Positivity of Linear Series and Vector Bundles (14w5056)

Arriving in Banff, Alberta Sunday, February 2 and departing Friday February 7, 2014


(University of Washington)

(Goethe Universität Frankfurt)

(Pedagogical University of Cracow)


Recent years have witnessed significant interest in various notions of positivity for line bundles and linear series, and a spectacular progress in their applications. Some of the highlights are the proof of the finite generation of canonical rings by Birkar, Cascini, Hacon, and Mtextsuperscript{c}Kernan, and the construction of moduli spaces for canonically polarized varieties by the cumulative work of Alexeev, Hassett, Keel, Koll'ar, Kov'acs, Mori, Shepherd-Barron, Viehweg, and others.This circle of ideas has many applications outside its immediate neighbourhood. Perhaps one of the most intriguing instances comes from arithmetic geometry: Lang's conjectures link the positivity of the canonical bundle of a variety to the behaviour of its rational points. In another direction, representation theory of classical groups is a place where ample divisors on homogeneous spaces play a distinguished role. Similar concepts have been long defined for vector bundles but they have been apparently much less studied. The main focus of the workshop is to study the existing positivity concepts of line bundles and vector bundles with an emphasis on the higher rank case.The goal is to bring together specialists in both areas, exchange information on recent developments and work together on concrete problems. One of the expected products of the workshop will be a list of important open problems and a road map to explore them. We also believe that an important aim of the workshop should be to introduce young scholars to modern concepts and current problems attached to notions of positivity for linear series and vector bundles.There exists a highly developed and widely used theory of positivity for line bundles, which forms one of the fundamental building blocks of modern projective geometry. With the recent shift in emphasis towards birational geometry, there is a heightened interest for the behaviour of big line bundles --- as opposed to ample ones, which would be a much stronger property, which describe the various birational models of the underlying variety. Both ample and big divisors have numerical, cohomological, and geometric characterizations, which makes for a quite satisfactory theory of positivity.This point of view is prevalent in most recent works in birational geometry, including the recent breakthrough of Birkar--Cascini--Hacon--Mtextsuperscript{c}Kernan on the finite generation of the canonical ring. The major new component as opposed to previous work in the area was the extensive use of multiplier ideals, which are again strongly related to positivity of divisors with fractional coefficients. Beside the intrinsic importance of this result for the field, it gave rise to numerous applications, which, for example include a fairly detailed knowledge of linear series on Fano varieties.Several of the positivity concepts defined for divisors such as ampleness, nefness, or bigness have been considered in the case of higher rank vector bundles. In addition to these there are various other concepts designed with vector bundles in mind, for instance Viehweg's weak semipositivity, Miyaoka's generic semipositivity and almost everywhere ampleness. These notions play a prominent role for example in the theory of moduli of higher-dimensional varieties, which is an active area of research with very significant recent progress due to Alexeev, Koll'ar, Kov'acs, and others.It has recently come to light in the thesis of Kelly Jabbusch that the two definitions of a big vector bundle put forward by Lazarsfeld and Viehweg, respectively, do not agree. This surprising but simple observation has opened the door to questions regarding the relationship between the various positivity concepts for vector bundles and their geometric consequences. We want to explore how the behaviour of positivity for line bundles extends or branches when considering higher-rank bundles.Historically, vector bundle techniques have played important role in understanding various properties of linear series, prominently seen in Green and Lazarsfeld's Koszul machinery for syzygy problems, Reider's work on adjoint line bundles and Fujita's conjecture on surfaces and recent results of Pareschi and Popa to mention but a few important examples.The specific goal of the proposed workshop would be to bring together the linear series community and researchers working on vector bundles to exchange information about recent developments in the respective fields, and start investigating the questions mentioned below.One area where there is hope for new developments soon is the study of Seshadri constants associated to vector bundles. There have been some initial results on the topic due to Hacon and Sommese among others, but the situation is much less understood than in the line bundle case.Some concrete sample problems here that one might want to attack are lower bounds on Seshadri constants of vector bundles in very general points and structure questions on multi-point Seshadri constants paralleling Nagata conjectures. Regarding the former question, one would like to capitalize on the ground-breaking work of Ein, K"uchle, and Lazarsfeld. Perhaps the direction in which the theory of linear series has developed at the highest speed in recent years is the theory of Newton--Okounkov bodies. Building on ideas of Okounkov, Kaveh--Khovanskii and Lazarsfeld--Mustac tu a developed a framework in which convex bodies are associated to big divisors on projective varieties.By now we know a certain amount about their behaviour --- it has been shown for example that Newton--Okounkov bodies are always polygons in dimension two --- and they have found applications out in representation theory and combinatorics. Newton--Okounkov bodies have also been successfully applied to understanding properties of the volume function defined on the big cone of an algebraic variety. To this day, arguably the most important result is the recent work of Harada and Kaveh, where Newton--Okounkov bodies are used to construct integrable systems on smooth projective varieties. It appears to be an intriguing line of research to investigate if one can meaningfully extend the theory of Newton--Okounkov bodies for vector bundles and recover similar or unexpected new properties of higher order asymptotic cohomology functions of vector bundles. The workshop would be the ideal place to start investigations in this direction.