Jörg Schürmann

Purity for graded potentials and quantum cluster positivity

Consider a smooth quasi-projective variety XX equipped with a C∗C∗-action, and a regular function f:X→Cf:X→C which is C∗C∗-equivariant with respect to a positive weight action on the base. We prove the purity of the mixed Hodge structure and the hard Lefschetz theorem on the cohomology of the vanishing cycle complex of ff on proper components of the critical locus of ff, generalizing a result of Steenbrink for isolated quasi-homogeneous singularities. Building on work by Kontsevich and Soibelman, Nagao, and Efimov, we use this result to prove the quantum positivity conjecture for cluster mutations for all quivers admitting a positively graded nondegenerate potential. We deduce quantum positivity for all quivers of rank at most 4; quivers with nondegenerate potential admitting a cut; and quivers with potential associated to triangulations of surfaces with marked points and nonempty boundary.

Characteristic Classes of Singular Toric Varieties

In this paper we compute the motivic Chern classes and homology Hirzebruch characteristic classes of (possibly singular) toric varieties, which in the context of complete toric varieties fit nicely with a generalized Hirzebruch-Riemann-Roch theorem. As important special cases, we obtain new (or recover well-known) formulae for the Baum-Fulton-MacPherson Todd (or MacPhersons Chern) classes of toric varieties, as well as for the Thom-Milnor L-classes of simplicial projective toric varieties. We present two different perspectives for the computation of these characteristic classes of toric varieties. First, we take advantage of the torus-orbit decomposition and the motivic properties of the motivic Chern and respectively homology Hirzebruch classes to express the latter in terms of dualizing sheaves and respectively the (dual) Todd classes of closures of orbits. This method even applies to torus-invariant subspaces of a given toric variety. The obtained formula is then applied to weighted lattice-point counting in lattice polytopes and their subcomplexes, yielding generalized Pick-type formulae. Second, in the case of simplicial toric varieties, we compute our characteristic classes by using the Lefschetz-Riemann-Roch theorem of Edidin-Graham in the context of the geometric quotient description of such varieties. In this setting, we define mock Hirzebruch classes of simplicial toric varieties (which specialize to the mock Chern, mock Todd, and mock L-classes of such varieties) and investigate the difference between the (actual) homology Hirzebruch class and the mock Hirzebruch class. We show that this difference is localized on the singular locus, and we obtain a formula for it in which the contribution of each singular cone is identified explicitly. Finally, the two methods of computing characteristic classes are combined for proving several characteristic class formulae originally obtained by Cappell and Shaneson in the early 1990s.© 2015 Wiley Periodicals, Inc.

Equivariant Characteristic Classes of Singular Complex Algebraic Varieties

Homology Hirzebruch characteristic classes for singular varieties have been recently defined by Brasselet, Schuurmann, and Yokura as an attempt to unify previously known characteristic class theories for singular spaces (e.g., MacPherson-Chern classes, Baum-Fulton-MacPherson Todd classes, and Goresky-MacPherson

Nearby cycles and characteristic classes of singular spaces

L

A general intersection formula for Lagrangian cycles

-classes). In this paper we define equivariant analogues of these classes for singular quasi-projective varieties acted upon by a finite group of algebraic automorphisms and show how these can be used to calculate the homology Hirzebruch classes of global quotient varieties. We also compute the new classes in the context of monodromy problems, e.g., for varieties that fiber equivariantly (in the complex topology) over a connected algebraic manifold. As another application, we discuss Atiyah-Meyer type formulae for twisted Hirzebruch classes of global orbifolds.

Lectures on Characteristic Classes of Constructible Functions

The purpose of this paper is applying minimality of hyperplane arrangements to local system cohomology groups. It is well known that twisted cohomology groups with coefficients in a generic rank one local system vanish except in the top degree, and bounded chambers form a basis of the remaining cohomology group. We determine precisely when this phenomenon happens for two-dimensional arrangements.Let f(z,¯) be a mixed polar homogeneous polynomial of n variables z = (z1, . . . , zn). It defines a projective real algebraic va- riety V := {(z) 2 CP n 1 | f(z,¯) = 0} in the projective space CP n 1 . The behavior is different from that of the projective hypersurface. The topology is not uniquely determined by the degree of the variety even if V is non-singular. We study a basic property of such a variety.In this paper we give an introduction to our recent work on characteristic classes of complex hypersurfaces based on some talks given at conferences in Strasbourg, Oberwolfach and Kagoshima. We explain the relation between nearby cycles for constructible functions or sheaves as well as for (relative) Grothendieck groups of algebraic varieties and mixed Hodge modules, and the specialization of characteristic classes of singular spaces like the Chern-, Todd-, Hirzebruch- and motivic Chern-classes. As an application we get a description of the differences between the corresponding virtual and functorial characteristic classes of complex hypersurfaces in terms of vanishing cycles related to the singularities of the hypersurface.

Spectral Hirzebruch–Milnor classes of singular hypersurfaces

We prove a generalization to the context of real geometry of an intersection formula for the vanishing cycle functor, which in the complex context is due to Dubson, Ginsburg, Le and Sabbah (after a conjecture of Deligne). It is also a generalization of similar results of Kashiwara-Schapira, where these authors work with a suitable assumption about the micro-support of the corresponding constructible complex of sheaves. We only use a similar assumption about the support of the corresponding characteristic cycle so that our result can be formulated in the language of constructible functions and Lagrangian cycles.

Hirzebruch–Milnor Classes and Steenbrink Spectra of Certain Projective Hypersurfaces

The following lectures were delivered at the Mini-School “Charac- teristic classes of singular varieties” in Banach Center, 23–27 April 2002, by Jorg Schurmann. These lectures discuss the calculus of characteristic classes associated with constructible functions on possibly singular varieties, and focus on the specialization properties. The point of view of characteristic classes of Lagrangian cycles is emphasized. A Verdier-type R.iemann-Roch theorem is discussed.1

Equivariant characteristic classes of external and symmetric products of varieties

We introduce spectral Hirzebruch–Milnor classes for singular hypersurfaces. These can be identified with Steenbrink spectra in the isolated singularity case, and may be viewed as their global analogues in general. Their definition uses vanishing cycles of mixed Hodge modules and the Todd class transformation. These are compatible with the pushforward by proper morphisms, and the classes can be calculated by using resolutions of singularities. Formulas for Hirzebruch–Milnor classes of projective hypersurfaces in terms of these classes are given in the case where the multiplicity of a generic hyperplane section is not 1. These formulas using hyperplane sections instead of hypersurface ones are easier to calculate in certain cases. Here we use the Thom–Sebastiani theorem for the underlying filtered D -modules of vanishing cycles, from which we can deduce the Thom–Sebastiani type theorem for spectral Hirzebruch–Milnor classes. For the Chern classes after specializing to