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  • 1.
    Abathun, Addisalem
    Stockholm University, Faculty of Science, Department of Mathematics.
    Asymptotic distribution of zeros of a certain class of hypergeometric polynomials2014Licentiate thesis, monograph (Other academic)
    Abstract [en]

    The thesis consists of two papers, both treating hypergeometric polynomials, and a short introduction. The main results are as follows.In the first paper,we study the asymptotic zero distribution of a family of hypergeometric polynomials in one complex variable as their degree goes to infinity,using the associated differential equations that hypergeometric polynomials satisfy.   We describe in particular the curve complex on which the zeros cluster, as level curves associated to integrals on an algebraic curve derived from the equation.   The new result is first of all that we are able to formulate results on the location of zeros of generalized hypergeometric polynomials in greater generality than before (earlier results are mainly concerned with the Gauss hypergeometric case.) Secondly, we are able to formulate a precise conjucture giving the asymptotic behaviour of zeros in the generalized case of our polynomials, which covers previous results.In the second paper we partly prove one of the  conjectures in the first paper by using Euler integral representation of the Gauss hypergeometric functions together with the Saddle point method.

  • 2.
    Alexandersson, Per
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hopkins, Sam
    Zaimi, Gjergji
    Restricted Birkhoff Polytopes and Ehrhart Period Collapse2023In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444Article in journal (Refereed)
    Abstract [en]

    We show that the polytopes obtained from the Birkhoff polytope by imposing additional inequalities restricting the “longest increasing subsequence” have Ehrhart quasi-polynomials which are honest polynomials, even though they are just rational polytopes in general. We do this by defining a continuous, piecewise-linear bijection to a certain Gelfand–Tsetlin polytope. This bijection is not an integral equivalence but it respects lattice points in the appropriate way to imply that the two polytopes have the same Ehrhart (quasi-)polynomials. In fact, the bijection is essentially the Robinson–Schensted–Knuth correspondence.

  • 3.
    Alm, Johan
    Stockholm University, Faculty of Science, Department of Mathematics.
    Universal algebraic structures on polyvector fields2014Doctoral thesis, monograph (Other academic)
    Abstract [en]

    The theory of operads is a conceptual framework that has become a kind of universal language, relating branches of topology and algebra. This thesis uses the operadic framework to study the derived algebraic properties of polyvector fields on manifolds.The thesis is divided into eight chapters. The first is an introduction to the thesis and the research field to which it belongs, while the second chapter surveys the basic mathematical results of the field.The third chapter is devoted to a novel construction of differential graded operads, generalizing an earlier construction due to Thomas Willwacher. The construction highlights and explains several categorical properties of differential graded algebras (of some kind) that come equipped with an action by a differential graded Lie algebra. In particular, the construction clarifies the deformation theory of such algebras and explains how such algebras can be twisted by Maurer-Cartan elements.The fourth chapter constructs an explicit strong homotopy deformation of polynomial polyvector fields on affine space, regarded as a two-colored noncommutative Gerstenhaber algebra. It also constructs an explicit strong homotopy quasi-isomorphism from this deformation to the canonical two-colored noncommmutative Gerstenhaber algebra of polydifferential operators on the affine space. This explicit construction generalizes Maxim Kontsevich's formality morphism.The main result of the fifth chapter is that the deformation of polyvector fields constructed in the fourth chapter is (generically) nontrivial and, in a sense, the unique such deformation. The proof is based on some cohomology computations involving Kontsevich's graph complex and related complexes. The chapter ends with an application of the results to properties of a derived version of the Duflo isomorphism.The sixth chapter develops a general mathematical framework for how and when an algebraic structure on the germs at the origin of a sheaf on Cartesian space can be "globalized" to a corresponding algebraic structure on the global sections over an arbitrary smooth manifold. The results are applied to the construction of the fourth chapter, and it is shown that the construction globalizes to polyvector fields and polydifferential operators on an arbitrary smooth manifold.The seventh chapter combines the relations to graph complexes, explained in chapter five, and the globalization theory of chapter six, to uncover a representation of the Grothendieck-Teichmüller group in terms of A-infinity morphisms between Poisson cohomology cochain complexes on a manifold.Chapter eight gives a simplified version of a construction of a family of Drinfel'd associators due to Carlo Rossi and Thomas Willwacher. Our simplified construction makes the connections to multiple zeta values more transparent--in particular, one obtains a fairly explicit family of evaluations on the algebra of formal multiple zeta values, and the chapter proves certain basic properties of this family of evaluations.

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  • 4.
    Backelin, Jörgen
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Oneto, Alessandro
    Stockholm University, Faculty of Science, Department of Mathematics.
    On a class of power ideals2015In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 219, no 8, p. 3158-3180Article in journal (Refereed)
    Abstract [en]

    In this paper we study the class of power ideals generated by the k(n) forms (x(0) + xi(g1) x(1) + ... + xi(gn) x(n))((k-1)d) where xi is a fixed primitive kth-root of unity and 0 <= g(j) <= k - 1 for all j. For k = 2, by using a Z(k)(n+1)-grading on C[x(0),..., x(n)], we compute the Hilbert series of the associated quotient rings via a simple numerical algorithm. We also conjecture the extension for k > 2. Via Macaulay duality, those power ideals are related to schemes of fat points with support on the k(n) points [1 : xi(g1) : ... : xi(gn)] in P-n. We compute Hilbert series, Betti numbers and Grobner basis for these 0-dimensional schemes. This explicitly determines the Hilbert series of the power ideal for all k: that this agrees with our conjecture for k > 2 is supported by several computer experiments.

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  • 5.
    Backman, Theo
    Stockholm University, Faculty of Science, Department of Mathematics.
    Configuration spaces, props and wheel-free deformation quantization2016Doctoral thesis, monograph (Other academic)
    Abstract [en]

    The main theme of this thesis is higher algebraic structures that come from operads and props.

    The first chapter is an introduction to the mathematical framework needed for the content of this thesis. The chapter does not contain any new results.

    The second chapter is concerned with the construction of a configuration space model for a particular 2-colored differential graded operad encoding the structure of two A algebras with two A morphisms and a homotopy between the morphisms. The cohomology of this operad is shown to be the well-known 2-colored operad encoding the structure of two associative algebras and of an associative algebra morphism between them.

    The third chapter is concerned with deformation quantization of (potentially) infinite dimensional (quasi-)Poisson manifolds. Our proof employs a variation on the transcendental methods pioneered by M. Kontsevich for the finite dimensional case. The first proof of the infinite dimensional case is due to B. Shoikhet. A key feature of the first proof is the construction of a universal L structure on formal polyvector fields. Our contribution is a simplification of B. Shoikhet proof by considering a more natural configuration space and a simpler choice of propagator. The result is also put into a natural context of the dg Lie algebras coming from graph complexes; the L structure is proved to come from a Maurer-Cartan element in the oriented graph complex.

    The fourth chapter also deals with deformation quantization of (quasi-)Poisson structures in the infinite dimensional setting. Unlike the previous chapter, the methods used here are purely algebraic. Our main theorem is the possibility to deformation quantize quasi-Poisson structures by only using perturbative methods; in contrast to the transcendental methods employed in the previous chapter. We give two proofs of the theorem via the theory of dg operads, dg properads and dg props. We show that there is a dg prop morphism from a prop governing star-products to a dg prop(erad) governing (quasi-)Poisson structures. This morphism gives a theorem about the existence of a deformation quantization of (quasi-)Poisson structure. The proof proceeds by giving an explicit deformation quantization of super-involutive Lie bialgebras and then lifting that to the dg properad governing quasi-Poisson structures. The prop governing star-products was first considered by S.A. Merkulov, but the properad governing quasi-Poisson structures is a new construction.

    The second proof of the theorem employs the Merkulov-Willwacher polydifferential functor to transfer the problem of finding a morphism of dg props to that of finding a morphism of dg operads.We construct an extension of the well known operad of A algebras such that the representations of it in V are equivalent to an A structure on V[[ħ]]. This new operad is also a minimal model of an operad that can be seen as the extension of the operad of associative algebras by a unary operation. We give an explicit map of operads from the extended associative operad to the operad we get when applying the Merkulov-Willwacher polydifferential functor to the properad of super-involutive Lie bialgebras. Lifting this map so as to go between their respective models gives a new proof of the main theorem.

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  • 6.
    Bergh, Daniel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Computations in the Grothendieck Group of Stacks2012Licentiate thesis, monograph (Other academic)
    Abstract [en]

    Given an algebraic group, one may consider the class of its classifying stackin the Grothendieck group of stacks. This is an invariant studied byEkedahl. For certain connected groups, called the special groups bySerre and Grothendieck, the invariant simply gives the inverse of the class ofthe group itself. It is natural to ask whether the same is true for otherconnected groups. We investigate this for the groups PGL(2) and PGL(3) under mild restrictions on the choice of base field.In the case of PGL(2), the question turns out to have a positive answer. In the case of PGL(3), we reduce the question to the computation of the invariant for thenormaliser of a maximal torus in PGL(3). The reduction involves determiningthe class of a certain gerbe over the moduli stack of elliptic curves.

  • 7.
    Bergh, Daniel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Destackification and Motivic Classes of Stacks2014Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis consists of three articles treating topics in the theory of algebraic stacks. The first two papers deal with motivic invariants. In the first, we show that the class of the classifying stack BPGLn is the inverse of the class of PGLn in the Grothendieck ring of stacks for n ≤ 3. This shows that the multiplicativity relation holds for the universal torsors, although it is known not to hold for torsors ingeneral for the groups PGL2 and PGL3.

    In the second paper, we introduce an exponential function which can be viewed as a generalisation of Kapranov's motivic zeta function. We use this to derive a binomial theorem for a power operation defined on the Grothendieck ring of varieties. As an application, we give an explicit expression for the motivic class of a universal quasi-split torus, which generalises a result by Rökaeus.

    The last paper treats destackification. We give an algorithm for removing stackiness from smooth, tame stacks with abelian stabilisers by repeatedly applying stacky blow-ups. As applications, we indicate how the result can be used for destackifying general Deligne–Mumford stacks in characteristic zero, and to obtain a weak factorisation theorem for such stacks.

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  • 8.
    Bergh, Daniel
    Stockholm University, Faculty of Science, Department of Mathematics.
    Functorial destackification of tame stacks with abelian stabilisersManuscript (preprint) (Other academic)
    Abstract [en]

    In this article, we study the problem of modifying smooth, algebraic stacks with finite, diagonalisable stabilisers such that their coarse spaces become smooth. The only modifications used are root stacks and blow-ups in smooth centres. If the generic stabiliser of the original stack is trivial, the canonical map from the resulting stack to its coarse space is also a root stack. Hence we can think of the process as removing stackiness from, or destackifying, a smooth stack with help of stacky blow-ups. The construction work over a general base and are functorial in the sense that they respect base change andcompositions with gerbes and smooth, stabiliser preserving maps. As applications, we indicate how this can be used for destackifying general Deligne-Mumford stacks with finite inertia in characteristic zero, and to obtain a weak factorisation theorem for such stacks. Over any field, the method can be used for desingularising locally simplicial toric varieties, without assuming the presence of toroidal structures.

  • 9.
    Bergh, Daniel
    Stockholm University, Faculty of Science, Department of Mathematics. Max Planck Institute for Mathematics, Germany.
    Motivic classes of some classifying stacks2016In: Journal of the London Mathematical Society, ISSN 0024-6107, E-ISSN 1469-7750, Vol. 93, no 1, p. 219-243Article in journal (Refereed)
    Abstract [en]

    We prove that the class of the classifying stack BPGL(n) is the multiplicative inverse of the class of the projective linear group PGL(n) in the Grothendieck ring of stacks K-0(Stack(k)) for n = 2 and n = 3 under mild conditions on the base field k. In particular, although it is known that the multiplicativity relation {T} = {S} . {PGL(n)} does not hold for all PGL(n)-torsors T -> S, it holds for the universal PGL(n)-torsors for said n.

  • 10.
    Bergh, Daniel
    Stockholm University, Faculty of Science, Department of Mathematics.
    The Binomial Theorem and motivic classes of universal quasi-split toriManuscript (preprint) (Other academic)
    Abstract [en]

    Taking symmetric powers of varieties can be seen as a functor from the category of varieties to the category of varieties with an action by the symmetric group. We study a corresponding map between the Grothendieck groups of these categories. In particular, we derive a binomial formula and use it to give explicit expressions for the classes of universal quasi-split tori in the equivariant Grothendieck group of varieties.

  • 11.
    Berglund, Alexander
    Stockholm University, Faculty of Science, Department of Mathematics.
    Homological perturbation theory for algebras over operads2014In: Algebraic and Geometric Topology, ISSN 1472-2747, E-ISSN 1472-2739, Vol. 14, no 5, p. 2511-2548Article in journal (Refereed)
    Abstract [en]

    We extend homological perturbation theory to encompass algebraic structures governed by operads and cooperads. The main difficulty is to find a suitable notion of algebra homotopy that generalizes to algebras over operads  O . To solve this problem, we introduce thick maps of  O –algebras and special thick maps that we call pseudo-derivations that serve as appropriate generalizations of algebra homotopies for the purposes of homological perturbation theory. 

       As an application, we derive explicit formulas for transferring  Ω(C) –algebra structures along contractions, where C  is any connected cooperad in chain complexes. This specializes to transfer formulas for  O ∞  –algebras for any Koszul operad O , in particular for A ∞  –,  C ∞  –,  L ∞  – and  G ∞  –algebras. A key feature is that our formulas are expressed in terms of the compact description of  Ω(C) –algebras as coderivation differentials on cofree C –coalgebras. Moreover, we get formulas not only for the transferred structure and a structure on the inclusion, but also for structures on the projection and the homotopy

  • 12.
    Berglund, Alexander
    Stockholm University, Faculty of Science, Department of Mathematics.
    Rational homotopy theory of mapping spaces via Lie theory for L-infinity algebrasArticle in journal (Refereed)
  • 13.
    Berglund, Alexander
    Stockholm University, Faculty of Science, Department of Mathematics.
    Rational homotopy theory of mapping spaces via Lie theory for L-infinity algebras2015In: Homology, Homotopy and Applications, ISSN 1532-0073, E-ISSN 1532-0081, Vol. 17, no 2, p. 343-369Article in journal (Refereed)
    Abstract [en]

    We calculate the higher homotopy groups of the Deligne–Getzler ∞-groupoid associated to a nilpotent L∞-algebra. As an application, we present a new approach to the rational homotopy theory of mapping spaces.

  • 14.
    Berglund, Alexander
    Stockholm University, Faculty of Science, Department of Mathematics.
    Shellability and the strong gcd-condition2009In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 16, no 2Article in journal (Refereed)
    Abstract [en]

    Shellability is a well-known combinatorial criterion on a simplicial complex for verifying that the associated Stanley-Reisner ring k[] is Cohen-Macaulay. Anotion familiar to commutative algebraists, but which has not received as muchattention from combinatorialists as the Cohen-Macaulay property, is the notion ofa Golod ring. Recently, J¨ollenbeck introduced a criterion on simplicial complexesreminiscent of shellability, called the strong gcd-condition, and he together with theauthor proved that it implies Golodness of the associated Stanley-Reisner ring. Thetwo algebraic notions were earlier tied together by Herzog, Reiner and Welker, whoshowed that if k[∨] is sequentially Cohen-Macaulay, where ∨ is the Alexanderdual of , then k[] is Golod. In this paper, we present a combinatorial companionof this result, namely that if ∨ is (non-pure) shellable then satisfies the stronggcd-condition. Moreover, we show that all implications just mentioned are strict ingeneral but that they are equivalences if is a flag complex.

  • 15.
    Berglund, Alexander
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Bergström, Jonas
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hirzebruch L-polynomials and multiple zeta values2018In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 372, no 1-2, p. 125-137Article in journal (Refereed)
    Abstract [en]

    We express the coefficients of the Hirzebruch L-polynomials in terms of certain alternating multiple zeta values. In particular, we show that every monomial in the Pontryagin classes appears with a non-zero coefficient, with the expected sign. Similar results hold for the polynomials associated to the Â-genus.

  • 16.
    Berglund, Alexander
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Börjeson, Kaj
    Stockholm University, Faculty of Science, Department of Mathematics.
    Free loop space homology of highly connected manifolds2017In: Forum mathematicum, ISSN 0933-7741, E-ISSN 1435-5337, Vol. 29, no 1, p. 201-228Article in journal (Refereed)
    Abstract [en]

    We calculate the homology of the free loop space of (n - 1)-connected closed manifolds of dimension at most 3 n - 2 (n >= 2), with the Chas-Sullivan loop product and loop bracket. Over a field of characteristic zero, we obtain an expression for the BV-operator. We also give explicit formulas for the Betti numbers, showing they grow exponentially. Our main tool is the connection between formality, coformality and Koszul algebras that was elucidated by the first author [6].

  • 17.
    Berglund, Alexander
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hess, Kathryn
    Homotopic Hopf-Galois extensions revisitedManuscript (preprint) (Other academic)
    Abstract [en]

    In this article we revisit the theory of homotopic Hopf-Galois extensions introduced in arXiv:0902.3393v2 [math.AT], in light of the homotopical Morita theory of comodules established in arXiv:1411.6517 [math.AT]. We generalize the theory to a relative framework, which we believe is new even in the classical context and which is essential for treating the Hopf-Galois correspondence in forthcoming work of the second author and Karpova. We study in detail homotopic Hopf-Galois extensions of differential graded algebras over a commutative ring, for which we establish a descent-type characterization analogous to the one Rognes provided in the context of ring spectra. An interesting feature in the differential graded setting is the close relationship between homotopic Hopf-Galois theory and Koszul duality theory. We show that nice enough principal fibrations of simplicial sets give rise to homotopic Hopf-Galois extensions in the differential graded setting, for which this Koszul duality has a familiar form.

  • 18.
    Berglund, Alexander
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hess, Kathryn
    Homotopical Morita theory for coringsManuscript (preprint) (Other academic)
    Abstract [en]

    A coring (A,C) consists of an algebra A and a coalgebra C in the monoidal category of A-bimodules. Corings and their comodules arise naturally in the study of Hopf-Galois extensions and descent theory, as well as in the study of Hopf algebroids. In this paper, we address the question of when two corings in a symmetric monoidal model category V are homotopically Morita equivalent, i.e., when their respective categories of comodules are Quillen equivalent. The category of comodules over the trivial coring (A,A) is isomorphic to the category of A-modules, so the question above englobes that of when two algebras are homotopically Morita equivalent. We discuss this special case in the first part of the paper, extending previously known results. To approach the general question, we introduce the notion of a 'braided bimodule' and show that adjunctions between A-Mod and B-Mod that lift to adjunctions between (A,C)-Comod and (B,D)-Comod correspond precisely to braided bimodules between (A,C) and (B,D). We then give criteria, in terms of homotopic descent, for when a braided bimodule induces a Quillen equivalence. In particular, we obtain criteria for when a morphism of corings induces a Quillen equivalence, providing a homotopic generalization of results by Hovey and Strickland on Morita equivalences of Hopf algebroids. To illustrate the general theory, we examine homotopical Morita theory for corings in the category of chain complexes over a commutative ring.

  • 19.
    Berglund, Alexander
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Madsen, Ib
    University of Copenhagen.
    Rational homotopy theory of automorphisms of highly connected manifoldsArticle in journal (Refereed)
  • 20.
    Berglund, Alexander
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Madsen, Ib
    Rational homotopy theory of automorphisms of highly connected manifoldsManuscript (preprint) (Other academic)
    Abstract [en]

    We study the rational homotopy types of classifying spaces of automorphism groups of 2d-dimensional (d-1)-connected manifolds (d > 2). We prove that the rational homology groups of the homotopy automorphisms and the block diffeomorphisms of the manifold #^g S^d x S^d relative to a disk stabilize as g increases. Via a theorem of Kontsevich, we obtain the striking result that the stable rational cohomology of the homotopy automorphisms comprises all unstable rational homology groups of all outer automorphism groups of free groups.

  • 21.
    Berglund, Alexander
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Saleh, Bashar
    Stockholm University, Faculty of Science, Department of Mathematics.
    A dg Lie model for relative homotopy automorphisms2020In: Homology, Homotopy and Applications, ISSN 1532-0073, E-ISSN 1532-0081, Vol. 22, no 2, p. 105-121Article in journal (Refereed)
    Abstract [en]

    We construct a dg" role="presentation" style="display: inline; line-height: normal; font-size: 17.3333px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(64, 64, 64); font-family: "Times New Roman", Times, serif; position: relative;">dgdg Lie algebra model for the universal cover of the classifying space of the grouplike monoid of homotopy automorphisms of a space that fix a given subspace. We derive the model from a known model for based homotopy automorphisms together with general result on rational models for geometric bar constructions.

  • 22.
    Bergström, Jonas
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Bergvall, Olof
    The equivariant Euler characteristic of A3[2]2020In: Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V, ISSN 0391-173X, E-ISSN 2036-2145, Vol. XX, no 4, p. 1345-1357Article in journal (Refereed)
    Abstract [en]

    We compute the weighted Euler characteristic, equivariant with respect to the action of the symplectic group of degree six over the field of two elements, of the moduli space of principally polarized Abelian threefolds together with a level two structure. 

  • 23.
    Bergström, Jonas
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Faber, Carel
    van der Geer, Gerard
    Siegel modular forms of degree three and the cohomology of local systems2014In: Selecta Mathematica, New Series, ISSN 1022-1824, E-ISSN 1420-9020, Vol. 20, no 1, p. 83-124Article in journal (Refereed)
    Abstract [en]

    We give an explicit conjectural formula for the motivic Euler characteristic of an arbitrary symplectic local system on the moduli space of principally polarized abelian threefolds. The main term of the formula is a conjectural motive of Siegel modular forms of a certain type; the remaining terms admit a surprisingly simple description in terms of the motivic Euler characteristics for lower genera. The conjecture is based on extensive counts of curves of genus three and abelian threefolds over finite fields. It provides a lot of new information about vector-valued Siegel modular forms of degree three, such as dimension formulas and traces of Hecke operators. We also use it to predict several lifts from genus 1 to genus 3, as well as lifts from and new congruences of Harder type.

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  • 24.
    Bergström, Jonas
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Minabe, Satoshi
    On the cohomology of moduli spaces of (weighted) stable rational curves2013In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 275, no 3-4, p. 1095-1108Article in journal (Refereed)
    Abstract [en]

    We give a recursive algorithm for computing the character of the cohomology of the moduli space of stable -pointed genus zero curves as a representation of the symmetric group on letters. Using the algorithm we can show a formula for the maximum length of this character. Our main tool is connected to the moduli spaces of weighted stable curves introduced by Hassett.

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  • 25.
    Bergström, Jonas
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Minabe, Satoshi
    On the cohomology of the Losev–Manin moduli space2014In: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 144, no 1, p. 241-252Article in journal (Refereed)
    Abstract [en]

    We determine the cohomology of the Losev--Manin moduli space $\overline{M}_{0, 2 | n}$ of pointed genus zero curves as a representation of the product of symmetric groups $\Sg_2 \times \Sg_n$.

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  • 26.
    Bergvall, Olof
    Stockholm University, Faculty of Science, Department of Mathematics.
    Cohomology of the moduli space of curves of genus three with level two structure2014Licentiate thesis, monograph (Other academic)
    Abstract [en]

    In this thesis we investigate the moduli space M3[2] of curves of genus 3 equipped with a symplectic level 2 structure. In particular, we are interested in the cohomology of this space. We obtain cohomological information by decomposing M3[2] into a disjoint union of two natural subspaces, Q[2] and H3[2], and then making S7- resp. S8-equivariantpoint counts of each of these spaces separately.

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  • 27. Campos, Ricardo
    et al.
    Ward, Benjamin C.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Gravity formality2018In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 331, p. 439-483Article in journal (Refereed)
    Abstract [en]

    We show that Willwacher's cyclic formality theorem can be extended to preserve natural Gravity operations on cyclic multivector fields and cyclic multidifferential operators. We express this in terms of a homotopy Gravity quasiisomorphism with explicit local formulas. For this, we develop operadic tools related to mixed complexes and cyclic homology and prove that the operad M(O )of natural operations on cyclic operators is formal and hence quasi-isomorphic to the Gravity operad.

  • 28. Carlini, Enrico
    et al.
    Oneto, Alessandro
    Stockholm University, Faculty of Science, Department of Mathematics. Polytechnic University of Turin, Italy.
    Monomials as Sums of k-th Powers of Forms2015In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 43, no 2, p. 650-658Article in journal (Refereed)
    Abstract [en]

    Motivated by recent results on the Waring problem for polynomial rings [4] and representation of monomial as sum of powers of linear forms [3], we consider the problem of presenting monomials of degree kd as sums of kth-powers of forms of degree d. We produce a general bound on the number of summands for any number of variables which we refine in the two variables case. We completely solve the k = 3 case for monomials in two and three variables.

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  • 29.
    Cooper, Simon
    Stockholm University, Faculty of Science, Department of Mathematics.
    Tautological rings of Shimura varieties2022Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    This licentiate thesis consists of two papers. In paper I the tautological ring of a Hilbert modular variety at an unramified prime is computed. The method of van der Geer in the case of A_{g} is extended to deal with the case of the Hilbert modular variety, which is more complicated. An example involving the unitary group is given which shows that this method cannot be used to compute the tautological rings of all Shimura varieties of Hodge type. In paper II we compute the pushforward map from a sub flag variety defined by a Levi subgroup to the Siegel flag variety. Specifically, this is the Levi factor of the parabolic associated with the maximal rational boundary component of the Siegel Shimura datum. The method involves an explicit understanding of the pullback map and an application of the self intersection formula.

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  • 30. Eklund, David
    et al.
    Jost, Christine
    Stockholm University, Faculty of Science, Department of Mathematics.
    Peterson, Chris
    A method to compute Segre classes of subschemes of projective space2013In: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, Vol. 12, no 2, p. 1250142-Article in journal (Refereed)
    Abstract [en]

    We present a method to compute the degrees of the Segre classes of a subscheme of complex projective space. The method is based on generic residuation and intersection theory. It has been implemented using the software system Macaulay2.

  • 31.
    Espic, Hadrien
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Saleh, Bashar
    Stockholm University, Faculty of Science, Department of Mathematics.
    On the group of homotopy classes of relative homotopy automorphismsManuscript (preprint) (Other academic)
    Abstract [en]

    We prove that the group of homotopy classes of relative homotopy automorphisms of a simply connected finite CW-complex is finitely presented and that the rationalization map from this group to its rational analogue has a finite kernel.

  • 32.
    Frostne, Isabel
    Stockholm University, Faculty of Science, Department of Mathematics and Science Education.
    Abstrakta och konkreta ting i geometrilandskapet: Varför elever i årskurs 7-9 har lätt och svårt i geometriområdet samt vad läraren gör för att underlätta elevernas förståelse2019Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    Geometry is an area in mathematics that is considered not abstract, on the contrary from other areas in mathematics. As geometry is considered an unabstracted area in mathematics, why has students around the world difficulties with geometry? TIMSS (Trends in International Mathematics and Science Study) has shown that Swedish students in 8th grade has difficulties in algebra and geometry. The study focuses on why secondary school students’ have easy to understand some parts in geometry and why they have difficulties in other parts. Furthermore the study focuses on strategies teachers use to facilitate understanding in geometry. The study is carried out by interviewing six teachers in secondary school. The interviews are recorded and transcribed for enabling thematic analysis. The result shows that teachers experiences that students have easy to understand the first dimension (length and perimeter) and easy to understand geometrical objects as for example rectangular shapes. The reason behind the easiness is that these elements in geometry is known for the students, easy for the teachers to explain and not abstract. The students have difficulties comprehending two and especially three dimensional objects, difficult geometrical objects as circular objects, objects where the height is “situated” outside the object and irregular figures, unit conversions and concepts in geometry. The reasons behind these difficulties are mainly: the elements and methods are unfamiliar and abstract to the students. The abstraction in geometry are shown as comprehending how big or small sizes are in two and three dimensions and difficulties to comprehend the big discrepancy between the numbers in unit conversion, Teachers also observe that students have difficulties in visualizing and manipulating objects. These results show that what is known and not abstract are the opposite for what students have difficulties with, i.e. unknown and abstract. The strategies the teachers use are mostly to concretize the difficulties in geometry and in that way show students why it is valid. Other strategies are concerning with building a strong foundation in geometry, to combine geometry with other subjects in school and using students prior knowledge to build new knowledge. The red articles agrees with the result from the study.

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  • 33.
    Hainaut, Louis
    Stockholm University, Faculty of Science, Department of Mathematics.
    Configuration spaces on a wedge of spheres and Hochschild-Pirashvili homologyManuscript (preprint) (Other academic)
    Abstract [en]

    We study the compactly supported rational cohomology of configuration spaces of points on wedges of spheres, equipped with natural actions of the symmetric group and the group Out(F_g) of outer automorphism of the free group. These representations are closely related to Hochschild-Pirashvili homology with coefficients in square-zero algebras, and they show up in seemingly unrelated parts of mathematics, from cohomology of moduli spaces of curves to polynomial functors on free groups.       

    We show that these cohomology representations form a polynomial functor, and use various geometric models to compute a substantial part of its composition factors. We further compute the composition factors completely for all configurations of n\leq 10 particles. An application of this analysis is a new super-exponential lower bound on the symmetric group action on the weight 0 component of H^*_c(M_{2,n}).

  • 34.
    Hainaut, Louis
    Stockholm University, Faculty of Science, Department of Mathematics.
    Some computations of compact support cohomology of configuration spaces2022Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    This licentiate thesis consists of two papers related to configuration spaces of points.

    In paper I a general formula for the Euler characteristic of configuration spaces on any topologically stratified space X is obtained in terms of geometric and combinatorial data about the strata. More generally this paper provides a formula for the Euler characteristic of the cohomology with compact support of these configuration spaces with coefficients in a constructible complex of sheaves K on X. The formula for the classical Euler characteristic is then obtained by taking K to be the dualizing complex of X. This formula generalizes similar results about configuration spaces on a manifold or on a simplicial complex, as well as another formula for any Hausdorff space X when the complex of sheaves K is trivial.

    In paper II we study the cohomology with compact support of configuration spaces on a wedge sum of spheres X, with rational coefficients. We prove that these cohomology groups are the coefficients of an analytic functor computing the Hochschild--Pirashvili homology of X with certain coefficients. Moreover, we prove that, up to a filtration, these same cohomology groups are a polynomial functor in the reduced cohomology of X, with coefficients not depending on X. Contrasting the information provided by two different models we are able to partially compute these coefficients, and in particular we obtained a complete answer for configurations of at most 10 points. The coefficients thus obtained can be used to compute the weight 0 part of the cohomology with compact support of the moduli space M_{2,n}.

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  • 35.
    Hainaut, Louis
    Stockholm University, Faculty of Science, Department of Mathematics.
    The Euler characteristic of configuration spaces2022In: Bulletin of the Belgian Mathematical Society Simon Stevin, ISSN 1370-1444, E-ISSN 2034-1970, Vol. 29, no 1, p. 87-96Article in journal (Refereed)
    Abstract [en]

    In this short note we present a generating function computing the compactly supported Euler characteristic  χc(F(x,n),K⊠n) of the configuration spaces on a topologically stratified space X, with K a constructible complex of sheaves on X, and we obtain as a special case a generating function for the Euler characteristic χ(F(X,n)). We also recall how to use existing results to turn our computation of the Euler characteristic into a computation of the equivariant Euler characteristic.

  • 36.
    Hainaut, Louis
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Gadish, Nir
    Configuration spaces on a wedge of spheres and Hochschild-Pirashvili homologyManuscript (preprint) (Other academic)
    Abstract [en]

    We study the compactly supported rational cohomology of configuration spaces of points on wedges of spheres, equipped with natural actions of the symmetric group and the group Out(F_g) of outer automorphism of the free group. These representations are closely related to Hochschild-Pirashvili homology with coefficients in square-zero algebras, and they show up in seemingly unrelated parts of mathematics, from cohomology of moduli spaces of curves to polynomial functors on free groups.       

    We show that these cohomology representations form a polynomial functor, and use various geometric models to compute a substantial part of its composition factors. We further compute the composition factors completely for all configurations of n\leq 10 particles. An application of this analysis is a new super-exponential lower bound on the symmetric group action on the weight 0 component of H^*_c(M_{2,n}).

  • 37. Honigs, Katrina
    et al.
    Lombardi, Luigi
    Tirabassi, Sofia
    Stockholm University, Faculty of Science, Department of Mathematics. University of Bergen, Norway.
    Derived equivalences of canonical covers of hyperelliptic and Enriques surfaces in positive characteristic2020In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 295, no 1-2, p. 727-749Article in journal (Refereed)
    Abstract [en]

    We prove that any Fourier–Mukai partner of an abelian surface over an algebraically closed field of positive characteristic is isomorphic to a moduli space of Gieseker-stable sheaves. We apply this fact to show that the set of Fourier–Mukai partners of a canonical cover of a hyperelliptic or Enriques surface over an algebraically closed field of characteristic greater than three is trivial. These results extend earlier results of Bridgeland–Maciocia and Sosna to positive characteristic.

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  • 38.
    Hosgood, Timothy
    Stockholm University, Faculty of Science, Department of Mathematics.
    Simplicial Chern–Weil theory for coherent analytic sheaves, part II2023In: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785Article in journal (Refereed)
    Abstract [en]

    In the previous part of this diptych, we defined the notion of an admissible simplicial connection, as well as explaining how H.I. Green constructed a resolution of coherent analytic sheaves by locally free sheaves on the Čech nerve. This paper seeks to apply these abstract formalisms, by showing that Green’s barycentric simplicial connection is indeed admissible, and that this condition is exactly what we need in order to be able to apply Chern–Weil theory and construct characteristic classes. We show that, in the case of (global) vector bundles, the simplicial construction agrees with what one might construct manually: the explicit Čech representatives of the exponential Atiyah classes of a vector bundle agree. Finally, we summarise how all the preceding theory fits together to allow us to define Chern classes of coherent analytic sheaves, as well as showing uniqueness in the compact case. 

  • 39.
    Jost, Christine
    Stockholm University, Faculty of Science, Department of Mathematics.
    A Macaulay2 package for characteristic classes and the topological Euler characteristic of complex projective schemesManuscript (preprint) (Other academic)
    Abstract [en]

    The Macaulay2 package CharacteristicClasses provides commands for the computation of the topological Euler characteristic, the degrees of the Chern classes and the degrees of the Segre classes of a closed subscheme of complex projective space. The computations can be done both symbolically and numerically, the latter using an interface to Bertini. We provide some background of the implementation and show how to use the package with the help of examples.

  • 40.
    Jost, Christine
    Stockholm University, Faculty of Science, Department of Mathematics.
    An algorithm for computing the topological Euler characteristic of complex projective varietiesManuscript (preprint) (Other academic)
    Abstract [en]

    We present an algorithm for the symbolic and numerical computation of the degrees of the Chern-Schwartz-MacPherson classes of a closed subvariety of projective space P^n. As the degree of the top Chern-Schwartz-MacPherson class is the topological Euler characteristic, this also yields a method to compute the topological Euler characteristic of projective varieties. The method is based on Aluffi's symbolic algorithm to compute degrees of Chern-Schwartz-MacPherson classes, a symbolic method to compute degrees of Segre classes, and the regenerative cascade by Hauenstein, Sommese and Wampler. The new algorithm complements the existing algorithms. We also give an example for using a theorem by Huh to compute an invariant from algebraic statistics, the maximum likelihood degree of an implicit model.

  • 41.
    Jost, Christine
    Stockholm University, Faculty of Science, Department of Mathematics.
    Globalizing L-infinity automorphisms of the Schouten algebra of polyvector fieldsIn: Differential geometry and its applications (Print), ISSN 0926-2245, E-ISSN 1872-6984Article in journal (Refereed)
    Abstract [en]

    Recently, Willwacher showed that the Grothendieck-Teichmuller group GRT acts by L-infinity-automorphisms on the Schouten algebra of polyvector fields T_poly(R^d) on affine space R^d. In this article, we prove that a large class of L-infinity-automorphisms on the Schouten algebra, including Willwacher's, can be globalized. That is, given an L-infinity-automorphism of T_poly(R^d) and a general smooth manifold M with the choice of a torsion-free connection, we give an explicit construction of an L-infinity-automorphism of the Schouten algebra T_poly(M) on the manifold M, depending on the chosen connection. The method we use is the Fedosov trick.

  • 42.
    Jost, Christine
    Stockholm University, Faculty of Science, Department of Mathematics.
    Topics in Computational Algebraic Geometry and Deformation Quantization2013Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis consists of two parts, a first part on computations in algebraic geometry, and a second part on deformation quantization. More specifically, it is a collection of four papers. In the papers I, II and III, we present algorithms and an implementation for the computation of degrees of characteristic classes in algebraic geometry. Paper IV is a contribution to the field of deformation quantization and actions of the Grothendieck-Teichmüller group.

    In Paper I, we present an algorithm for the computation of degrees of Segre classes of closed subschemes of complex projective space. The algorithm is based on the residual intersection theorem and can be implemented both symbolically and numerically.

    In Paper II, we describe an algorithm for the computation of the degrees of Chern-Schwartz-MacPherson classes and the topological Euler characteristic of closed subschemes of complex projective space, provided an algorithm for the computation of degrees of Segre classes. We also explain in detail how the algorithm in Paper I can be implemented numerically. Together this yields a symbolical and a numerical version of the algorithm.

    Paper III describes the Macaulay2 package CharacteristicClasses. It implements the algorithms from papers I and II, as well as an algorithm for the computation of degrees of Chern classes.

    In Paper IV, we show that L-infinity-automorphisms of the Schouten algebra T_poly(R^d) of polyvector fields on affine space R^d which satisfy certain conditions can be globalized. This means that from a given L-infinity-automorphism of T_poly(R^d) an L-infinity-automorphism of T_poly(M) can be constructed, for a general smooth manifold M. It follows that Willwacher's action of the Grothendieck-Teichmüller group on T_poly(R^d) can be globalized, i.e., the Grothendieck-Teichmüller group acts on the Schouten algebra T_poly(M) of polyvector fields on a general manifold M.

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  • 43.
    Källström, Rolf
    et al.
    University of Gävle, Department of Mathematics.
    Tadesse, Yohannes
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hilbert series of modules over Lie algebroidsArticle in journal (Refereed)
    Abstract [en]

    We consider modules M over Lie algebroids g_A which are of  finite type over a local noetherian ring A.  Using ideals  J\subseteq A such that g_A . J \subseteq J  and the length  l_{g_A}(M/JM)< \infty we can define in a natural way the  Hilbert series of M with respect to the defining ideal J.  This  notion is in particular studied for modules over the Lie algebroid  of k-linear derivations g_A=T_A(I) that preserve an ideal  I\subseteq A, for example when A is the ring of convergent  power series.

  • 44.
    Leigh, Oliver
    Stockholm University, Faculty of Science, Department of Mathematics. The University of Melbourne, Australia; The University of British Columbia, Canada .
    Unweighted Donaldson-Thomas Theory of the Banana 3fold with Section Classes2020In: Quarterly Journal of Mathematics, ISSN 0033-5606, E-ISSN 1464-3847, Vol. 71, no 3, p. 867-942Article in journal (Refereed)
    Abstract [en]

    We further the study of the Donaldson–Thomas theory of the banana 3-folds which were recently discovered and studied by Bryan [3]. These are smooth proper Calabi–Yau 3-folds which are fibred by Abelian surfaces such that the singular locus of a singular fibre is a non-normal toric curve known as a ‘banana configuration’. In [3], the Donaldson–Thomas partition function for the rank 3 sub-lattice generated by the banana configurations is calculated. In this article, we provide calculations with a view towards the rank 4 sub-lattice generated by a section and the banana configurations. We relate the findings to the Pandharipande–Thomas theory for a rational elliptic surface and present new Gopakumar–Vafa invariants for the banana 3-fold.

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  • 45.
    Leites, Dimitry
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Bernstein, Joseph
    Department of mathematics, Tel-Aviv University, Israel.
    Molotkov, Vladimir
    Institute of Nuclear Research and Nuclear Energy, Sofia, Bulgaria.
    Shander, Vladimir
    Seminar of Supersymmetries: volume 1 (edited by D. Leites)2011Book (Refereed)
    Abstract [ru]

    Supermanifold theory is a relatively new branch of mathematics. Ideas of supersymmetry appeared to resolve several hitherto seemingly unsolvable problems of theoretical physics and quickly flourished into a rich blend of differential and algebraic geometers with own deep problems. In this book there are presented basics of linear and general algebra in superspaces, elements of algebraic and differential geometers on supermanifolds. The book is saturated by open questions of various degree of complexity and will be useful to researchers (theoretical physicists and mathematicians) as well as (research) students.

  • 46. Libgober, Anatoly
    et al.
    Shapiro, Boris Z.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Meromorphic functions without real critical values and related braids2023In: European Journal of Mathematics, ISSN 2199-675X, E-ISSN 2199-6768, Vol. 9, no 3, article id 70Article in journal (Refereed)
    Abstract [en]

    We study the open subset of the Hurwitz space, consisting of meromorphic functions of a given degree defined on closed Riemann surfaces of a given genus and having no real critical values, and enumerate its connected components in terms of braids. Specifically, to a function in this open set, we assign a braid in the braid group of the underlying closed surface and characterize all braids which might appear using this construction. We introduce the equivalence relation among these braids such that the braids corresponding to the meromorphic functions from the same connected component of the above Hurwitz space are equivalent while non-equivalent braids correspond to distinct connected components. Several special families of meromorphic functions, some applications, and further problems are discussed.

  • 47.
    Lindell, Erik
    Stockholm University, Faculty of Science, Department of Mathematics.
    Abelian cycles in the homology of the Torelli groupManuscript (preprint) (Other academic)
    Abstract [en]

    The Torelli group of an orientable smooth surface is the group of isotopy classes of orientation preserving diffeomorphisms that act trivially on the first homology group of the surface. In this paper, we study the rational homology groups of Torelli groups of smooth, compact and orientable surfaces. More specifically, we study a certain class of stable homology classes, i.e. classes that exist for sufficiently large genus, and explicitly describe the image of these classes under a higher degree version of the Johnson homomorphism, as a representation of the symplectic group. This gives a lower bound on the dimension of the stable homology of the group, as well as providing some further evidence that these homology groups satisfy representation stability for symplectic groups, in the sense of Church and Farb.

  • 48.
    Lindell, Erik
    Stockholm University, Faculty of Science, Department of Mathematics.
    Stable phenomena for some automorphism groups in topology2021Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    This licentiate thesis consists of two papers about topics related to representation stability for different automorphisms groups of topological spaces and manifolds.

    In Paper I, we study the rational homology groups of \textit{Torelli groups} of smooth, compact and orientable surfaces. The Torelli group of a smooth surface is the group of isotopy classes of orientation preserving diffeomorphisms that act trivially on the first homology group of the surface. In the paper, we study a certain class of stable homology classes, i.e. classes that exist for sufficiently large genus, and explicitly describe the image of these classes under a higher degree version of the \textit{Johnson homomorphism}, as a representation of the symplectic group. This gives a lower bound on the dimension of the stable homology of the group, as well as providing some further evidence that these homology groups satisfy representation stability for symplectic groups, in the sense of Church and Farb.

    In Paper II, we study pointed homotopy automorphisms of iterated wedge sums of spaces as well as boundary relative homotopy automorphisms of iterated connected sums of manifolds with a disk removed. We prove that the rational homotopy groups of these, for simply connected CW-complexes and closed manifolds respectively,  satisfy representation stability for symmetric groups, in the sense of Church and Farb.

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  • 49.
    Lindell, Erik
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Saleh, Bashar
    Representation stability for homotopy automorphismsManuscript (preprint) (Other academic)
    Abstract [en]

    We study pointed homotopy automorphisms of iterated wedge sums of spaces as well as boundary relative homotopy automorphisms of iterated connected sums of manifolds with a disk removed. We prove that the rational homotopy groups of these, for simply connected CW-complexes and closed manifolds respectively,  satisfy representation stability for symmetric groups, in the sense of Church and Farb.

  • 50. Lopes, Margarida Mendes
    et al.
    Pardini, Rita
    Tirabassi, Sofia
    Stockholm University, Faculty of Science, Department of Mathematics.
    A footnote to a theorem of Kawamata2023In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 296, no 10, p. 4739-4744Article in journal (Refereed)
    Abstract [en]

    Kawamata has shown that the quasi-Albanese map of a quasi-projective variety with log-irregularity equal to the dimension and log-Kodaira dimension 0 is birational. In this note, we show that under these hypotheses the quasi-Albanese map is proper in codimension 1 as conjectured by Iitaka. 

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