Geometry, I,II, Springer, 2004.[22] R. Lazarsfeld, M. Mustaţă, Convex bodies associated to linear series. Ann. Sci. Éc. Norm.Supér. (4) (2009), 783-835.[23] C. Li, Greatest lower bounds on Ricci curvature for toric Fano manifolds, Adv. Math. 226(2011), 4921–4932.[24] G. Liu, Kähler manifolds with Ricci curvature lower bound. Asian J. Math. (2014), 69-99.[25] Y. Liu, The volume of singular Kähler–Einstein Fano varieties. Compos. Math. (2018),1131-1158.[26] Y. Liu, Z. Zhuang, Characterization of projective spaces by Seshadri constants. Math. Z. (2018), 25-38.[27] J. Park, J. Shin, Seshadri constants and Okounkov bodies revisited, arXiv:1812.07261 (2018). [28] Y.A. Rubinstein, Some discretizations of geometric evolution equations and the Ricci iterationon the space of Kähler metrics, Adv. Math. (2008), 1526–1565.[29] Y.A. Rubinstein, On the construction of Nadel multiplier ideal sheaves and the limitingbehavior of the Ricci flow, Trans. Amer. Math. Soc. 361 (2009), 5839–5850.[30] J. Song, X. Wang, The greatest Ricci lower bound, conical Einstein metrics and Chern numberinequality, Geom. Topol. 20 (2016), 49–102.[31] G. Székelyhidi, Greatest lower bounds on the Ricci curvature of Fano manifolds, CompositioMath. 147 (2011), 319–331.[32] G. Tian, On stability of the tangent bundles of Fano varieties, Internat. J. Math. 3 (1992),401–413.[33] F. Wang, A volume stability theorem on toric manifolds with positive Ricci curvature. Proc.Amer. Math. Soc. (2015), 3613-3618.[34] K. Zhang, Continuity of delta invariants and twisted Kähler–Einstein metrics,arXiv:2003.11858 (2020). Beijing International Center for Mathematical Research, Peking University.
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