A. G. Khovanskii

Newton polyhedra and toroidal varieties

The toroidal compactification (C~0)~f ~ plays the same role as the projective compactification ~ P ~ in the classical case. Toroidal varieties are well known [2, 3]. It is almost as easy to handle them as projective spaces. In a subsequent paper the geometry of toroidal varieties will be used for the calculation of the arithmetic genus and Euler characteristic of variety X. Here we discuss the connection of this geometry with the elementary geometry of integral polyhedra.

Algebraic Equations and Convex Bodies

The well-known Bernstein–Kushnirenko theorem from the theory of Newton polyhedra relates algebraic geometry and the theory of mixed volumes. Recently, the authors have found a far-reaching generalization of this theorem to generic systems of algebraic equations on any algebraic variety. In the present note we review these results and their applications to algebraic geometry and convex geometry.

Convex bodies and multiplicities of ideals

We associate convex regions in ℝn to m-primary graded sequences of subspaces, in particular m-primary graded sequences of ideals, in a large class of local algebras (including analytically irreducible local domains). These convex regions encode information about Samuel multiplicities. This is in the spirit of the theory of Gröbner bases and Newton polyhedra on the one hand, and the theory of Newton-Okounkov bodies for linear systems on the other hand. We use this to give a new proof as well as a generalization of a Brunn-Minkowski inequality for multiplicities due to Teissier and Rees-Sharp.

Rectification of circles

A set of curves in the plane is called rectifiable near the point a if there exists a neighborhood U of a and a diffeomorphism of U taking the curves in the set (more precisely, the portions of the curves contained in the region U) into lines (more precisely, into portions of lines lying in the image of the region U). A bundle of curves with center at the point a is any set of curves passing through a. A bundle is called simple if curves in the bundle having identical tangents at the point a coincide identically in some neighborhood U of a. If a bundle of curves with center at the point a is rectifiable near a then the bundle is simple. We are interested in the behavior of the curves in a rectifiable bundle near the center. We identify curves which coincide identically in some neighborhood of a. The curves I~ of a bundle will be regarded as the graphs of functions y~ -y~ (x). The parameter c~ for the curves I~ in a simple bundle can be taken to be the tangent of the angle of inclination of the tangent to the curve I~ at the point a.

Moment polytopes, semigroup of representations and Kazarnovskii’s theorem

Two representations of a reductive group G are spectrally equivalent if the same irreducible representations appear in both of them. The semigroup of finite-dimensional representations of G with tensor product and up to spectral equivalence is a rather complicated object. We show that the Grothendieck group of this semigroup is more tractable and we give a description of it in terms of moment polytopes of representations. As a corollary, we give a proof of the Kazarnovskii theorem on the number of solutions in G of a system f1 = · · · = fm = 0, where m = dim(G) and each fi is a generic function in the space of matrix elements of a representation πi of G.

On the Continuability of Multivalued Analytic Functions to an Analytic Subset

In the paper, it is shown that a germ of a many-valued analytic function can be continued analytically along the branching set at least until the topology of this set is changed. This result is needed to construct the many-dimensional topological version of Galois theory. The proof heavily uses Whitneys stratification.

On the Monodromy of a Multivalued Function along Its Ramification Locus

We consider multivalued analytic functions in ℂn) whose set of singular points occupies a very small part of ℂn). Under a mapping of a topological space Y into ℂn), such a function f can induce a multivalued function on Y. This is possible even if the image of Y entirely lies in the ramification set of f. We estimate the monodromy group of the induced function via the monodromy group of f.

Okounkov Bodies and Applications

The theory of Newton-Okounkov bodies, also called Okounkov bodies, is a relatively new connection between algebraic geometry and convex geometry. It generalizes the well-known and extremely rich correspondence between geometry of toric varieties and combinatorics of convex integral polytopes. Following a successful MFO Mini-workshop on this topic in August 2011, the MFO Half-Workshop 1422b, “Okounkov bodies and applications”, held in May 2014, explored the development of this area in recent years, with particular attention to applications and relationships to other areas such as number theory and tropical geometry. Mathematics Subject Classification (2010): 14C20, 14M25, 51M20, 14N25 , 14T05. Introduction by the Organisers Okounkov bodies were first introduced by Andrei Okounkov, in a construction motivated by a question of Khovanskii concerning convex bodies govering the multiplicities of representations. Recently, Kaveh-Khovanskii and Lazarsfeld-Mustata have generalized and systematically developed Okounkov’s construction, showing the existence of convex bodies which capture much of the asymptotic information about the geometry of (X,D) where X is an algebraic variety and D is a big divisor. This theory of Newton-Okounkov bodies can be viewed as a vast generalization of the well-known theory of toric varieties. The study of Okounkov bodies is a new research area with many open questions, and the purpose of the HalfWorkshop 1422b Okounkov bodies and applications, organised by Megumi Harada (McMaster), Kiumars Kaveh (Pittsburgh), and Askold Khovanskii (Toronto), was 1460 Oberwolfach Report 27/2014 to explore the many recent (and potential new) applications of this theory to other research areas. The Half-Workshop was well attended with over 20 participants, with broad geographic representation from all continents. The group of participants was a nice blend of researchers with various backgrounds such as tropical geometry, representation theory, toric topology, symplectic topology, integrable systems, and number theory. In addition to the senior participants, there were 2 participants supported through the Oberwolfach Leibniz Graduate Students” program. The workshop consisted of 18 research talks in total. In the remaining part of this introduction we briefly describe some of the topics discussed at the workshop. One of the major themes of the workshop was to define functions to and from Newton-Okounkov bodies. Functions from Newton-Okounkov bodies were discussed by Alex Kuronya, with a view towards applications in the study of big divisors and positivity of linear series on algebraic varieties. Functions to NewtonOkounkov bodies were discussed by David Witt-Nystrom in his talk on joint work with Julius Ross, in which they define a kind of analogue of a ‘moment map’ to a Newton-Okounkov body. (Witt-Nystrom also gave another talk on transforming metrics of a line bundle which provided some background on his work on moment maps.) Several of the talks reported on recent progress in the theory of NewtonOkounkov bodies. Victor Lozovanu reported on recent joint work with Kuronya on positivity of linear series on surfaces, and Kiumars Kaveh presented joint work with Khovanskii on the theory of local Newton-Okounkov bodies. One of the junior participants Takuya Murata, invited through the Oberwolfach Leibniz Graduate Students program, was given the opportunity to present his Ph.D. thesis results (supervised by one of the organizers, Kiumars Kaveh) on the asymptotic behavior of multiplicities of reductive group actions. Another major purpose of the workshop was to explore possible connections with other research areas. In this spirit, Huayi Chen gave a talk outlining possible avenues of applications of Newton-Okounkov bodies to arithmetic. Similarly, Sam Payne gave a talk on tropical methods for the study of linear series and Buchstaber gave a presentation on (2n, k)-manifolds; in both talks, many themes overlapped with those arising in the study of Newton-Okounkov bodies. Furthermore, Boris Kazarnovskii talked about an extension of the theory of Newton-Okounkov bodies to the non-algebraic setting of exponential sums, and about a very surprising relation of this non-algebraic subject to modern algebraic geometry. Symplectic geometry, symplectic topology, integrable systems, and toric degenerations also played a main role in the workshop. In this direction, both Chris Manon and Johan Martens reported on their recent work on the Vinberg monoid, while Yuichi Nohara gave a talk on toric degenerations of integrable systems on the Grassmannian and an application to the computation of the potential functions arising in symplectic topology. Continuing the theme of symplectic topology, Kaoru Ono gave a talk on Lagrangian tori in S × S. Okounkov Bodies and Applications 1461 The relation between Newton-Okounkov bodies and Schubert calculus was also a strong theme of the workshop. Valentina Kiritchenko spoke about a ‘geometric mitosis’ operation on Newton-Okounkov polytopes (associated to flag varieties) which give rise to collections of faces of the polytope representing a Schubert cycle. Vladlen Timorin gave a talk on counting vertices of Gel’fand-Cetlin polytopes, which are a special case of Newton-Okounkov bodies of flag varieties. June Huh presented his results on positivity of Chern classes of Schubert cells and varieties, concluding with open questions in this area which relate to Newton-Okounkov bodies. Finally, in the last talk of the workshop, Dave Anderson spoke on computing the effective cone of Bott-Samelson varieties, which arise naturally in the study of Newton-Okounkov bodies due to their central role in representation theory and the geometry of flag varieties. Acknowledgement: The MFO and the workshop organizers would like to thank the National Science Foundation for supporting the participation of junior researchers in the workshop by the grant DMS-1049268, “US Junior Oberwolfach Fellows”. Okounkov Bodies and Applications 1463 Workshop: Okounkov Bodies and Applications

Note on the Grothendieck Group of Subspaces of Rational Functions and Shokurov's Cartier b-divisors

In a previous paper the authors developed an intersection theory for subspaces of rational functions on an algebraic variety X over k = C. In this short note, we first extend this intersection the- ory to an arbitrary algebraically closed ground field k. Secondly we give an isomorphism between the group of Cartier b-divisors on the birational class of X and the Grothendieck group of the semigroup of subspaces of rational functions on X. The constructed isomorphism moreover preserves the inter- section numbers. This provides an alternative point of view on Cartier b-divisors and their intersection theory.

Variations on the theme of solvability by radicals

We discuss the problem of representability and nonrepresentability of algebraic functions by radicals. We show that the Riemann surfaces of functions that are the inverses of Chebyshev polynomials are determined by their local behavior near branch points. We find lower bounds on the degrees of equations to which sufficiently general algebraic functions can be reduced by radicals. We also begin to classify rational functions of prime degree whose inverses are representable by radicals.