Bi-Frobenius Algebra Structures on Quantum Complete Intersections

This paper is to look for bi-Frobenius algebra structures on quantum complete intersections over field k.We find a class of comultiplications,such that if√−1∈k,then a quantum complete intersection becomes a bi-Frobenius algebra with comultiplication of this form if and only if all the parameters qij=±1.Also,it is proved that if√−1∈k then a quantum exterior algebra in two variables admits a bi-Frobenius algebra structure if and only if the parameter q=±√1.While if−1/∈k,then the exterior algebra with two variables admits no bi-Frobenius algebra structures.We prove that the quantum complete intersections admit a bialgebra structure if and only if it admits a Hopf algebra structure,if and only if it is commutative,the characteristic of k is a prime p,and every ai a power of p.This also provides a large class of examples of bi-Frobenius algebras which are not bialgebras(and hence not Hopf algebras).In commutative case,other two comultiplications on complete intersection rings are given,such that they admit non-isomorphic bi-Frobenius algebra structures.

Elliptic genera of level N for complete intersections

We focus on the elliptic genera of level N at the cusps of a congruence subgroup for any complete intersection.Writing the first Chern class of a complete intersection as a product of an integral coefficient c1 and a generator of the 2nd integral cohomology group,we mainly discuss the values of the elliptic genera of level N for the complete intersection in the cases of c_(1)>,=,or<0.In particular,the values about the Todd genus,Â-genus,and A_(k)-genus can be derived from the elliptic genera of level N.

Diffeomorphism Classification of Smooth Weighted Complete Intersections

Xn（d1,...,dr-1,dr.;w） and Xn（e1,...,er-1,dr;w） are two complex odd-dimensional smooth weighted complete intersections defined in a smooth weighted hypersurfaee

Geometrical Realization of Low-Dimensional Complete Intersections

This paper aims to give some examples of diffeomorphic(or homeomorphic) lowdimensional complete intersections, which can be considered as a geometrical realization of classification theorems about complete intersections. A conjecture of Libgober and Wood(1982) will be confirmed by one of the examples.

Complete Intersection Monomial Curves and the Cohen-Macaulayness of Their Tangent Cones

Let C(n)be a complete intersection monomial curve in the 4-dimensional affine space.In this paper we study the complete intersection property of the monomial curve C(n+wv),where w>0 is an integer and v ∈ N^4.In addition,we investigate the Cohen-Macaulayness of the tangent cone of C(n+wv).

POLES OF ZETA FUNCTIONS OF COMPLETE INTERSECTIONS

A vanishing theorem is proved for e-adic cohomology with compact support on an affine (singular) complete intersection. As an application, it is shown that for an affine complete intersection defined over a finite field of q elements, the reciprocal "poles" of the zeta function are always divisible by q as algebraic integers. A p-adic proof is also given, which leads to further q-divisibiliy of the poles or equivalently an improvement of the polar part of the AxKatz theorem for an affine complete intersection. Similar results hold for a projective complete intersection.

Finite Indeterminacy of Homogenous Polynomial Germs under Some Subgroups R_(I_r) of R

Using the finite determinacy relation with the regular sequence in the Ring Theory and the complete intersection in Analytic Geometry, the finite indeterminacy of homogeneous polynomial germs under some subgroups R1（r） of R in both real and complex case is proven by the homogeneity of the polynomial germs. It results in the finite determinacy of homogeneous polynomial germs needn＇t be discussed respectively.

Symplectic Conditions on Grassmannian,Flag,and Schubert Varieties

In this paper,a description of the set-theoretical defining equations of symplectic(type C)Grassmannian/flag/Schubert varieties in corresponding(type A)algebraic varieties is given as linear polynomials in Plucker coordinates,and it is proved that such equations generate the defining ideal of variety of type C in those of type A.As applications of this result,the number of local equations required to obtain the Schubert variety of type C from the Schubert variety of type A is computed,and further geometric properties of the Schubert variety of type C are given in the aspect of complete intersections.Finally,the smoothness of Schubert variety in the non-minuscule or cominuscule Grassmannian of type C is discussed,filling gaps in the study of algebraic varieties of the same type.

Orbifold Gromov-Witten theory of weighted blowups

Consider a compact symplectic sub-orbifold groupoid S of a compact symplectic orbifold groupoid(X,ω).LetXabe the weight-a blowup of X along S,and Da=PNa be the exceptional divisor,where N is the normal bundle of S in X.In this paper we show that the absolute orbifold Gromov-Witten theory ofXacan be effectively and uniquely reconstructed from the absolute orbifold Gromov-Witten theories of X,S and Da,the natural restriction homomorphism HCR^*(X)→HCR*(S)and the first Chern class of the tautological line bundle over DQ.To achieve this we first prove similar results for the relative orbifold Gromov-Witten theories of(Xa|Da)and(Na|Da).As applications of these results,we prove an orbifold version of a conjecture of Maulik and Pandharipande(Topology,2006)on the Gromov-Witten theory of blowups along complete intersections,a conjecture on the Gromov-Witten theory of root constructions and a conjecture on the Leray-Hirsch result for the orbifold Gromov-Witten theory of Tseng and You(J Pure Appl Algebra,2016).