Character Sheaves for Classical Graded Lie Algebras

In this note we study character sheaves for graded Lie algebras arising from inner automorphisms of special linear groups and Vinberg’s type Ⅱ classical graded Lie algebras.

Nadel-type multiplier ideal sheaves on complex spaces with singularities

In this article,we introduce multiplier ideal sheaves on complex spaces with singularities(not necessarily normal)via Ohsawa’s extension measure,as a special case of which,it turns out to be the socalled Mather-Jacobian multiplier ideals in the algebro-geometric setting.Inspired by Nadel’s coherence and Guan-Zhou’s strong openness properties of the multiplier ideal sheaves,we discuss similar properties for the generalized multiplier ideal sheaves.As applications,we obtain a reasonable generalization of(algebraic)adjoint ideal sheaves to the analytic setting and establish some extension theorems on K?hler manifolds from singular hypersurfaces.Relying on our multiplier and adjoint ideals,we also give characterizations for several important classes of singularities of pairs associated with plurisubharmonic functions.

Sheaves on non-reduced curves in a projective surface

Sheaves on non-reduced curves can appear in moduli spaces of 1-dimensional semistable sheaves over a surface and moduli spaces of Higgs bundles as well.We estimate the dimension of the stack M_(X)(nC,χ)of pure sheaves supported at the non-reduced curve nC(n≥2)with C an integral curve on X.We prove that the Hilbert-Chow morphism h_(L,χ):M_(X)^(H)(L,χ)→|L|sending each semistable 1-dimensional sheaf to its support has all its fibers of the same dimension for X Fano or with the trivial canonical line bundle and|L|contains integral curves.

Wear Detection System for Elevator Traction Sheave

In order to achieve accurate non-contact measurement of the mechanical wear of the elevator traction sheave grooves,a wear detection system based on machine vision was developed.The industrial camera was fixed through a special fixture,and the images were collected by aligning each groove.In this paper,target groove is extracted based on normalized correlation coefficient matching.Corner points are extracted to describe the contour of the traction wheel groove.The inflection point of the wheel groove boundary is determined by straight boundary fitting,and the amount of rope groove wear is calculated by geometric knowledge.Based on the physical model structure,a mathematical model is established to eliminate the unavoidable occlusion error.The experimental results show that this system can carry out an accurate quantitative analysis and realize an accurate measurement of the wear of the traction sheave rope groove with convenience and accuracy.

Seattering Diagrams,Sheaves,and Curves

We review the recent proof of the N.Takahashi's conjecture on genus 0 Gromov Witten invariants of(P^(2),E),where E is a smooth cubic curve in the complex projective plane P^(2).The main idea is the use of the algebraic notion of scattering diagram as a bridge between the world of Gromov-Witten invariants of(P^(2),E)and the world of moduli spaces of coherent sheaves on P^(2).Using this bridge,the N.Takahashi's conjecture can be translated into a manageable question about moduli spaces of coherent sheaves onP^(2)This survey is based on a three hours lecture series given as part of the Beijing-Zurich moduli workshop in Beijing,9-12 September 2019.

Derived Categories in Langlands Geometrical Ramifications: Approaching by Penrose Transforms

Some derived categories and their deformed versions are used to develop a theory of the ramifications of field studied in the geometrical Langlands program to obtain the correspondences between moduli stacks and solution classes represented cohomologically under the study of the kernels of the differential operators studied in their classification of the corresponding field equations. The corresponding D-modules in this case may be viewed as sheaves of conformal blocks (or co-invariants) (images under a version of the Penrose transform) naturally arising in the framework of conformal field theory. Inside the geometrical Langlands correspondence and in their cohomological context of strings can be established a framework of the space-time through the different versions of the Penrose transforms and their relation between them by intertwining operators (integral transforms that are isomorphisms between cohomological spaces of orbital spaces of the space-time), obtaining the functors that give equivalences of their corresponding categories.(For more information,please refer to the PDF version.)

Bilinear forms with trace functions over arbitrary sets and applications to Sato-Tate

We prove non-trivial upper bounds for general bilinear forms with trace functions of bountiful sheaves,where the supports of two variables can be arbitrary subsets in F_(p) of suitable sizes.This essentially recovers the Polya-Vinogradov range,and also applies to symmetric powers of Kloosterman sums and Frobenius traces of elliptic curves.In the case of hyper-Kloosterman sums,we can beat the Pólya-Vinogradov barrier by combining additive combinatorics with a deep result of Kowalski,Michel and Sawin(2017) on sum-products of Kloosterman sheaves.Two Sato-Tate distributions of Kloosterman sums and Frobenius traces of elliptic curves in sparse families are also concluded.

Langton's Type Theorem on Algebraic Orbifolds

In this paper,we prove the Langton’s type theorem on separatedness and properness for the moduli functor of torsion free semistable sheaves on algebraic orbifolds over an algebraically closed field k.

The Coherent-Constructible Correspondence and Fourier-Mukai Transforms

As evidence for his conjecture in birational log geometry, Kawamata constructed a family of derived equivalences between toric orbifolds. In a previous paper, the authors showed that the derived category of a toric orbifold is naturally identified with a category of polyhedrally-constructible sheaves on R^n. In this paper we investigate and reprove some of Kawamata＇s results from this perspective.

Factorization of Generalized Theta Functions Revisited

This survey is based on my lectures given in the last few years. As a reference, constructions of moduli spaces of parabolic sheaves and generalized parabolic sheaves are provided. By a refinement of the proof of vanishing theorems, we show, without using vanishing theorems, a new observation that dim H0（Uc, θuc） is independent of all of the choices for any smooth curves. The estimate of various codimensions and computation of canonical line bundle of moduli space of generalized parabolic sheaves on a reducible curve are provided in Section 6, which is completely new.

A finite dimensional proof of the Verlinde formula

We prove two recurrence relations among dimensions Dg(r,d,ω):=dim H^0(UC,ω,ΘUC,ω)of spaces of generalized theta functions on the moduli spaces UC,ω.By using these recurrence relations,an explicit formula(the Verlinde formula)of Dg(r,d,ω)is proved(see Theorem 4.3).

New procedure to generate multipliers in complex Neumann problem and effective Kohn algorithm

The purpose of this paper is threefold.(i) To explain the effective Kohn algorithm for multipliers in the complex Neumann problem and its difference with the full-real-radical Kohn algorithm, especially in the context of an example of Catlin-D'Angelo concerning the ineffectiveness of the latter.(ii) To extend the techniques of multiplier ideal sheaves for the complex Neumann problem to general systems of partial differential equations.(iii) To present a new procedure of generation of multipliers in the complex Neumann problem as a special case of the multiplier ideal sheaves techniques for general systems of partial differential equations.

Tautological Integrals on Symmetric Products of Curves

We propose a conjecture on the generating series of Chern numbers of tautological bundles on symmetric products of curves and establish the rank 1 and rank -1 case of this conjecture. Thus we compute explicitly the generating series of integrals of Segre classes of tautological bundles of line bundles on curves, which has a similar structure as Lehn＇s conjecture for surfaces.

Generalized Canonical Isomorphisms on Determinant

In this paper we define tensor modules （sheaves） of Schur type and of generalized Schur type associated with given modules （sheaves）, using the so-called Schur functors. According to the functorial property, we give a series of tensor modules （sheaves） of Schur types in a categorical description. The main conclusion is that, by using basic ideas of algebraic geometry, there exists a canonical isomorphism of different tensor modules （sheaves） of Schur types if the original sheaf is locally free, which is in fact a generalization of results in linear algebra into locally free sheaves.