Quantum cohomology of the Hilbert scheme of points in the plane

We determine the ring structure of the equivariant quantum cohomology of the Hilbert scheme of points of ℂ². The operator of quantum multiplication by the divisor class is a nonstationary deformation of the quantum Calogero-Sutherland many-body system. A relationship between the quantum cohomology of the Hilbert scheme and the Gromov-Witten/Donaldson-Thomas correspondence for local curves is proven.     

The Caporaso–Harris formula and plane relative Gromov-Witten invariants in tropical geometry

Some years ago Caporaso and Harris have found a nice way to compute the numbers N(d, g) of complex plane curves of degree d and genus g through 3d + g − 1 general points with the help of relative Gromov-Witten invariants. Recently, Mikhalkin has found a way to reinterpret the numbers N(d, g) in terms of tropical geometry and to compute them by counting certain lattice paths in integral polytopes. We relate these two results by defining an analogue of the relative Gromov-Witten invariants and rederiving the Caporaso–Harris formula in terms of both tropical geometry and lattice paths. H. Markwig has been funded by the DFG grant Ga 636/2.     

BPS invariants for resolutions of polyhedral singularities

We study the BPS invariants of the preferred Calabi–Yau resolution of ADE polyhedral singularities ${\mathbb{C}^3/G}$ given by Nakamura’s G-Hilbert schemes. Genus 0 BPS invariants are defined by means of the moduli space of torsion sheaves as proposed by Katz (J Differ Geom 79(2):185–195, 2008). We show that these invariants are equal to half the number of certain positive roots of an ADE root system associated to G. This is in agreement with the prediction given in Bryan and Gholampour (Invent Math, in press) via Gromov–Witten theory.     

On groups of central type,non-degenerate and bijective cohomology classes

A finite group G is of central type (in the non-classical sense) if it admits a non-degenerate cohomology class [c] ∈ H ²(G, ℂ*) (G acts trivially on ℂ*). Groups of central type play a fundamental role in the classification of semisimple triangular complex Hopf algebras and can be determined by their representation-theoretical properties. Suppose that a finite group Q acts on an abelian group A so that there exists a bijective 1-cocycle π ∈ Z ¹(Q,Ǎ), where Ǎ = Hom(A, ℂ*) is endowed with the diagonal Q-action. Under this assumption, Etingof and Gelaki gave an explicit formula for a non-degenerate 2-cocycle in Z ²(G, ℂ*), where G:= A × Q. Hence, the semidirect product G is of central type. In this paper, we present a more general correspondence between bijective and non-degenerate cohomology classes. In particular, given a bijective class [π] ∈ H ¹(Q,Ǎ) as above, we construct non-degenerate classes [cπ] ∈ H ²(G,ℂ*) for certain extensions 1 → A → G → Q → 1 which are not necessarily split. We thus strictly extend the above family of central type groups.     

Weighted Bergman spaces¶associated with causal symmetric spaces

Let H\G be a causal symmetric space sitting inside its complexification H _ℂ\G _ℂ. Then there exist certain G-invariant Stein subdomains Ξ of H _ℂ\G _ℂ. The Haar measure on H _ℂ\G _ℂ gives rise to a G-invariant measure on Ξ. With respect to this measure one can define the Bergman space B ²(Ξ) of square integrable holomorphic functions on Ξ. The group G acts unitarily on the Hilbert space B ²(Ξ) by left translations in the arguments. The main result of this paper is the Plancherel Theorem for B ²(Ξ), i.e., the disintegration formula for the left regular representation into irreducibles. Received: Received: 23 November 1998     

Linear bounds for levels of stable rationality

Let G be one of the groups SL _n (ℂ), Sp_2n (ℂ), SO _m (ℂ), O _m (ℂ), or G ₂. For a generically free G-representation V, we say that N is a level of stable rationality for V/G if V/G × ℙ ^N is rational. In this paper we improve known bounds for the levels of stable rationality for the quotients V/G. In particular, their growth as functions of the rank of the group is linear for G being one of the classical groups.     

Relative Gromov-Witten invariants and the mirror formula

Let X be a smooth complex projective variety, and let be a smooth very ample hypersurface such that is nef. Using the technique of relative Gromov-Witten invariants, we give a new short and geometric proof of (a version of) the “mirror formula”, i.e. we show that the generating function of the genus zero 1-point Gromov-Witten invariants of Y can be obtained from that of X by a certain change of variables (the so-called “mirror transformation”). Moreover, we use the same techniques to give a similar expression for the (virtual) numbers of degree-d plane rational curves meeting a smooth cubic at one point with multiplicity 3d, which play a role in local mirror symmetry. Received: 11 July 2001 / Published online: 4 February 2003 Funded by the DFG scholarships Ga 636/1–1 and Ga 636/1–2.     

Crystallography and Riemann surfaces

The level set of an elliptic function is a doubly periodic point set in ℂ. To obtain a wider spectrum of point sets, we consider, more generally, a Riemann surface S immersed in ℂ² and its sections (“cuts”) by ℂ More specifically, we consider surfaces S defined in terms of a fundamental surface element obtained as a conformai map of triangular domains in ℂ. The discrete group of isometries of ℂ² generated by reflections in the triangle edges leaves S invariant and generalizes double-periodicity. Our main result concerns the special case of maps of right triangles, with the right angle being a regular point of the map. For this class of maps we show that only seven Riemann surfaces, when cut, form point sets that are discrete in ℂ. Their isometry groups all have a rank 4 lattice subgroup, but only three of the corresponding point sets are doubly periodic in ℂ. The remaining surfaces form quasiperiodic point sets closely related to the vertex sets of quasiperiodic tilings. In fact, vertex sets of familiar tilings are recovered in all cases by applying the construction to a piecewise flat approximation of the corresponding Riemann surface. The geometry of point sets formed by cuts of Riemann surfaces is no less “rigid” than the geometry determined by a tiling, and has the distinct advantage in having a regular behavior with respect to the complex parameter which specifies the cut.     

Confined subgroups in periodic simple finitary linear groups

A subgroupX of the locally finite groupG is said to beconfined, if there exists a finite subgroupF≤G such thatX ^g∩F≠1 for allg∈G. Since there seems to be a certain correspondence between proper confined subgroups inG and non-trivial ideals in the complex group algebra ℂG, we determine the confined subgroups of periodic simple finitary linear groups in this paper. Dedicated to the memory of our friend and collaborator Richard E. Phillips     

Explicit expression of Cartan’s connection for Levi-nondegenerate 3-manifolds in complex surfaces,and identification of the Heisenberg sphere

We study effectively the Cartan geometry of Levi-nondegenerate C ⁶-smooth hypersurfaces M ³ in ℂ². Notably, we present the so-called curvature function of a related Tanaka-type normal connection explicitly in terms of a graphing function for M, which is the initial, single available datum. Vanishing of this curvature function then characterizes explicitly the local biholomorphic equivalence of such M ³ ⊂ ℂ² to the Heisenberg sphere ℍ³, such M’s being necessarily real analytic.     

Characterizations of G-tube domains

Let G be a real connected Lie group for which the universal complexification G ^ℂ has a polar decomposition G ^ℂ≅G exp(i?), where ? denotes the Lie algebra of G. The present paper is concerned with Riemann G-domains over the complex group G ^ℂ viewed as a G-manifold via the left multiplication. Such a Riemann domain X is said to be of Reinhardt type if G contains a discrete cocompact subgroup $\Gamma$ for whichG/Γ is a Stein manifold. Here the following is proved: Every Riemann G-domain of Reinhardt type is schlicht, hence a G-tube domain, i.e., a G-invariant subdomain of G ^ℂ. As an application one obtains conditions for a holomorphically separable G-manifold to be a G-tube domain. Received: 22 October 1998     

The augmentation ideal in group near-rings

The definition of the group near-ring R[G] of the near-ring R over the group G as a near-ring of mappings from R ^(G) to itself is due to Le Riche et al. (Arch Math 52:132–139, 1989). In this paper we consider the augmentation ideal Δ of R[G]. If the exponent of G is not 2, then the structure of ΔR ^(G) is determined in terms of commutators and distributors. This is then used to show that Δ is nilpotent if and only if R is weakly distributive, has characteristic p ⁿ for some prime p and G is a finite p-group for the same prime p.      

Bichromatic and Equichromatic Lines in ℂ2 and ℝ2

Let G and R each be a finite set of green and red points, respectively, such that |G|=n, |R|=n−k, G∩R=∅, and the points of G∪R are not all collinear. Let t be the total number of lines determined by G∪R. The number of equichromatic lines (a subset of bichromatic) is at least (t+2n+3−k(k+1))/4. A slightly weaker lower bound exists for bichromatic lines determined by points in ℂ². For sufficiently large point sets, a proof of a conjecture by Kleitman and Pinchasi is provided. A lower bound of (2t+14n−k(3k+7))/14 is demonstrated for bichromatic lines passing through at most six points. Lower bounds are also established for equichromatic lines passing through at most four, five, or six points.     

Prime ideals in crossed products of finite groups

LetR * G be a crossed product of the finite groupG over the ringR. In this paper we discuss the relationship between the prime ideals ofR*G and theG-prime ideals ofR. In particular, we show that Incomparability and Going Down hold in this situation. In the course of the proof, we actually completely describe all the prime idealsP ofR*G such thatP∩R is a fixedG-prime ideal ofR. As an application, we prove that ifG is a finite group of automorphisms ofR, then the prime (primitive) ranks ofR and of the fixed ringR ^G are equal provided •G•⁻∈R. In an appendix, we extend some of these 3 results to crossed products of the infinite cyclic group.     

The signature of generalized flag manifolds and applications

Let G be a linear algebraic group over C and P be a parabolic subgroup. We determine the signatures of the flag manifold G/P. As an application, we prove that the nonsingular hypersurfaces of degree 2 in CP^n are prime if n satisfies certain conditions.     

Isomorphisms preserving invariants

Let V and W be finite dimensional real vector spaces and let G ì GL(V){G \subset {\rm GL}(V)} and H ì GL(W){H \subset {\rm GL}(W)} be finite subgroups. Assume for simplicity that the actions contain no reflections. Let Y and Z denote the real algebraic varieties corresponding to \mathbbR[V]^G{\mathbb{R}[V]^G} and \mathbbR[W]^H{\mathbb{R}[W]^H}, respectively. If V and W are quasi-isomorphic, i.e., if there is a linear isomorphism L : V → W such that L sends G-orbits to H-orbits and L ⁻¹ sends H-orbits to G-orbits, then L induces an isomorphism of Y and Z. Conversely, suppose that f : Y → Z is a germ of a diffeomorphism sending the origin of Y to the origin of Z. Then we show that V and W are quasi-isomorphic, This result is closely related to a theorem of Strub [8], for which we give a new proof. We also give a new proof of a result of Kriegl et al. [3] on lifting of biholomorphisms of quotient spaces.     

On the algebraic independence in the selberg class

We prove that a functionF of the Selberg class ℐ is ab-th power in ℐ, i.e.,F=H ^b for someHσ ℐ, if and only ifb divides the order of every zero ofF and of everyp-componentF _p. This implies that the equationF ^a=G^b with (a, b)=1 has the unique solutionF=H ^b andG=H ^a in ℐ. As a consequence, we prove that ifF andG are distinct primitive elements of ℐ, then the transcendence degree of ℂ[F,G] over ℂ is two.     

Realisierungen des idealen Randes einer Riemannschen Fläche unter konformen Abschließungen

Let a noncompact Riemann surface R of positive finite genus g be given. If f : R → R′ is a conformal mapping of R into a compact Riemann surface R′ of genus g, we have a realization of the ideal boundary of R on the surface R′. We consider (for the fixed R) all the possible R′ and the associated conformal mappings, and study how large the realized boundary can be. To this aim we pass to the (common) universal space ℂ ^g of the Jacobi variety of any R′ and show that the image sets of the ideal boundary of R in ℂ ^g are uniformly bounded.

     

Domains with Prescribed Automorphism Group

Let G be the automorphism group of a bounded strictly pseudoconvex domain D⊂ℂ ^N with a smooth ( C^￥\mathcal{C}^{\infty} ) boundary. Let H be a closed subgroup of G. Pertaining to the question whether it is possible to realize H as the automorphism group of a strictly pseudoconvex domain D′ which is an arbitrarily small perturbation of D in C^￥\mathcal{C}^{\infty} topology, we give a partial answer by describing sufficient conditions for D and G.     

Isoperimetric profile comparisons and Yamabe constants

We estimate from below the isoperimetric profile of S² ×\mathbb R²{S^2 \times {\mathbb R}^2} and use this information to obtain lower bounds for the Yamabe constant of S² ×\mathbb R²{S^2 \times {\mathbb R}^2} . This provides a lower bound for the Yamabe invariants of products S ² ×  M ² for any closed Riemann surface M. Explicitly we show that Y (S ² ×  M ²) >  (2/3)Y(S ⁴).