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Tightness of Liouville first passage percolation for γ ∈ ( 0 , 2 ) $\gamma \in (0,2)$ Publ. math. IHES (IF 1.222) Pub Date : 2020-11-04 Jian Ding, Julien Dubédat, Alexander Dunlap, Hugo Falconet
We study Liouville first passage percolation metrics associated to a Gaussian free field \(h\) mollified by the two-dimensional heat kernel \(p_{t}\) in the bulk, and related star-scale invariant metrics. For \(\gamma \in (0,2)\) and \(\xi = \frac{\gamma }{d_{\gamma }}\), where \(d_{\gamma }\) is the Liouville quantum gravity dimension defined in Ding and Gwynne (Commun. Math. Phys. 374:1877–1934,
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Deviations of ergodic sums for toral translations II. Boxes Publ. math. IHES (IF 1.222) Pub Date : 2020-10-19 Dmitry Dolgopyat, Bassam Fayad
We study the Kronecker sequence \(\{n\alpha \}_{n\leq N}\) on the torus \({\mathbf {T}}^{d}\) when \(\alpha \) is uniformly distributed on \({\mathbf {T}}^{d}\). We show that the discrepancy of the number of visits of this sequence to a random box, normalized by \(\ln ^{d} N\), converges as \(N\to \infty \) to a Cauchy distribution. The key ingredient of the proof is a Poisson limit theorem for the
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Generic regularity of free boundaries for the obstacle problem Publ. math. IHES (IF 1.222) Pub Date : 2020-07-02 Alessio Figalli, Xavier Ros-Oton, Joaquim Serra
The goal of this paper is to establish generic regularity of free boundaries for the obstacle problem in \(\mathbf {R}^{n}\). By classical results of Caffarelli, the free boundary is \(C^{\infty }\) outside a set of singular points. Explicit examples show that the singular set could be in general \((n-1)\)-dimensional—that is, as large as the regular set. Our main result establishes that, generically
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Explicit spectral gaps for random covers of Riemann surfaces Publ. math. IHES (IF 1.222) Pub Date : 2020-06-25 Michael Magee, Frédéric Naud
We introduce a permutation model for random degree \(n\) covers \(X_{n}\) of a non-elementary convex-cocompact hyperbolic surface \(X=\Gamma \backslash \mathbf {H}\). Let \(\delta \) be the Hausdorff dimension of the limit set of \(\Gamma \). We say that a resonance of \(X_{n}\) is new if it is not a resonance of \(X\), and similarly define new eigenvalues of the Laplacian. We prove that for any \(\epsilon
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Complexity of parabolic systems Publ. math. IHES (IF 1.222) Pub Date : 2020-05-15 Tobias Holck Colding, William P. Minicozzi
We first bound the codimension of an ancient mean curvature flow by the entropy. As a consequence, all blowups lie in a Euclidean subspace whose dimension is bounded by the entropy and dimension of the evolving submanifolds. This drastically reduces the complexity of the system. We use this in a major application of our new methods to give the first general bounds on generic singularities of surfaces
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Period integrals from wall structures via tropical cycles, canonical coordinates in mirror symmetry and analyticity of toric degenerations Publ. math. IHES (IF 1.222) Pub Date : 2020-05-14 Helge Ruddat, Bernd Siebert
We give a simple expression for the integral of the canonical holomorphic volume form in degenerating families of varieties constructed from wall structures and with central fiber a union of toric varieties. The cycles to integrate over are constructed from tropical 1-cycles in the intersection complex of the central fiber. One application is a proof that the mirror map for the canonical formal families
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Discrete series multiplicities for classical groups over Z$\mathbf {Z}$ and level 1 algebraic cusp forms Publ. math. IHES (IF 1.222) Pub Date : 2020-03-05 Gaëtan Chenevier, Olivier Taïbi
The aim of this paper is twofold. First, we introduce a new method for evaluating the multiplicity of a given discrete series representation in the space of level 1 automorphic forms of a split classical group \(G\) over \(\mathbf {Z}\), and provide numerical applications in absolute rank \(\leq 8\). Second, we prove a classification result for the level one cuspidal algebraic automorphic representations
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Riemannian hyperbolization Publ. math. IHES (IF 1.222) Pub Date : 2020-02-28 Pedro Ontaneda
The strict hyperbolization process of Charney and Davis produces a large and rich class of negatively curved spaces (in the geodesic sense). This process is based on an earlier version introduced by Gromov and later studied by Davis and Januszkiewicz. If M is a manifold its Charney-Davis strict hyperbolization is also a manifold, but the negatively curved metric obtained is very far from being Riemannian
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Quasimap wall-crossings and mirror symmetry Publ. math. IHES (IF 1.222) Pub Date : 2020-02-07 Ionuţ Ciocan-Fontanine, Bumsig Kim
We state a wall-crossing formula for the virtual classes of \({\varepsilon }\)-stable quasimaps to GIT quotients and prove it for complete intersections in projective space, with no positivity restrictions on their first Chern class. As a consequence, the wall-crossing formula relating the genus \(g\) descendant Gromov-Witten potential and the genus \(g\)\({\varepsilon }\)-quasimap descendant potential
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E2$E_{2}$ -cells and mapping class groups Publ. math. IHES (IF 1.222) Pub Date : 2019-06-17 Søren Galatius, Alexander Kupers, Oscar Randal-Williams
We prove a new kind of stabilisation result, “secondary homological stability,” for the homology of mapping class groups of orientable surfaces with one boundary component. These results are obtained by constructing CW approximations to the classifying spaces of these groups, in the category of \(E_{2}\)-algebras, which have no \(E_{2}\)-cells below a certain vanishing line.
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The period-index problem for real surfaces Publ. math. IHES (IF 1.222) Pub Date : 2019-05-28 Olivier Benoist
We study when the period and the index of a class in the Brauer group of the function field of a real algebraic surface coincide. We prove that it is always the case if the surface has no real points (more generally, if the class vanishes in restriction to the real points of the locus where it is well-defined), and give a necessary and sufficient condition for unramified classes. As an application
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Separation for the stationary Prandtl equation Publ. math. IHES (IF 1.222) Pub Date : 2019-09-05 Anne-Laure Dalibard, Nader Masmoudi
In this paper, we prove that separation occurs for the stationary Prandtl equation, in the case of adverse pressure gradient, for a large class of boundary data at \(x=0\). We justify the Goldstein singularity: more precisely, we prove that under suitable assumptions on the boundary data at \(x=0\), there exists \(x^{*}>0\) such that \(\partial _{y} u_{|y=0}(x) \sim C \sqrt{x^{*} -x}\) as \(x\to x^{*}\)
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A local model for the trianguline variety and applications Publ. math. IHES (IF 1.222) Pub Date : 2019-08-22 Christophe Breuil, Eugen Hellmann, Benjamin Schraen
We describe the completed local rings of the trianguline variety at certain points of integral weights in terms of completed local rings of algebraic varieties related to Grothendieck’s simultaneous resolution of singularities. We derive several local consequences at these points for the trianguline variety: local irreducibility, description of all local companion points in the crystalline case, combinatorial
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Covariantly functorial wrapped Floer theory on Liouville sectors Publ. math. IHES (IF 1.222) Pub Date : 2019-08-23 Sheel Ganatra, John Pardon, Vivek Shende
We introduce a class of Liouville manifolds with boundary which we call Liouville sectors. We define the wrapped Fukaya category, symplectic cohomology, and the open-closed map for Liouville sectors, and we show that these invariants are covariantly functorial with respect to inclusions of Liouville sectors. From this foundational setup, a local-to-global principle for Abouzaid’s generation criterion
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Foliations with positive slopes and birational stability of orbifold cotangent bundles Publ. math. IHES (IF 1.222) Pub Date : 2019-04-18 Frédéric Campana, Mihai Păun
Let \(X\) be a smooth connected projective manifold, together with an snc orbifold divisor \(\Delta \), such that the pair \((X, \Delta )\) is log-canonical. If \(K_{X}+\Delta \) is pseudo-effective, we show, among other things, that any quotient of its orbifold cotangent bundle has a pseudo-effective determinant. This improves considerably our previous result (Campana and Păun in Ann. Inst. Fourier
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Joinings of higher rank torus actions on homogeneous spaces Publ. math. IHES (IF 1.222) Pub Date : 2019-02-14 Manfred Einsiedler, Elon Lindenstrauss
We show that joinings of higher rank torus actions on \(S\)-arithmetic quotients of semi-simple or perfect algebraic groups must be algebraic.
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Topological Hochschild homology and integral p $p$ -adic Hodge theory Publ. math. IHES (IF 1.222) Pub Date : 2019-04-17 Bhargav Bhatt, Matthew Morrow, Peter Scholze
In mixed characteristic and in equal characteristic \(p\) we define a filtration on topological Hochschild homology and its variants. This filtration is an analogue of the filtration of algebraic \(K\)-theory by motivic cohomology. Its graded pieces are related in mixed characteristic to the complex \(A\Omega\) constructed in our previous work, and in equal characteristic \(p\) to crystalline cohomology
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Categorical actions on unipotent representations of finite unitary groups Publ. math. IHES (IF 1.222) Pub Date : 2019-03-08 O. Dudas, M. Varagnolo, E. Vasserot
Using Harish-Chandra induction and restriction, we construct a categorical action of a Kac-Moody algebra on the category of unipotent representations of finite unitary groups in non-defining characteristic. We show that the decategorified representation is naturally isomorphic to a direct sum of level 2 Fock spaces. From our construction we deduce that the Harish-Chandra branching graph coincides with
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Fourier interpolation on the real line Publ. math. IHES (IF 1.222) Pub Date : 2018-09-17 Danylo Radchenko, Maryna Viazovska
In this paper we construct an explicit interpolation formula for Schwartz functions on the real line. The formula expresses the value of a function at any given point in terms of the values of the function and its Fourier transform on the set \(\{0, \pm\sqrt{1}, \pm\sqrt{2}, \pm\sqrt{3},\dots\}\). The functions in the interpolating basis are constructed in a closed form as an integral transform of
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Integral p $p$ -adic Hodge theory Publ. math. IHES (IF 1.222) Pub Date : 2019-01-16 Bhargav Bhatt, Matthew Morrow, Peter Scholze
We construct a new cohomology theory for proper smooth (formal) schemes over the ring of integers of \(\mathbf {C}_{p}\). It takes values in a mixed-characteristic analogue of Dieudonné modules, which was previously defined by Fargues as a version of Breuil–Kisin modules. Notably, this cohomology theory specializes to all other known \(p\)-adic cohomology theories, such as crystalline, de Rham and
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Measure concentration and the weak Pinsker property Publ. math. IHES (IF 1.222) Pub Date : 2018-02-15 Tim Austin
Let \((X,\mu)\) be a standard probability space. An automorphism \(T\) of \((X,\mu)\) has the weak Pinsker property if for every \(\varepsilon > 0\) it has a splitting into a direct product of a Bernoulli shift and an automorphism of entropy less than \(\varepsilon \). This property was introduced by Thouvenot, who asked whether it holds for all ergodic automorphisms. This paper proves that it does
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Integral models of Shimura varieties with parahoric level structure Publ. math. IHES (IF 1.222) Pub Date : 2018-04-30 M. Kisin, G. Pappas
For a prime \(p > 2\), we construct integral models over \(p\) for Shimura varieties with parahoric level structure, attached to Shimura data \((G,X)\) of abelian type, such that \(G\) splits over a tamely ramified extension of \({\mathbf {Q}}_{\,p}\). The local structure of these integral models is related to certain “local models”, which are defined group theoretically. Under some additional assumptions
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La conjecture du facteur direct Publ. math. IHES (IF 1.222) Pub Date : 2017-12-07 Yves André
M. Hochster a conjecturé que pour toute extension finie \(S\) d’un anneau commutatif régulier \(R\), la suite exacte de \(R\)-modules \(0\to R \to S \to S/R\to0\) est scindée. En nous appuyant sur sa réduction au cas d’un anneau local régulier \(R\) complet non ramifié d’inégale caractéristique, nous proposons une démonstration de cette conjecture dans le contexte de la théorie perfectoïde de P. Scholze
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Le lemme d’Abhyankar perfectoide Publ. math. IHES (IF 1.222) Pub Date : 2017-12-07 Yves André
Nous étendons le théorème de presque-pureté de Faltings-Scholze-Kedlaya-Liu sur les extensions étales finies d’algèbres perfectoïdes au cas des extensions ramifiées, sans restriction sur le lieu de ramification. Nous déduisons cette version perfectoïde du lemme d’Abhyankar du théorème de presque-pureté, par un passage à la limite mettant en jeu des versions perfectoïdes du théorème d’extension de Riemann
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Invariant and stationary measures for the action on Moduli space Publ. math. IHES (IF 1.222) Pub Date : 2018-04-17 Alex Eskin, Maryam Mirzakhani
We prove some ergodic-theoretic rigidity properties of the action of on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of is supported on an invariant affine submanifold. The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner’s seminal work.
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A viscosity method in the min-max theory of minimal surfaces Publ. math. IHES (IF 1.222) Pub Date : 2017-10-26 Tristan Rivière
We present the min-max construction of critical points of the area using penalization arguments. Precisely, for any immersion of a closed surface \(\Sigma \) into a given closed manifold, we add to the area Lagrangian a term equal to the \(L^{q}\) norm of the second fundamental form of the immersion times a “viscosity” parameter. This relaxation of the area functional satisfies the Palais–Smale condition
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Percolation of random nodal lines Publ. math. IHES (IF 1.222) Pub Date : 2017-09-18 Vincent Beffara,Damien Gayet
We prove a Russo-Seymour-Welsh percolation theorem for nodal domains and nodal lines associated to a natural infinite dimensional space of real analytic functions on the real plane. More precisely, let \(U\) be a smooth connected bounded open set in \(\mathbf{R}^{2}\) and \(\gamma, \gamma '\) two disjoint arcs of positive length in the boundary of \(U\). We prove that there exists a positive constant
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Calabi-Yau manifolds with isolated conical singularities Publ. math. IHES (IF 1.222) Pub Date : 2017-08-25 Hans-Joachim Hein,Song Sun
Let \(X\) be a complex projective variety with only canonical singularities and with trivial canonical bundle. Let \(L\) be an ample line bundle on \(X\). Assume that the pair \((X,L)\) is the flat limit of a family of smooth polarized Calabi-Yau manifolds. Assume that for each singular point \(x \in X\) there exist a Kähler-Einstein Fano manifold \(Z\) and a positive integer \(q\) dividing \(K_{Z}\)
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C*-simplicity and the unique trace property for discrete groups Publ. math. IHES (IF 1.222) Pub Date : 2017-06-28 Emmanuel Breuillard,Mehrdad Kalantar,Matthew Kennedy,Narutaka Ozawa
A discrete group is said to be C*-simple if its reduced C*-algebra is simple, and is said to have the unique trace property if its reduced C*-algebra has a unique tracial state. A dynamical characterization of C*-simplicity was recently obtained by the second and third named authors. In this paper, we introduce new methods for working with group and crossed product C*-algebras that allow us to take
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Meromorphic tensor equivalence for Yangians and quantum loop algebras Publ. math. IHES (IF 1.222) Pub Date : 2017-06-28 Sachin Gautam,Valerio Toledano Laredo
Let \(\mathfrak{g}\) be a complex semisimple Lie algebra, and \(Y_{\hbar }(\mathfrak{g})\), \(U_{q}(L\mathfrak{g})\) the corresponding Yangian and quantum loop algebra, with deformation parameters related by \(q=e^{\pi \iota \hbar }\). When \(\hbar \) is not a rational number, we constructed in Gautam and Toledano Laredo (J. Am. Math. Soc. 29:775, 2016) a faithful functor \(\Gamma \) from the category
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On the hyperbolicity of general hypersurfaces Publ. math. IHES (IF 1.222) Pub Date : 2017-06-27 Damian Brotbek
In 1970, Kobayashi conjectured that general hypersurfaces of sufficiently large degree in \(\mathbf {P}^{n}\) are hyperbolic. In this paper we prove that a general sufficiently ample hypersurface in a smooth projective variety is hyperbolic. To prove this statement, we construct hypersurfaces satisfying a property which is Zariski open and which implies hyperbolicity. These hypersurfaces are chosen
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Double ramification cycles on the moduli spaces of curves Publ. math. IHES (IF 1.222) Pub Date : 2017-05-10 F. Janda,R. Pandharipande,A. Pixton,D. Zvonkine
Curves of genus \(g\) which admit a map to \(\mathbf {P}^{1}\) with specified ramification profile \(\mu\) over \(0\in \mathbf {P}^{1}\) and \(\nu\) over \(\infty\in \mathbf {P}^{1}\) define a double ramification cycle \(\mathsf{DR}_{g}(\mu,\nu)\) on the moduli space of curves. The study of the restrictions of these cycles to the moduli of nonsingular curves is a classical topic. In 2003, Hain calculated
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Geometric presentations of Lie groups and their Dehn functions Publ. math. IHES (IF 1.222) Pub Date : 2016-12-20 Yves Cornulier,Romain Tessera
We study the Dehn function of connected Lie groups. We show that this function is always exponential or polynomially bounded, according to the geometry of weights and of the 2-cohomology of their Lie algebras. Our work, which also addresses algebraic groups over local fields, uses and extends Abels’ theory of multiamalgams of graded Lie algebras, in order to provide workable presentations of these
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Diffeomorphisms with positive metric entropy Publ. math. IHES (IF 1.222) Pub Date : 2016-10-18 A. Avila,S. Crovisier,A. Wilkinson
We obtain a dichotomy for \(C^{1}\)-generic, volume-preserving diffeomorphisms: either all the Lyapunov exponents of almost every point vanish or the volume is ergodic and non-uniformly Anosov (i.e. nonuniformly hyperbolic and the splitting into stable and unstable spaces is dominated). This completes a program first put forth by Ricardo Mañé.
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Teichmüller curves in genus three and just likely intersections in \(\mathbf{G}_{m}^{n}\times\mathbf{G}_{a}^{n}\) Publ. math. IHES (IF 1.222) Pub Date : 2016-06-15 Matt Bainbridge,Philipp Habegger,Martin Möller
We prove that the moduli space of compact genus three Riemann surfaces contains only finitely many algebraically primitive Teichmüller curves. For the stratum \(\Omega\mathcal{M}_{3}(4)\), consisting of holomorphic one-forms with a single zero, our approach to finiteness uses the Harder-Narasimhan filtration of the Hodge bundle over a Teichmüller curve to obtain new information on the locations of
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Gaussian asymptotics of discrete \(\beta \)-ensembles Publ. math. IHES (IF 1.222) Pub Date : 2016-06-14 Alexei Borodin,Vadim Gorin,Alice Guionnet
We introduce and study stochastic \(N\)-particle ensembles which are discretizations for general-\(\beta \) log-gases of random matrix theory. The examples include random tilings, families of non-intersecting paths, \((z,w)\)-measures, etc. We prove that under technical assumptions on general analytic potential, the global fluctuations for such ensembles are asymptotically Gaussian as \(N\to \infty
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Relative Stanley–Reisner theory and Upper Bound Theorems for Minkowski sums Publ. math. IHES (IF 1.222) Pub Date : 2016-03-23 Karim A. Adiprasito,Raman Sanyal
In this paper we settle two long-standing questions regarding the combinatorial complexity of Minkowski sums of polytopes: We give a tight upper bound for the number of faces of a Minkowski sum, including a characterization of the case of equality. We similarly give a (tight) upper bound theorem for mixed facets of Minkowski sums. This has a wide range of applications and generalizes the classical
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Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces Publ. math. IHES (IF 1.222) Pub Date : 2016-03-02 Mohammed Abouzaid,Denis Auroux,Ludmil Katzarkov
We consider mirror symmetry for (essentially arbitrary) hypersurfaces in (possibly noncompact) toric varieties from the perspective of the Strominger-Yau-Zaslow (SYZ) conjecture. Given a hypersurface \(H\) in a toric variety \(V\) we construct a Landau-Ginzburg model which is SYZ mirror to the blowup of \(V\times \mathbf {C}\) along \(H\times0\), under a positivity assumption. This construction also
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On the Fukaya category of a Fano hypersurface in projective space Publ. math. IHES (IF 1.222) Pub Date : 2016-02-15 Nick Sheridan
This paper is about the Fukaya category of a Fano hypersurface \(X \subset \mathbf {CP}^{n}\). Because these symplectic manifolds are monotone, both the analysis and the algebra involved in the definition of the Fukaya category simplify considerably. The first part of the paper is devoted to establishing the main structures of the Fukaya category in the monotone case: the closed–open string maps, weak
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Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs Publ. math. IHES (IF 1.222) Pub Date : 2016-01-18 Caucher Birkar,De-Qi Zhang
For every smooth complex projective variety \(W\) of dimension \(d\) and nonnegative Kodaira dimension, we show the existence of a universal constant \(m\) depending only on \(d\) and two natural invariants of the very general fibres of an Iitaka fibration of \(W\) such that the pluricanonical system \(|mK_{W}|\) defines an Iitaka fibration. This is a consequence of a more general result on polarized
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