Centre de Mathématiques Laurent Schwartz

Demailly's Holomorphic Morse Inequalities.- Characterization of Moishezon Manifolds.- Holomorphic Morse Inequalities on Non-compact Manifolds.- Asymptotic Expansion of the Bergman Kernel.- Kodaira Map.- Bergman Kernel on Non-compact Manifolds.- Toeplitz Operators.- Bergman Kernels on Symplectic Manifolds.

The purpose of the present paper is to set up a formalism inspired from\nnon-Archimedean geometry to study K-stability. We first provide a detailed\nanalysis of Duistermaat-Heckman measures in the context of test configurations,\ncharacterizing in particular the trivial case. For any normal polarized variety\n(or, more generally, polarized pair in the sense of the Minimal Model Program),\nwe introduce and study the non-Archimedean analogues of certain classical\nfunctionals in K\\"ahler geometry. These functionals are defined on the space of\ntest configurations, and the Donaldson-Futaki invariant is in particular\ninterpreted as the non-Archimedean version of the Mabuchi functional, up to an\nexplicit error term. Finally, we study in detail the relation between uniform\nK-stability and singularities of pairs, reproving and strengthening Y. Odaka's\nresults in our formalism. This provides various examples of uniformly K-stable\nvarieties.\n

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We give an algebraic construction of the positive intersection products of pseudo-effective classes and use them to prove that the volume function on the Néron–Severi space of a projective variety is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C Superscript 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathcal {C}^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -differentiable, expressing its differential as a positive intersection product. We also relate the differential to the restricted volumes. We then apply our differentiability result to prove an algebro-geometric version of the Diskant inequality in convex geometry, allowing us to characterize the equality case of the Khovanskii–Teissier inequalities for nef and big classes.

We study the large eigenvalue limit for the eigenfunctions of the Laplacian, on a compact manifold of negative curvature -in fact, we only assume that the geodesic flow has the Anosov property. In the semi-classical limit, we prove that the Wigner measures associated to eigenfunctions have positive metric entropy. In particular, they cannot concentrate entirely on closed geodesics.

We give an elementary proof of the existence of an asymptotic expansion in powers of k of the Bergman kernel associated to Lk, where L is a positive line bundle over a compact complex manifold. We also give an algorithm for computing the coefficients in the expansion.

Multi-soliton solutions, i.e. solutions behaving as the sum of N given solitons as t \to +\infty , were constructed for the L^2 critical and subcritical (NLS) and (gKdV) equations in previous works (see [Merle, F.: Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity. Comm. Math. Phys. 129 (1990), no. 2, 223-240], [Martel, Y.: Asymptotic N -soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations. Amer. J. Math. 127 (2005), no. 5, 1103-1140] and [Martel, Y. and Merle, F.: Multi solitary waves for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), 849-864]). In this paper, we extend the construction of multi-soliton solutions to the L^2 supercritical case both for (gKdV) and (NLS) equations, using a topological argument to control the direction of instability.

Bihermitian complex surfaces are oriented conformal four-manifolds admitting two independent compatible complex structures. Non-anti-self-dual bihermitian structures on R4 and the four-dimensional torus T4 have recently been discovered by P. Kobak. We show that an oriented compact 4-manifold, admitting a non-anti-self-dual bihermitian structure, is a torus or K3 surface in the strongly bihermitian case (when the two complex structures are independent at each point) or, otherwise, must be obtained from the complex projective plane or a minimal ruled surface of genus less than 2 by blowing up points along some anti-canonical divisor (but the actual existence of bihermitian structures in the latter case is still an open question). The paper includes a general method for constructing non-anti-self-dual bihermitian structures on tori, K3 surfaces and S1 × S3. Further properties of compact bihermitian surfaces are also investigated. 1991 Mathematics Subject Classification: 53C12, 53C55, 32J15.

In this paper we develop a theory of Grothendieck's six operations of lisse-étale constructible sheaves on Artin stacks locally of finite type over certain excellent schemes of finite Krull dimension. We also give generalizations of the classical base change theorems and Kunneth formula to stacks, and prove new results about cohomological descent for unbounded complexes.

31 pages, 4 figures.

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We consider a classical equation known as the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi Superscript 4"> <mml:semantics> <mml:msup> <mml:mi> ϕ </mml:mi> <mml:mn>4</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\phi ^4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> model in one space dimension. The kink, defined by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H left-parenthesis x right-parenthesis equals hyperbolic tangent left-parenthesis x slash StartRoot 2 EndRoot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>tanh</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H(x)=\tanh (x/{\sqrt {2}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , is an explicit stationary solution of this model. From a result of Henry, Perez and Wreszinski it is known that the kink is orbitally stable with respect to small perturbations of the initial data in the energy space. In this paper we show asymptotic stability of the kink for odd perturbations in the energy space. The proof is based on Virial-type estimates partly inspired from previous works of Martel and Merle on asymptotic stability of solitons for the generalized Korteweg-de Vries equations. However, this approach has to be adapted to additional difficulties, pointed out by Soffer and Weinstein in the case of general Klein-Gordon equations with potential: the interactions of the so-called internal oscillation mode with the radiation, and the different rates of decay of these two components of the solution in large time.

We consider the quintic generalized Korteweg–de Vries equation (gKdV) ut+(uxx+u5)x=0,which is a canonical mass critical problem, for initial data in H1 close to the soliton. In earlier works on this problem, finite- or infinite-time blow up was proved for non-positive energy solutions, and the solitary wave was shown to be the universal blow-up profile, see [16], [26] and [20]. For well-localized initial data, finite-time blow up with an upper bound on blow-up rate was obtained in [18]. In this paper, we fully revisit the analysis close to the soliton for gKdV in light of the recent progress on the study of critical dispersive blow-up problems (see [31], [39], [32] and [33], for example). For a class of initial data close to the soliton, we prove that three scenarios only can occur: (i) the solution leaves any small neighborhood of the modulated family of solitons in the scale invariant L2 norm; (ii) the solution is global and converges to a soliton as t → ∞; (iii) the solution blows up in finite time T with speed ‖ux(t)‖L2∼C(u0)T-tast→T.Moreover, the regimes (i) and (iii) are stable. We also show that non-positive energy yields blow up in finite time, and obtain the characterization of the solitary wave at the zero-energy level as was done for the mass critical non-linear Schrödinger equation in [31].

In this book, Claire Voisin provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The volume is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by Voisin. The book focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by Voisin looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

We give a variational proof of a version of the Yau-Tian-Donaldson conjecture for twisted Kähler-Einstein currents, and use this to express the greatest (twisted) Ricci lower bound in terms of a purely algebro-geometric stability threshold. Our approach does not involve the continuity method or Cheeger-Colding-Tian theory, and uses instead pluripotential theory and valuations. Along the way, we study the relationship between geodesic rays and non-Archimedean metrics.

We study the asymptotic of the Bergman kernel of the spin c Dirac operator on high tensor powers of a line bundle.

We give a variational proof of a version of the Yau–Tian–Donaldson conjecture for twisted Kähler–Einstein currents, and use this to express the greatest (twisted) Ricci lower bound in terms of a purely algebro-geometric stability threshold. Our approach does not involve a continuity method or the Cheeger–Colding–Tian theory, and uses instead pluripotential theory and valuations. Along the way, we study the relationship between geodesic rays and non-Archimedean metrics.

Inspir\'es par un argument de C. Voisin, nous montrons l'existence d'hypersurfaces quartiques lisses dans ${\bf P}^4_{\mathbb C}$ qui ne sont pas stablement rationnelles, plus pr\'ecis\'ement dont le groupe de Chow de degr\'e z\'ero n'est pas universellement \'egal \`a $\mathbb Z$. --- There are (many) smooth quartic hypersurfaces in ${\bf P}^4_{\mathbb C}$ which are not stably rational. More precisely, their degree zero Chow group is not universally equal to $\mathbb Z$. The proof uses a variation of a specialisation method due to C. Voisin.

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We show that piecewise isometries, i.e. non-necessarily invertible maps defined on a finite union of polytopes and coinciding with an isometry on the interior of each polytope, have zero topological entropy in any dimension. This had been conjectured by a number of authors. The proof is by an induction on the dimension and uses a device (the differential of a piecewise linear map) introduced by M. Tsujii.