S. Boucksom, J.-P. Demailly, M. Paun and Th. Peternell proved that the cone
of mobile curves ME(X) of a projective complex manifold X is dual to the cone
generated by classes of effective divisors and conjectured an extension of this
duality in the Kaehler set-up. We show that their conjecture implies that ME(X)
coincides with the cone of integer classes represented by closed positive
smooth (n-1,n-1)-forms. Without assuming the validity of the conjecture we
prove that this equality of cones still holds at the level of degree functions

Toma, Matei

English

arXiv

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 348 (2010) 71–73Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comAnalytic GeometryA Note on the cone of mobile curvesUne Note sur le cône des courbes mobilesMatei Toma a,b,∗a Institut Élie-Cartan, Nancy-Université, CNRS, INRIA, B.P. 239, 54506 Vandoeuvre-lès-Nancy cedex, Franceb Institute of Mathematics of the Romanian Academy, Romaniaa r t i c l e i n f o a b s t r a c tArticle history:Received 4 September 2009Accepted after revision 10 November 2009Available online 24 December 2009Presented by Jean-Pierre DemaillyS. Boucksom, J.-P. Demailly, M. Păun and Th. Peternell proved that the cone of mobilecurves ME(X) of a projective complex manifold X is dual to the cone generated by classesof effective divisors and conjectured an extension of this duality in the Kähler set-up. Weshow that their conjecture implies that ME(X) coincides with the cone of integer classesrepresented by closed positive smooth (n − 1,n − 1)-forms. Without assuming the validityof the conjecture we prove that this equality of cones still holds at the level of degreefunctions.© 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éS. Boucksom, J.-P. Demailly, M. Păun et Thomas Peternell ont montré que le cône descourbes mobiles ME(X) d’une variété projective complexe X est le dual du cône engendrépar les classes de diviseurs effectifs, et ils ont conjecturé que cette dualité pouvait s’étendredans le contexte kählerien. Nous montrons que cette conjecture implique que ME(X)coïncide avec le cône des classes entières représentées par des formes positives ferméesde type (n − 1,n − 1) et de classe C∞ . Sans supposer que cette conjecture soit vraie,nous montrons que cette égalité de cônes a lieu en tout cas au niveau des fonctions degréassociées.© 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Let X be a smooth complex projective variety of dimension n. A curve C on X is called mobile if it is the memberof an algebraic family of (generically) irreducible curves covering X . Let ME(X) denote the closed convex cone generatedby classes of mobile curves inside N1(X) := (Hn−1,n−1R (X) ∩ H2n−2(X,Z)/Tors) ⊗Z R. We shall call the elements of ME(X)mobile classes.In [2] it is shown that the following cones in N1(X) coincide:(1) the cone ME(X) of mobile curves,(2) the cone MNS := M ∩ N1, where M ⊂ Hn−1,n−1R (X) is the closure of the convex cone generated by cohomology classesof currents of the type ν∗(ω̃1 ∧ · · · ∧ ω̃n−1) for Kähler forms ω̃1, . . . , ω̃n−1 on a modification ν : X̃ → X on X ,(3) the dual cone (ENS)∨ of the cone ENS of pseudo-effective divisors on X .* Address for correspondence: Institut Élie-Cartan, Nancy-Université, CNRS, INRIA, B.P. 239, 54506 Vandoeuvre-lès-Nancy cedex, France.E-mail address: Matei.Toma@iecn.u-nancy.fr.1631-073X/$ – see front matter © 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2009.11.00372 M. Toma / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 71–73It was known from [4] that ENS = E ∩ NSR , where E is the cone of classes of positive closed currents of type (1,1) andNSR(X) := (H1,1R (X)∩ H2(X,Z)/Tors)⊗Z R. In [2, Conjecture 2.3] it is further conjectured that the cones M and E are dual.In this Note we compare ME(X) to the closed convex cone Pn−1,n−1 in Hn−1,n−1R(X) generated by closed positive smooth(n−1,n−1)-forms. From the above statements it is clear that Pn−1,n−1 ∩ N1(X) ⊂ ME(X). The converse inclusion will followfrom our arguments if we admit the conjecture of Boucksom, Demailly, Păun and Peternell. If not we still get an equality atthe level of degree functions as follows.Any mobile class α gives rise to a degree functiondegα : Pic(X) → R, degα L := c1(L)αand further to a notion of stability of torsion-free sheaves on X which generalizes the classical case α = Hn−1 for a class Hof an ample divisor, cf. [3]. On the other side we consider semi-Kähler metrics on X , i.e. such that their associated Kählerforms ω satisfy dωn−1 = 0. Such a metric gives likewise a degree function:degω : Pic(X) → R, degω L := c1(L)[ωn−1].Then we can state:Theorem. For any mobile class α in the interior of the mobile cone ME(X) there exists a semi-Kähler metric on X with associatedKähler form ω such that degα = degω on Pic(X).By [6, Corollary 5.3.9] this has the following consequence on the moduli space of stable vector bundles:Corollary. If α is a class in the interior of ME(X), then the smooth part of the moduli space of stable vector bundles with respect to αadmits a natural Kähler structure.Note that the statement of the theorem would be easy to prove by a direct averaging process if we knew that mobilecurves are given by submersive families. But in general focal points could create singularities in the direct image of theintegral mean value. As the referee suggests this could be possibly overcome by picking smoothly varying densities on thecurves in such a way that their global support on the family of curves avoid critical points. We will use instead a technicallyeasier duality argument.We start by introducing some more notation: We denote by D′ p,q the space of currents of bidegree (p,q) (and bidimen-sion (n − p,n − q)) on X . LetV d = V d(X) :={T ∈ D′1,1R∣∣ dT = 0}/ddc D′0,0R,V ddc = V ddc (X) :={T ∈ D′1,1R∣∣ ddc T = 0}/{∂̄ S + ∂ S̄ ∣∣ S ∈ D′1,0},V +d , V+ddc the cones generated by positive currents in V d and V ddc respectively and V+d,NS , V+ddc ,NS their intersections withNSR(X) := (H1,1R (X) ∩ H2(X,Z)/Tors) ⊗Z R.The following lemma makes use of the duality theorem of [2] and of a result of Alessandrini and Bassanelli on pull-backsof pluriharmonic currents [1].Lemma. The natural map j : V d → V ddc induces a bijection between positive conesV +d,NS → V +ddc ,NS.Proof. It is known that in the case of compact Kähler manifolds the map j, which associates to a class of a bidegree (1,1)closed current {T }d ∈ V d its class {T }ddc ∈ V ddc , is well defined and bijective.The inclusion j(V +d ) ⊂ V +ddc being obvious, we consider a ddc-closed positive current T with {T }ddc ∈ V ddc ,NS . Let η :=j−1({T }ddc ) ∈ V d. We shall show that η ∈ V +d .By the cited result of [2] it suffices to check that for any modification ν : X̃ → X and Kähler forms ω̃1, . . . , ω̃n−1 on X̃one hasην∗([ω̃1 ∧ · · · ∧ ω̃n−1])  0.Butην∗([ω̃1 ∧ · · · ∧ ω̃n−1]) = ν∗(η)[ω̃1 ∧ · · · ∧ ω̃n−1] = ν∗({T }ddc )[ω̃1 ∧ · · · ∧ ω̃n−1]and Theorem 3 of [1] asserts that ν∗({T }ddc ) ∈ V +ddc ( X̃), whence the desired inequality. We can now give the proof of the theorem.M. Toma / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 71–73 73Proof. Let α be an element in the interior of the cone of mobile curves. We denote by D′ 1,1+ the cone of positive currentsinside the space D′ 1,1 of bidegree (1,1) currents on X . We fix a Kähler form σ on X and set D′ 1,1+,σ := {T ∈ D′ 1,1+ |∫X T ∧σ n−1 = 1}. This is a compact set for the weak topology on D′ 1,1 [5, III.1.23]. Let β1, . . . , βk ∈ Hn−1,n−1R (X) be such thatV ddc ,NS = {t ∈ V ddc | tβ1 = 0, . . . , tβk = 0} and set W := {T ∈ D′ 1,1 | ddc T = 0, {T }ddc α = 0, {T }ddc β1 = 0, . . . , {T }ddc βk = 0}.Remark that W and D′ 1,1+,σ are disjoint. Indeed, if T ∈ D′ 1,1+,σ were ddc-closed and {T } ∈ V +ddc ,NS , then by the lemma therewould exist a d-closed positive current S ∈ D′ 1,1+ such that {T }ddc = j({S}d) and in particular{T }ddc α = {S}dα > 0.The Hahn–Banach theorem then implies the existence of a functional on D′ 1,1, which vanishes on W and is positive onD′ 1,1+,σ . This functional is thus given by a real (n − 1,n − 1)-form u on X . The form u is strictly positive on X since thefunctional is positive on D′ 1,1+,σ . The vanishing on W implies that u is also d-closed. As functionals on NSR(X), [u] and αhave the same kernel, hence they coincide up to some multiplicative constant.It is enough now to take a positive (n − 1)-st root ω of u. For the convenience of the reader we show how this is done.Remark first that(i∑1i, jnaij dzi ∧ dz̄ j)n−1= (n − 1)!i(n−1)2∑1i, jn(−1)i+ jc ji dẑi ∧ d ˆ̄z j, (1)where we denoted by ci j the cofactor of aij in the matrix A = (aij)1i, jn and dẑi := dz1 ∧ · · · ∧ dzi−1 ∧ dzi+1 ∧ · · · ∧ dzn ,ˆdz̄ j := dz̄1 ∧ · · · ∧ dz̄ j−1 ∧ dz̄ j+1 ∧ · · · ∧ dz̄n . The relation t C A = det(A)In for the matrix of cofactors C = (ci j)1i, jn impliesA = n−1√det(C) t C−1when A is positive definite. When the matrix C is positive definite, one obtains a unique positive definite solution A ofEq. (1). AcknowledgementI wish to thank Sébastien Boucksom and the anonymous referee for their comments on this Note.References[1] L. Alessandrini, G. Bassanelli, Modifications of compact balanced manifolds, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995) 1517–1522.[2] S. Boucksom, J.-P. Demailly, M. Păun, Th. Peternell, The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension,arXiv:math/0405285.[3] F. Campana, Th. Peternell, Geometric stability of the cotangent bundle and the universal cover of a projective manifold, arXiv:math/0405093.[4] J.-P Demailly, Regularization of closed positive currents and intersection theory, J. Algebraic Geom. 1 (1992) 361–409.[5] J.P. Demailly, Complex analytic and algebraic geometry, http://www-fourier.ujf-grenoble.fr/~demailly/books.html.[6] M. Lübke, A. Teleman, The Kobayashi–Hitchin Correspondence, World Scientific Publishing Co., Inc., River Edge, NJ, 1995.

A Note on the cone of mobile curves

Matei Toma

Comptes Rendus Mathématique

https://comptes-rendus.academie-sciences.fr/mathematique/item/10.1016/j.crma.2009.11.003.pdf

A note on the cone of mobile curves

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