RESEARCH STATEMENT TURGAY BAYRAKTAR

Transcription

1 RESEARCH STATEMENT TURGAY BAYRAKTAR My research focuses on complex dynamics in several variables. More precisely, I investigate the dynamical properties of meromorphic mappings of compact complex manifolds by using tools from complex geometry and pluripotential theory. I consider problems of constructing invariant currents to analyze the dynamics of these mappings. The dynamical study of rational maps f : P P of the Riemann sphere was initiated by Fatou and Julia in the early 900 s and has developed into a very rich and beautiful subject. There is a dichotomy between the Julia set where the dynamics is chaotic and the Fatou set on which the dynamics is predictable. A basic tool in the iteration theory of rational maps of P is Montel s theorem on normal families. In higher dimensions, however, the analogues of Montel s theorem do not have the appropriate adaptability. Therefore, we needed to have a different approach. On the other hand, the theory of iteration of meromorphic mappings of complex manifolds in higher dimensions has been studied intensively in the past two decades. One of the central themes under investigation was the asymptotic equidistribution of pre-images of linear subspaces under the iterates of a rational mapping [BS92, FS95, RS97]. In the case of codimension one, for meromorphic maps of compact Kähler manifolds, one of the main results in [Bay] provides a general construction of an f -invariant Green current which represents an expanding direction for f in the cohomology and has good convergence properties. Such a current is useful in the sense that if the ambient space is projective then it describes the asymptotic equidistribution of pre-images of zeros of random holomorphic sections whenever the expanded class is the first Chern class of a semi-positive line bundle [Bay2]. Furthermore, as a special case, if f : X X is meromorphic map of a Riemann surface and the topological degree of the map is larger than one then the Green current coincides with the unique f-invariant measure of maximal entropy. Below, first I explain the general setting and then I describe the details of my research projects [Bay, Bay2].. Green Current.. Notation. Let X be a compact Kähler manifold of dimension k and f : X X be a meromorphic map. In particular, any rational map of a projective variety is a meromorphic map. We let I f denote the indeterminacy locus of f, i.e. the set of points where f is not holomorphic. We say that f is dominant if its image contains an open set. The term map is loosely used here, since f is ill-defined at every point of I f. Therefore, many natural concepts (such as pull-back of a smooth form by f) that are defined for smooth maps, require additional interpretation in the meromorphic setting. Date: May 22, 202.

2 2 TURGAY BAYRAKTAR Although a meromorphic map is not smooth it induces a linear action on the cohomology f : H, (X, R) H, (X, R). In general, this linear action is not compatible with the dynamics of the map. We say that f is -regular if (f n ) = (f ) n for each n =, 2... For instance, if f is a rational map of P k then f may be identified with the algebraic degree of f. This allows us to give a more general definition of degree. Namely, the dynamical degree is defined by λ f := lim inf ( (f n X\I n ) ω w k ) n f n where ω is a Kähler form on X. If f is -regular then λ f coincides with spectral radius of f. A positive closed current is a measure coefficient form acting (by integration) on the test forms of the complementary bidegree. A motivating example of a positive closed current is the current of integration along a closed smooth submanifold. The set of pseudo-effective (psef) cohomology classes H, psef (X, R) are the ones that can be represented by positive closed currents. The set of so-called classes H, (X, R) is the interior of H, psef (X, R). These notions coincide with the classical ones in algebraic geometry if the ambient space is projective and α H 2 (X, Z)..2. Green Currents. As a part of my research project [Bay], under mild dynamical assumptions I was able to construct invariant currents for meromorphic maps of compact Kähler manifolds. I also proved that these currents have good convergence properties. This extends earlier results of [BS9, FS95, FG0, Can0, DF0, Gue04, DG09, DDG0] obtained in various settings. Theorem.. [Bay] Let f : X X be a dominant -regular meromorphic map and α H, psef (X, R) such that f α = λα for some λ >. We also let T min := σ + i π ϕ min α denote a positive closed current with minimal singularities. If ( ) λ n ϕ min f n 0 in L (X) then for every smooth form θ α we have the existence of the limit T α := lim n λ n (f n ) θ which depends only on the class α. In addition, T α is a positive closed (, ) current satisfying f T α = λt α. Furthermore, () T α is minimally singular among the positive closed currents S whose de Rham class is α and satisfy f S = λs. (2) T α is extreme within the cone of positive closed currents whose cohomology classes belong to R + α. In the literature, the current T α is referred as the Green current. The property (2) of Theorem. is a form of ergodicity. The dynamical assumption ( ) holds in a quite general setting [Bay]. For instance, if X is a complex homogenous manifold (i.e. automorphism group of X acts transitively e.g. X = P k or (P ) k ) then ( ) holds. Another case where condition ( ) holds is that if the class α is Kähler. If the ambient space is projective then we have the following algebraic criterion to check if the condition ( ) holds :

3 RESEARCH STATEMENT 3 Theorem.2. [Bay] Let X be a projective manifold and f : X X be a dominant -regular rational map. Assume that λ f > is a simple eigenvalue of f with f α f = λ f α f If α f C 0 for every algebraic irreducible curve C f(i f ), where f(i f ) X denotes the total transform of I f, then ϕ min f n 0 in L (X). λ n f Notice that the algebraic criterion in Theorem.2 holds for every nef class. In [Bay], among other things I provided a family of birational mappings which fall into the framework of Theorem.2. However, there exists curves which have negative intersection with the class α f. In particular, such a class it is not nef therefore it can not be represented by a positive form nor approximated by the classes represented by positive forms. Green currents are interesting from various point of views. In certain cases, the support of the Green current plays the role of the Julia set classically defined for rational maps of P. In the case of holomorphic mappings f of P k, the Green current T f has locally continuous potentials. Therefore, following Bedford-Taylor theory one can define the exterior powers of T f which are also dynamically interesting currents. In particular, the top degree intersection yields the unique f-invariant measure of maximal entropy. Dynamical Green currents demonstrate also various interesting geometric properties. I will elaborate this point-view more in section Equidistribution towards Green Current Let f : P P be a rational map of degree d 2 and ν be any probability measure on P. A theorem of Brolin [Bro65], Lyubich [Lju83] and Freire-Lopez- Mañé [FLM83] asserts that the pre-images d n (f n ) ν converges weakly to the measure of maximal entropy µ f if and only if ν(e f ) = 0 where E f is an (possibly empty) exceptional set. An equidistribution problem in the setting of meromorphic mappings of compact Kähler manifolds in codimension one can be formulated as follows: Let (X, f, α) be as in Theorem. and assume that λ >. Provide necessary and sufficient conditions on a positive closed current S α so that the sequence λ n (f n ) S converges to the Green current T α. 2.. Equidistribution of Zeros of Sections. Let X be a projective manifold and π : L X be a holomorphic line bundle. We denote the semi-group N(L) := {m N : H 0 (X, L m ) 0}. Given m N(L), we consider the canonical map induced by the complete linear series L m Φ m : X P Nm x [s 0 (x) : s (x) : : s Nm (x)] where the identification P H 0 (X, L m ) = P Nm is determined by the choice of the basis s 0,..., s Nm for H 0 (X, L m ). Note that Φ m is a rational map which is holomorphic on the complement of the base locus B L m := s H 0 (X,L m ) s (0). Recall that a holomorphic line bundle L X is called semi-ample if B L m = for some m > 0. A line bundle L is called if the mapping Φ m is birational onto its image for sufficiently large m > 0.

4 4 TURGAY BAYRAKTAR Finally, a set K P k is called pluripolar if K {φ = } for some quassiplurisubharmonic function φ. These are small sets; for instance Lebesgue measure of a pluripolar set is zero. Theorem 2.. [Bay2] Let f : X X be a -regular dominant rational map such that λ f > simple with f α = λ f α where α = c (L) and L X is a line bundle. We assume that L C 0 for every algebraic irreducible curve C f(i f ), where f(i f ) X denotes the total transform of I f. If the algebra R(L) = m 0 H 0 (X, L m ) is finitely generated then for sufficiently large m > 0 there exists a pluripolar set E m L m such that for every H E m in the sense of currents as n. λ n (f n ) ( m [H]) T α If X = P k and L is the hyperplane bundle O() then the corresponding result follows from [RS97] and [FS92]. Recall that for a line bundle L the algebra R(L) is not always finitely generated. In fact, if L is and nef then R(L) is finitely generated if and only if L is semi-ample Equidistribution of Currents with Mild Singularities. In this section X denotes merely compact Kähler manifold. The next result indicates that if the class α f can be represented by a positive closed current with mild singularities (in the sense of Lelong numbers) then any such current equidistributes towards the Green current. Roughly speaking, Lelong number of a positive closed current S at a point x, denoted by ν(s, x), quantifies the singularity of a local potential of S at the point x. For example, Lelong number coincides with the algebraic multiplicity in the case of a current of integration. Theorem 2.2. [Bay2] Let f : X X be a dominant -regular meromorphic map. Assume that λ f > is a simple eigenvalue of f with f α f = λ f α f. Furthermore, we assume that ν(t min, x) = 0 for every x X where T min = σ + i π ϕ min α is a minimally singular current. Then for every positive closed (, ) current S α such that the Lelong numbers ν(s, x) = 0 for each x X the sequence λ n f (f n ) S converges weakly to the Green current T α. Recall that any class α H, nef (X, R) H, (X, R) can be represented by a positive closed current with identically zero Lelong numbers. Theorem 2.2 interpolates between asymptotic equidistribution of smooth forms (Theorem.) and that of current of integrations along zero divisors of sections (Theorem 2.). 3. Invariant Classes and Classification of Varieties In present section, we consider the relationship between the cohomological properties of f -invariant classes and the underlying manifold X. In particular, we have the following result: Theorem 3.. [Bay2] Let X be a compact Kähler surface and f : X X be a dominant meromorphic map. If λ f > d top (f) and α H, (X, R) such that f α = λ f α then X is rational.

5 RESEARCH STATEMENT 5 If f : X X be a bimeromorphic map of a compact Kähler surface X with λ f > then a theorem of Diller and Favre [DF0] asserts that (f, X) is birationally conjugate to an automorphism (g, Y ) for some compact Kähler surface Y if and only if the invariant class α f H, (X, R). However, this is no longer true in higher dimensions: Proposition 3.2. [Bay2] There exists a discrete family rational threefolds X τ and birational mappings f τ : X τ X τ with positive topological entropy such that λ fτ > is simple eigenvalue of f with f α H, f = λ f α f and α f H, (Xτ ) (X, R). However, (f τ, X τ ) is not birationally conjugate to an automorphism of a compact complex threefold. 4. Research in Progress 4.. Laminar Currents. Geometric structure of Green currents play a central role in the study of ergodic properties of rational mappings of projective surfaces [BLS93, Duj03, Duj06]. Laminar currents were introduced by Bedford, Smilie and Lyubich [BLS93] as a class of geometric currents in complex dimension two and have been studied intensively since then. A positive (, ) current on X is called laminar if it is locally described by integration over families of graphs with respect to a transverse measure in which integrants have no isolated intersections. A laminar current T is called uniformly laminar if the graphs form a lamination of some open subset of X and T is a foliation cycle associated to this lamination. It was proved by Dujardin [Duj03] that if f is a birational map of a projective surface then the Green current of f is laminar. More recently, this result was extended to the case of small topological degree [DDG0]. In the case of a projective surface, laminar structure of Green currents allows one to define the measure of maximal entropy as the geometric intersection of the Green currents of forward and backward mapping [Duj06] without using pluripotential theory. More recently, Dinh [Din05] introduced woven currents, a weaker form of laminarity, to study the geometric properties of Green currents in higher dimensions. Therefore, laminarity has been an important property of dynamically defined currents. One of the projects that I am carrying on is based on the following question: Does the Green current given by Theorem. has woven, or even laminar, structure? 4.2. Regular Polynomial Automorphisms. A Hénon map is a polynomial automorphism of C 2 which has the form f(x, y) = (y, x 2 + c ax). Dynamics of these maps is very rich and studied extensively [BS92, FS92, BLS93]. In a joint work with R. Roeder (IUPUI), we investigate dynamical properties of a family of Skew-products of Hénon maps on C 4. That is, these maps are of the form (f, g) where f : C 2 C 2 is a Hénon map and g : (x, y) C 2 f(x, y) C 2 is also a Hénon map acting on the fibers of f. We consider those skew products that induce regular polynomial automorphisms in the sense of Sibony [Sib99]. This means that the indeterminacy locus of the meromorphic extensions of f and f to P 4 are disjoint. We consider C 4 as an open coordinate chart in P 4 and denote the hyperplane at infinity by H := P 4 C 4. The set I f := f(h I f ) where I f is the indeterminacy locus of f, is invariant under the meromorphic extension of f. In our case, I f is isomorphic to P and the meromorphic extension of f induces a hyperbolic holomorphic dynamical system on I f. Therefore, there exists a unique invariant measure of maximal entropy which we denote by µ f. A major theme

6 6 TURGAY BAYRAKTAR in this project to explore interplay between the Green current T f and the measure µ f. Similar problems were considered in [Jon99, BJ00] for regular endomorphisms of C k that is those are the mappings which extends to P k holomorphically. The innovation in this project is the presence of the indeterminacy locus and collapsing directions at infinity. References [Bay] T. Bayraktar. Green currents for meromorphic maps of compact Kähler manifolds. J. Geom. Anal., 202, 0.007/s [Bay2] T. Bayraktar. Equidistribution towards green currents in cohomology classes, arxiv:math/ v, 202. [BJ00] E. Bedford and M. Jonsson. Dynamics of regular polynomial endomorphisms of C k. Amer. J. Math., 22():53 22, [BLS93] E. Bedford, M. Lyubich, and J. Smillie. Polynomial diffeomorphisms of C 2. IV. The measure of maximal entropy and laminar currents. Invent. Math., 2():77 25, 993. [Bro65] H. Brolin. Invariant sets under iteration of rational functions. Ark. Mat., 6:03 44 (965), 965. [BS9] E. Bedford and J. Smillie. Polynomial diffeomorphisms of C 2 : currents, equilibrium measure and hyperbolicity. Invent. Math., 03():69 99, 99. [BS92] E. Bedford and J. Smillie. Polynomial diffeomorphisms of C 2. III. Ergodicity, exponents and entropy of the equilibrium measure. Math. Ann., 294(3): , 992. [Can0] S. Cantat. Dynamique des automorphismes des surfaces K3. Acta Math., 87(): 57, 200. [DDG0] J. Diller, R. Dujardin, and V. Guedj. Dynamics of meromorphic maps with small topological degree i: from cohomology to currents. Indiana Univ. Math. J., 59:52 562, 200. [DF0] J. Diller and C. Favre. Dynamics of bimeromorphic maps of surfaces. Amer. J. Math., 23(6):35 69, 200. [DG09] J. Diller and V. Guedj. Regularity of dynamical Green s functions. Trans. Amer. Math. Soc., 36(9): , [Din05] T.-C. Dinh. Suites d applications méromorphes multivaluées et courants laminaires. J. Geom. Anal., 5(2): , [DPS0] J.-P. D ly, T. Peternell, and M. Schneider. Pseudo-effective line bundles on compact Kähler manifolds. Internat. J. Math., 2(6):689 74, 200. [DS06] T.-C. Dinh and N. Sibony. Distribution des valeurs de transformations méromorphes et applications. Comment. Math. Helv., 8():22 258, [Duj03] R. Dujardin. Laminar currents in P 2. Math. Ann., 325(4): , [Duj06] R. Dujardin. Laminar currents and birational dynamics. Duke Math. J., 3(2):29 247, [FG0] C. Favre and V. Guedj. Dynamique des applications rationnelles des espaces multiprojectifs. Indiana Univ. Math. J., 50(2):88 934, 200. [FLM83] A. Freire, A. Lopes, and R. Mañé. An invariant measure for rational maps. Bol. Soc. Brasil. Mat., 4():45 62, 983. [FS92] John Erik Fornæss and Nessim Sibony. Complex Hénon mappings in C 2 and Fatou- Bieberbach domains. Duke Math. J., 65(2): , 992. [FS95] J. E. Fornaess and N. Sibony. Complex dynamics in higher dimension. II. In Modern methods in complex analysis (Princeton, NJ, 992), volume 37 of Ann. of Math. Stud., pages Princeton Univ. Press, Princeton, NJ, 995. [Gue04] V. Guedj. Decay of volumes under iteration of meromorphic mappings. Ann. Inst. Fourier (Grenoble), 54(7): (2005), [Jon99] Mattias Jonsson. Dynamics of polynomial skew products on C 2. Math. Ann., 34(3): , 999. [Lju83] M. Ju. Ljubich. Entropy properties of rational endomorphisms of the Riemann sphere. Ergodic Theory Dynam. Systems, 3(3):35 385, 983. [RS97] A. Russakovskii and B. Shiffman. Value distribution for sequences of rational mappings and complex dynamics. Indiana Univ. Math. J., 46(3): , 997.

7 RESEARCH STATEMENT 7 [Sib99] N. Sibony. Dynamique des applications rationnelles de P k. In Dynamique et géométrie complexes (Lyon, 997), volume 8 of Panor. Synthèses, pages ix x, xi xii, Soc. Math. France, Paris, 999. Mathematics Department, Indiana University Indiana, USA address: tbayrakt@indiana.edu