Projective Bundles and Blowing-Ups Ⅱ

We study the blowing-up X of a smooth projective variety X along a smooth center B that is equipped with a projective bundle structure over a variety Z.If the Picard number p(X)is 1 and dim X is at most 4,we classify all such pairs(X,B).If X is a projective space P_(n)(n≥5)and dim B is 2,we show that B is a linear subspace in X.

The non-abelian Hodge correspondence on some non-K?hler manifolds

The non-abelian Hodge correspondence was established by Corlette(1988),Donaldson(1987),Hit chin(1987)and Simpson(1988,1992).It states that on a compact Kahler manifold(X,ω),there is a one-to-one correspondence between the moduli space of semi-simple flat complex vector bundles and the moduli space of poly-stable Higgs bundles with vanishing Chern numbers.In this paper,we extend this correspondence to the projectively flat bundles over some non-Kahler manifold cases.Firstly,we prove an existence theorem of Poisson metrics on simple projectively flat bundles over compact Hermitian manifolds.As its application,we obtain a vanishing theorem of characteristic classes of projectively flat bundles.Secondly,on compact Hermitian manifolds which satisfy Gauduchon and astheno-K?hler conditions,we combine the continuity method and the heat flow method to prove that every semi-stable Higgs bundle withΔ(E,?E)·[ωn-2]=0 must be an extension of stable Higgs bundles.Using the above results,over some compact non-Kahler manifolds(M,ω),we establish an equivalence of categories between the category of semi-stable(poly-stable)Higgs bundles(E,?E,φ)withΔ(E,?E)·[ωn-2]=0 and the category of(semi-simple)projectively flat bundles(E,D)with(-1)(1/2)FD=α■IdE for some real(1,1)-formα.

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).

Commuting Involutions with Fixed Point Set of Constant Codimension

Special generators of the unoriented cobordism ring MO* are constructed to determine the groups Jn,kτ of n-dimensional cobordism classes in MOn containing a representative Mn admitting a （Z2）k -action with fixed point set of constant codimension.

Commuting Involutions with Fixed Point Set of Variable Codimension

Special generators of the unoriented cobordism ring MO* are constructed to determine some groups Jn,kl1,l2,…,lm of cobordism classes in MOn containing a representative Mn admitting a （Z2）k-action with the fixed point set of（n-li）-dimensional submanifolds of Mn.