The authors would like to give special thanks to N. Boston, K. Buzzard, and B. Conrad for providing so much valuable feedback on earlier versions of this paper. They are also grateful to A. Agboola, M. Bertolini, B. Edixhoven, J. Fearnley, R. Gross, L. Guo, F. Jarvis, H. Kisilevsky, E. Liverance, J. Manoharmayum, K. Ribet, D. Rohrlich, M. Rosen, R. Schoof, J.-P. Serre, C. Skinner, D. Thakur, J. Tilouine, J. Tunnell, A. Van der Poorten, and L. Washington for their helpful comments. Darmon thanks the members of CICMA and of the Quebec-Vermont Number Theory Seminar for many stimulating conversations on the topics of this paper, particularly in the Spring of 1995. For the same reason Diamond is grateful to the participants in an informal seminar at Columbia University in 1993-94, and Taylor thanks those attending the Oxford Number Theory Seminar in the Fall of 1995.

/pdf/fermat-s-last-theorem-2jvvp9fxkd.pdf

Fermat’s Last Theorem

We introduce complex singularity exponents of plurisubharmonic functions and prove a general semi-continuity result for them. This concept contains as a special case several similar concepts which have been considered e.g. by Arnold and Varchenko, mostly for the study of hypersurface singularities. The plurisubharmonic version is somehow based on a reduction to the algebraic case, but it also takes into account more quantitative informations of great interest for complex analysis and complex differential geometry. We give as an application a new derivation of criteria for the existence of Kahler–Einstein metrics on certain Fano orbifolds, following Nadel's original ideas (but with a drastic simplication in the technique, once the semi-continuity result is taken for granted). In this way, three new examples of rigid Kahler–Einstein Del Pezzo surfaces with quotient singularities are obtained.

/pdf/semi-continuity-of-complex-singularity-exponents-and-kahler-d6ljq73f8n.pdf

Semi-continuity of complex singularity exponents and Kähler–Einstein metrics on Fano orbifolds

Galois representations into GL2(Zp[[X]]) attached to ordinary cusp forms.

We introduce complex singularity exponents of plurisubharmonic functions and prove a general semi-continuity result for them. This concept contains as a special case several similar concepts which have been considered e.g. by Arnold and Varchenko, mostly for the study of hypersurface singularities. The plurisubharmonic version is somehow based on a reduction to the algebraic case, but it also takes into account more quantitative informations of great interest for complex analysis and complex differential geometry. We give as an application a new derivation of criteria for the existence of K\"ahler-Einstein metrics on certain Fano orbifolds, following Nadel's original ideas (but with a drastic simplication in the technique, once the semi-continuity result is taken for granted). In this way, 3 new examples of rigid K\"ahler-Einstein Del Pezzo surfaces with quotient singularities are obtained.

Semi-continuity of complex singularity exponents and K\"ahler-Einstein metrics on Fano orbifolds

The main result of this paper shows that "test configurations" give new lower bounds on the $L^{2}$ norm of the scalar curvature on a Kahler manifold. This is closely analogous to the analysis of the Yang-Mills functional over Riemann surfaces by Atiyah and Bott. The proof uses asymptotic approximation by finite-dimensional problems: the essential ingredient being the Tian-Zelditch-Lu expansion of the "density of states" function.

/pdf/lower-bounds-on-the-calabi-functional-5cws044isp.pdf

Lower bounds on the Calabi functional

Assuming uniform bounds for the curvature, the exponential convergence of the Kahler-Ricci flow is established under two con- ditions which are a form of stability: the Mabuchi energy is bound- ed from below, and the dimension of the space of holomorphic vector fields in an orbit of the diffeomorphism group cannot jump up in the limit.

/pdf/on-stability-and-the-convergence-of-the-kahler-ricci-flow-50vmiszasc.pdf

On stability and the convergence of the Kähler-Ricci flow

It is shown that geodesics in the space of K\\\"ahler potentials can be uniformly approximated by geodesics in the spaces of Bergman metrics. Two important tools in the proof are the Tian-Yau-Zelditch approximation theorem for K\\\"ahler potentials and the pluripotential theory of Bedford-Taylor, suitably adapted to K\\\"ahler manifolds.

The Monge-Ampère operator and geodesics in the space of Kähler potentials

Special Values of Zeta Functions, and Eisenstein Series of Half Integral Weight

Multiplier ideal sheaves are constructed as obstructions to the convergence of the K\"ahler-Ricci flow on Fano manifolds, following earlier constructions of Kohn, Siu, and Nadel, and using the recent estimates of Kolodziej and Perelman

Multiplier Ideal Sheaves and the K\"ahler-Ricci Flow

The Dirichlet problem for a Monge–Ampere equation corresponding to a non-negative, possible degenerate cohomology class on a Kahler manifold with boundary is studied. C1,α estimates away from a divisor are obtained, by combining techniques of Blocki, Tsuji, Yau and pluripotential theory. In particular, C1,α geodesic rays in the space of Kahler potentials are constructed for each test configuration.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cag/2010/0018/0001/CAG-2010-0018-0001-a006.pdf

Jacob Sturm

Papers

On stability and the convergence of the Kähler-Ricci flow

The Monge-Ampère operator and geodesics in the space of Kähler potentials

Special Values of Zeta Functions, and Eisenstein Series of Half Integral Weight

Multiplier Ideal Sheaves and the K\"ahler-Ricci Flow

The Dirichlet problem for degenerate complex Monge–Ampere equations