Search

This paper studies the virtual $\chi_{-y}$-genera of Grothendieck's Quot schemes on surfaces, thus refining the calculations of the virtual Euler characteristics by Oprea-Pandharipande. We first prove a structural result expressing the equivariant virtual $\chi_{-y}$-genera of Quot schemes universally in terms of the Seiberg-Witten invariants. The formula is simpler for curve classes of Seiberg-Witten length $N$, which are defined in the paper. By way of application, we give complete answers in the following cases: (i) arbitrary surfaces for the zero curve class, (ii) relatively minimal elliptic surfaces for rational multiples of the fiber class, (iii) minimal surfaces of general type with $p_g>0$ for any curve classes. Furthermore, a blow up formula is obtained for curve classes of Seiberg-Witten length $N$. As a result of these calculations, we prove that the generating series of the virtual $\chi_{-y}$-genera are given by rational functions for all surfaces with $p_g>0$, addressing a conjecture of Oprea-Pandharipande. In addition, we study the reduced $\chi_{-y}$-genera for $K3$ surfaces and primitive curve classes with connections to the Kawai-Yoshioka formula., Comment: 48 pages. Reformulation of Theorem 1 for readability. Extra explanation for the mixed terms in section 2.2.3. Reference updates

Given an iterated function system (IFS) of contractive similitudes, the theory of Gromov hyperbolic graph on the IFS has been established recently. In the paper, we introduce a notion of simple augmented tree which is a Gromov hyperbolic graph. By generalizing a combinatorial device of rearrangeable matrix, we show that there exists a near-isometry between the simple augmented tree and the symbolic space of the IFS, so that their hyperbolic boundaries are Lipschitz equivalent. We then apply this to consider the Lipschitz equivalence of self-similar sets with or without assuming the open set condition. Moreover, we also provide a criterion for a self-similar set to be a Cantor-type set which completely answers an open question raised in \cite{LaLu13}. Our study extends the previous works.

We investigate suitable conditions for a C0-semigroup (T(t))t≥0 of Hilbert space contractions to be unitarily equivalent to the restriction of the adjoint shift semigroup (S∗γ(t))t≥0 to an invariant subspace of the standard weighted Bergman space Aγ−2(C+,K). It turns out that (T(t))t≥0 admits a model by (S∗γ(t))t≥0 if and only if its cogenerator is γ-hypercontractive and limt→0T(t)=0 in strong operator topology. We then discuss how such semigroups can be characterized without involving the cogenerator. A sufficient condition is that, for each t>0, the operator T(t) is γ-hypercontractive. Surprisingly, this condition is necessary if and only if γ is integer. The paper is concluded with a conjecture that would imply a more symmetric characterization. (Less)

If f is in the Eremenko–Lyubich class B (transcendental entire functions with bounded singular set), then Ω = {z : |f (z)| >R } and f|Ω must satisfy certain simple topological conditions when R is sufficiently large. A model (Ω ,F ) is an open set Ω and a holomorphic function F on Ω that satisfy these same conditions. We show that any model can be approximated by an Eremenko– Lyubich function in a precise sense. In many cases, this allows the construction of functions in B with a desired property to be reduced to the construction of a model with that property, and this is often much easier to do. The singular set of a entire function f is the closure of its critical values and finite asymptotic values and is denoted by S(f ). The Eremenko–Lyubich class B consists of functions such that S(f ) is a bounded set (such functions are also called bounded type). The Speiser class S⊂ B (also called finite type) is those functions for which S(f ) is a finite set. In [9], Eremenko and Lyubich showed that if S(f ) ⊂ DR = {z : |z| R } is a disjoint union of analytic, unbounded, Jordan domains, and that f acts as a covering map f :Ω j →{ x : |z| >R } on each component Ωj of Ω. Building examples where Ω has a certain geometry is important for applications to dynamics. We would like to start with a model, that is, a choice of Ω and a covering map f :Ω →{ |z| > 1} and ask whether f can be approximated by an entire function F in B or S. In this paper, we deal with approximation by functions in B. It turns out that if Ω satisfies some obviously necessary topological conditions, then the approximation by Eremenko–Lyubich functions is always possible in a sense strong enough to imply that the functions f and F have the same dynamical behavior on their Julia sets. This allows us to build entire functions in B with certain behaviors by simply exhibiting a model with that behavior (this is often much easier to do). In [4], we deal with the analogous question for the Speiser class; again the approximation is always possible, but in a slightly weaker sense (dynamically, given any model we can build a function in the Speiser class that has the model’s dynamics on some subset of its Julia set). In the next few paragraphs, we introduce some notation to make these remarks more precise. Suppose that Ω = � j Ωj is a disjoint union of unbounded simply connected domains

A well known theorem of Shuzo Izumi, strengthened by David Rees, asserts that all the divisorial valuations centered in an analytically irreducible local noetherian ring are linearly comparable to each other. In the present paper we generalize this theorem to the case of Abhyankar valuations with archimedian value semigroup. Indeed, we prove that in a certain sense linear equivalence of topologies characterizes Abhyankar valuations with archimedian semigroups, centered in analytically irreducible local noetherian rings. Then we show that some of the classical results on equivalence of topologies in noetherian rings can be strengthened to include linear equivalence of topologies. We also prove a new comparison result between the Krull topology and the topology defined by the symbolic powers of an arbitrary ideal.