J. and A. ,

?. , By Theorem B it follows that M NA (X triv , A triv ) = DF(X triv

, It remains to prove that (X , A) is trivial if J NA (X , A) = 0. To see this, note that in this situation t ? J(? t ) is convex, and moreover J(? 0 ) = 0 and 0 ? J(? t ) < C for some C > 0 and all t ? [0, +?), other words

, Let X be a compact Kähler manifold with discrete automorphism group, and let ? ? H 1,1 (X, R) a Kähler class on X

, K-semistability is known (Corollary 5.2.3). From Proposition 5.3.1 we also have "uniform K-semistability

, ? Because of Theorem 5.3.2 it is clear that uniform K-stability implies K-stability, as expected

, The final part of the proof of Theorem 5.3.2, using the injectivity lemma 5.1.2, yields also a simple and direct proof of K-stability of cscK manifolds

, The reader may compare with the convex combinations of algebraic test configurations used in A. Isopoussus's thesis

, Proof of Lemma 6.1.2. Let (X , A) be a relatively Kähler smooth and dominating test configuration for (X, ? X )

, Now if ? X ? Kah(X), let ? > 0 be sufficiently small so that µ * p * 1 ? X + ?[E] is relatively Kähler on X. Then (X , A t ) := (X , µ * p *, p.1

, The condition A 0 = A is satisfied, and the subpoints (3) and (4) follow from the intersection theoretic interpretations of DF and J NA

, The coefficients of the above polynomials are straightforward to compute explicitly, using the intersection theoretic expressions for DF and J NA. Vanishing of the Futaki invariant and extremal metrics

, Suppose that (X, ? X ) is K-unstable. Then for any Kähler class ? X on X there is an 0 > 0 such that (X, (1 ? t)? X + t? X ) is K-unstable for

, By Proposition 3.2.20 there is then a smooth and dominating relatively Kähler test configuration (X , A) for (X, ? X ) such that DF(X , A) < 0. Fix any Kähler class ? X on X. By Theorem 6.1.2 there is for each t ? [0, 1] a normal and relatively Kähler test configuration (X , A t ) for (X, (1 ? t)? X + t? X )

, inclusion cscK locus ? K-semistable locus is strict in general, and says something slightly more than simply having an example of when this happens. Indeed, the cscK locus is relatively open in the set of Kähler classes with vanishing Futaki invariant, while the K-semistable locus is 'closed' in the sense of Corollary 6.1.8. This argument moreover holds even for non-projective compact Kähler manifolds and for projective Kähler manifolds

, The Yau-Tian-Donaldson conjecture for Kähler-Einstein metrics twisted by a transcendental class The twisted Kähler-Einstein equation has been extensively studied

, )-current (possibly non-positive and singular) on X. A solution of this equation is called a twisted Kähler-Einstein metric, and is realized as critical points of a twisted Mabuchi energy, or alternatively, of the twisted Ding energy functional (see e.g. [Rub08] and Chi Li's thesis). When ? is smooth

, is the current of integration of a divisor, the equation (6.1) was studied in

, the ordinary Kähler-Einstein equation on Fano manifolds. In [BBJ15] it was then proven that uniformly K-stable Fano manifolds with discrete automorphism group admit cscK metric. This is the most difficult direction of the Yau-TianDonaldson conjecture, and yields a variational proof in the Fano case. Based on the variational nature of the proof in [BBJ15] one could expect to be able to make use of our transcendental formalism (cf. Secion 4) in the case of a twisted Kähler-Einstein metrics

, )-current on X. We then ask the following: Question 6.2.2. Can the variational proof of [BBJ15] be adapted to show that, if X is uniformly twisted K-stable (in an appropriate sense

V. Apostolov, D. M. Calderbank, P. Gauduchon, and C. W. Tonnesen-friedman, Hamiltonian 2-forms in Kähler geometry III. Extremal metrics and stability, Invent, Math, vol.173, issue.3, pp.547-601, 2008.

J. Aroca, H. Hironaka, and J. Vicente, The theory of maximal contact, Memorias de Matematica del, p.29, 1975.

, Desingularization theorems, Memorias de Matematica del, p.30, 1977.

T. Aubin, Équations du type de Monge-Ampère sur les variétés Kähleriennes compactes, C. R. Acad. Sci. Paris, vol.283, pp.119-121, 1976.

R. Berman and B. Berndtsson, Convexity of the K-energy on the space of Kähler metrics and uniqueness of extremal metrics, 2014.

R. Berman, S. Boucksom, V. Guedj, and A. Zeriahi, A variational approach to complex Monge-Ampère equations, Publ. Math. de l'IHES, vol.117, pp.179-245, 2013.

R. Berman, S. Boucksom, and M. Jonsson, A variational approach to the YauTian-Donaldson Conjecture, 2015.

R. Berman, T. Darvas, and C. Lu, Convexity of the extended K-energy and the large time behaviour of the weak Calabi flow, 2015.

, Regularity of weak minimizers of the K-energy and applications to properness and K-stability, 2016.

R. Berman, From Monge-Ampère equations to envelopes and geodesic rays in the zero temperature limit, 2013.

K. Of, Fano varieties admitting Kähler-Einstein metrics, vol.203, pp.1-53, 2016.

S. Boucksom and V. Guedj, Regularizing properties of the Kähler-Ricci flow, An introduction to the, Lecture notes in mathematics 2086, pp.189-238, 2013.

S. Boucksom, T. Hisamoto, and M. Jonsson, Uniform K-stability, DuistermaatHeckman measures and singularities of pairs, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01708671

, Uniform K-stability and asymptotics of energy functionals in Kähler geometry, 2016.

W. P. Barth, K. Hulek, C. A. Peters, and A. Van-de-ven, Ergebnisse der Mathematik und ihrer Grenzgebiete. A series of Modern Surveys in Mathematics, vol.4, 2004.

S. Boucksom and M. Jonsson, Tropical and non-Archimedean limits of degenerating families of volume forms, J. Ec. Polytech. Math, vol.4, pp.87-139, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01708670

Z. Blocki, On geodesics in the space of Kähler metrics. advances in geometric analysis, Adv. Lect. Math. (ALM), vol.21, pp.3-19, 2012.

, The Calabi-Yau theorem. Complex Monge-Ampère equations and geodesics in the space of Kähler metrics, vol.2038, 2013.

E. Bierstone and P. D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math, vol.128, pp.207-302, 1997.

S. Boucksom, Monge-Ampère equations on complex manifolds with boundary, Complex Monge-Ampère equations and geodesics in the space of Kähler metrics, Lecture notes in mathematics 2038, pp.257-282, 2012.

C. Le-brun and S. R. Simanca, Extremal Kähler metrics and complex deformation theory, Geometric and functional analysis, vol.4, pp.298-336, 1994.

E. Bedford and B. A. Taylor, The Dirichlet problem for the complex Monge-Ampère operator, Invent. Math, vol.37, pp.1-44, 1976.

, A new capacity for plurisubharmonic functions, Acta Math, vol.149, pp.1-40, 1982.

E. Calabi, Extremal Kähler metrics, Seminar on Differential Geometry, vol.102, pp.259-290, 1982.

, Extremal Kähler metrics II, Differential Geometry and Complex Analysis, pp.95-114, 1985.

X. X. Chen, S. Donaldson, and S. Sun, Kähler-Einstein metrics on Fano manifolds I: approximation of metrics with cone singularities, J. Amer. Math. Soc, vol.28, pp.183-197, 2015.

, Kähler-Einstein metrics on Fano manifolds II: limits with cone angle less than 2pi, J. Amer. Math. Soc, vol.28, pp.199-234, 2015.

, Kähler-Einstein metrics on Fano manifolds III: limits as cone angle approaches 2pi and completion of the main proof, J. Amer. Math. Soc, vol.28, pp.235-278, 2015.

X. X. Chen, On the lower bound of the Mabuchi energy and its application, Int. Math. Res. Not, issue.12, pp.607-623, 2000.

, The space of Kähler metrics, J. Diff. Geom, vol.56, issue.12, pp.189-234, 2000.

L. Caffarelli, J. J. Kohn, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations II. Complex Monge-Ampère, and uniformly elliptic, equations, C.P.A.M, vol.38, issue.2, pp.209-252, 1985.

X. X. Chen, L. Li, and M. Paun, Approximation of weak geodesics and subharmonicity of Mabuchi energy, Ann. Fac. Sci. Toulouse. Math, vol.25, issue.5, pp.935-957, 2016.

T. C. Collins and G. Szekelyhidi, The twisted Kähler-Ricci flow, J. Reine Aingew. Math, vol.716, pp.179-205, 2014.

X. X. Chen, S. Sun, B. Wang, and K. Flow, Kähler-Einstein metric, and K-stability

J. Chu, V. Tosatti, and B. Weinkove, On the C 1,1 regularity of geodesics in the space of Kähler metrics

T. Darvas, The mabuchi completion of the space of Kähler potentials, to appear in Amer, J. Math, 2014.

, The Mabuchi geometry of finite energy classes, Adv. Math, issue.285, pp.182-219, 2015.

, Geometric pluripotential theory on Kähler manifolds, 2017.

, Weak geodesic rays in the space of Kähler potentials and the class

E. , Journal of the Institute of Mathematics of Jussieu, vol.16, issue.4, pp.837-858, 2017.

J. Demailly, Mésures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines, Mémoires de la SMF, 1984.

, Complex analytic and differential geometry, Open source, 2012.

R. Dervan, Relative K-stability for Kähler manifolds, 2016.

, Uniform stability of twisted constant scalar curvature Kähler metrics, Int. Math. Res. Not, issue.15, 2016.

T. Darvas and L. Lempert, Weak geodesic rays in the space of Kähler metrics, Math. Research Letters, vol.19, issue.5, 2012.

S. K. Donaldson, Anti-self-dual Yang-Mills connections over complex surfaces and stable vector bundles, Proc. London Math. Soc, vol.50, pp.1-26, 1985.

, Symmetric spaces, Kähler geometry and Hamiltonian dynamics, Northern California Symplectic Geometry Seminar, vol.2, pp.13-33, 1999.

, Scalar curvature and stability of toric varieties, J. Diff. Geom, issue.62, pp.289-349, 2002.

, Lower bounds on the Calabi functional, J. Diff. Geom, vol.70, issue.3, pp.453-472, 2005.

, Scalar curvature and projective embeddings, II, J Math, vol.56, issue.3, pp.345-356, 2005.

, Kähler metrics with cone singularities along a divisor, Essays in Mathematics and its Applications, pp.49-79, 2012.

, Stability of algebraic varieties and Kähler geometry, 2017.

R. Dervan, J. Ross, and K. For-kähler-manifolds, Math. Res. Lett, 2016.

T. Darvas and Y. A. Rubinstein, Tian's properness conjecture and Finsler geometry of the space of Kähler metrics, J. Amer. Math. Soc, vol.30, issue.2, pp.347-387, 2017.

V. Datar and G. Szekelyhidi, Kähler-Einstein metrics along the smooth continuity method, Geom. Funct. Analysis, vol.26, issue.4, pp.975-1010, 2016.

W. Y. Ding and G. Tian, The generalized Moser-Trudinger inequality, Nonlinear Analysis and Microlocal Analysis (K, pp.57-70, 1992.

R. Elkik, Métriques sur les fibrés d'intersection, Duke math, journal, issue.1, pp.303-328, 1990.

G. Fischer, Complex Analytic Geometry, Lecture Notes in Mathematics, vol.538, 1976.

J. E. Fornaess and R. Narasimhan, The Levi problem on complex spaces with singularities, Math. Ann, issue.248, pp.47-72, 1980.

F. Forstneric, Actions of (R, +) and (C, +) on complex manifolds, Math. Z, issue.223, pp.123-153, 1996.

A. Futaki, An obstruction to the existence of Einstein-Kähler metrics, Invent. Math, vol.73, pp.437-443, 1983.

, Kähler-Einstein metrics and Integral Invariants, vol.1314, 1988.

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 1983.

B. Guan, The Dirichlet problem for complex Monge-Ampère equations and regularity of the pluri-complex Green function, Comm. Anal. Geom, vol.6, issue.4, pp.687-703, 1998.

V. Guedj and A. Zeriahi, Degenerate complex Monge-Ampère equations, EMS Tracts in Mathematics, vol.26, 2017.

H. Hironaka, Introduction to the theory of infinitely near singular points, Memorias de Matematica del, p.28, 1974.

A. Isopoussu, K-stability of relative flag varieties, 2013.

T. D. Jeffres, R. Mazzeo, and Y. A. Rubinstein, Kähler-Einstein metrics with edge singularities, Annals of Mathematics, vol.183, pp.95-176, 2016.

J. Keller, About projectivisation of Mumford semistable bundles over a curve, Journal of the London Math. Society, vol.93, issue.1, pp.159-174, 2016.

G. Kempf, F. Knudsen, D. Mumford, and B. Saint-donat, , 1973.

J. Kollár, J. Nicaise, and C. Xu, Semi-stable extensions over 1-dimensional bases, 2017.

K. Kodaira, On Kähler varieties of restricted type (an intrinsic characterization of algebraic varieties), Ann. of Math, vol.60, pp.28-48, 1954.

J. Kollár, Lectures on resolution of singularities, vol.166, 2007.

, Singularities of the minimal model program, Cambridge Tracts in Mathematics, 2013.

J. Keller and J. Ross, A note on Chow stability of the projectivization of Gieseker stable bundles, Journal of Geom, pp.1-21, 2011.
URL : https://hal.archives-ouvertes.fr/hal-01282439

C. Li, Constant scalar curvature Kähler metric obtains the minimum of the Kenergy, Int. Math. Res. Notices, issue.9, pp.2161-2175, 2011.

A. Lichnerowicz, Sur les transformations analytiques des varietes kähleriennes compactes, C.R. Acad. Sci. Paris, vol.244, pp.3011-3013, 1957.

C. Li and C. Xu, Special test configurations and K-stability of Fano varieties, Annals of Math, vol.180, pp.197-232, 2014.

T. Mabuchi, A functional integrating Futaki invariants, Proc. Japan Acad, vol.61, pp.119-120, 1985.

, K-energy maps integrating Futaki invariants, Tohoku Math. J, vol.38, issue.4, pp.575-593, 1986.

, Some symplectic geometry on compact Kähler manifolds I, Osaka J. Math, vol.24, pp.227-252, 1987.

, stability of constant scalar curvature polarization, 2008.

Y. Matsushima, Sur la structure du groupe d'homeomorphismes analytiques d'une certaine variete kählerienne, Nagoya Math. J, vol.11, pp.145-150, 1957.

D. Mumford, J. Fogarty, and F. Kirwan, Geometric Invariant Theory, 1994.

A. Moriwaki, The continuity of Delignes pairing, Int. Math. Res. Notices, issue.19, pp.1057-1066, 1999.

D. W. Nystrom, Test configurations and Okounkov bodies, Compos. Math, vol.148, issue.6, pp.1736-1756, 2010.

Y. Odaka, A generalization of the Ross Thomas slope theory, Osaka J. of Math, vol.50, issue.1, pp.171-185, 2013.

D. H. Phong, J. Ross, and J. Sturm, Deligne pairings and the Knudsen-Mumford expansion, J. Diff. Geom, vol.78, issue.3, pp.475-496, 2008.

D. H. Phong and J. Sturm, The Dirichlet problem for degenerate complex MongeAmpère equations, Communications in Analysis and Geometry, vol.18, issue.1, pp.145-170, 2010.

S. Paul and G. Tian, CM stability and the generalized Futaki invariant I, 2006.

, CM stability and the generalized Futaki invariant II, Astérisque, issue.328, pp.339-354, 2009.

J. Ross and R. P. Thomas, An obstruction to the existence of constant scalar curvature Kähler metrics, J. Diff. Geom, vol.72, pp.429-466, 2006.

, A study of the Hilbert-Mumford criterion for the stability of projective varieties, J. Algebraic Geom, vol.16, issue.2, pp.201-255, 2007.

Y. A. Rubinstein, On energy functionals, Kähler-Einstein metrics, and the MoserTrudinger-Onofri neighborhood, J. Funct. Anal, vol.255, pp.2641-2660, 2008.

, Smooth and singular Kähler-Einstein metrics, Geometric and Spectral Analysis, Contemp. Math, pp.45-138, 2014.

Z. Sjöström-dyrefelt, K-semistability of cscK manifolds with transcendental cohomology class, 2016.

, On K-polystability of cscK manifolds with transcendental cohomology class, 2017.

S. Semmes, Complex Monge-Ampère equations and symplectic manifolds, Amer. J. Math, vol.114, pp.495-550, 1992.

J. Serre, Geometrie algebrique et geometrie analytique, Ann. Inst. Fourier, vol.6, pp.1-42, 1956.

Y. Shu, Compact complex surfaces and constant scalar curvature Kähler metrics, Geom Dedicata, vol.138, issue.1, pp.151-172, 2009.

J. Stoppa, K-stability of constant scalar curvature Kähler manifolds, Advances in Mathematics, vol.221, pp.1397-1408, 2009.

, Twisted constant scalar curvature Kähler metrics and Kähler slope stability, J. Diff. Geom, vol.83, pp.663-691, 2009.

G. Szekelyhidi, Introduction to extremal Kähler metrics, vol.152, 2014.

, With an appendix by S, Filtrations and test configurations, vol.362, pp.451-484, 2015.

A. Teleman, Symplectic stability, analytic stability in non-algebraic complex geometry, Int. J. Math, vol.15, issue.2, pp.183-209, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00881709

G. Tian, The K-energy on hypersurfaces and stability, Comm. Anal. Geom, vol.2, pp.239-265, 1994.

, Kähler-Einstein metrics with positive scalar curvature, Invent. Math, vol.130, issue.1, pp.1-37, 1997.

, Discrete and continuous dynamical systems, Chern forms and geometric stability, vol.6, pp.211-220, 2000.

, Canonical metrics in Kähler geometry, Lectures in Mathematics ETH Zurich, 2000.

, Existence of Einstein metrics on Fano manifolds, Metric and Differential Geometry, pp.119-159, 2012.

K. Kähler-einstein-metrics, Communications on Pure and Applied Math, vol.68, pp.1085-1156, 2015.

G. Tian and X. Zhu, A nonlinear inequality of Moser-Trudinger type, Calc. Var. PDE, vol.10, pp.349-354, 2000.

C. Voisin, On the homotopy types of Kähler compact and complex projective manifolds, Inventiones. Math, vol.157, issue.2, pp.329-343, 2004.

, Recent progress in Kähler and complex algebraic geometry, 4ECM Stockholm, 2004.

, On the homotopy types of Kähler manifolds and the birational kodaira problem, J. Differential. Geom, vol.72, issue.1, pp.43-71, 2006.

X. Wang, Height and GIT weight, Math. Res. Lett, vol.19, issue.04, pp.909-926, 2012.

J. Wlodarczyk, Resolution of singularities of analytic spaces, Proceedings of Gukova Geometry-Topology Conference, pp.31-63, 2008.

S. Yau, On the Ricci curvature of a compact Kähler manifolds and the complex Monge-Ampère equation, Comm. Pure Appl. Math, vol.31, pp.339-411, 1978.

S. Zhang, Heights and reductions of semi-stable varieties, Compositio Math, vol.104, pp.77-105, 1996.

B. Zhou and X. Zhu, Relative K-stability and modified K-energy on toric manifolds, Adv. Math, vol.219, pp.1327-1362, 2008.