分歧覆盖与等变凝聚层范畴

Focus on weighted projective lines and the categories of coherent sheaves on them, we study ramified coverings on projective varieties and the categories of equivariant coherent sheaves induced by group actions on projective varieties, including: study the corresponding relationships between weighted projective lines and some special ramified coverings on projective varieties, describe the relationships among weighted projective lines, and the relationships between weighted projective lines and some special projective varieties, and then extend the ideas to Geigle-Lenzing projective spaces; furthermore, study the ramified coverings which can be represented by the quotient maps from a topological space to the resulting quotient space by group action, then the categories and the derived categories of induced equivariant coherent sheaves; observe the existence of equivariant Serre duality；discuss the compatibility between equivariantizations and the constructions of Lie algebras; consider the tensor product on the category of equivariant coherent sheaves and determine whether the category of equivariant vector bundles become an abelian monoidal category. The above studies involve algebraic representation theory, algebraic geometry, algebraic number theory, Lie algebras, and several other study fields of algebras. The practice and completion of the research plans will further improve the theory of ramified coverings on projective varieties and invariant theory on the category of equivariant coherent sheaves, establish the link between weighted projective lines and projective varietieses, promote the study of singularity of surfaces.

以加权射影线及其凝聚层范畴作为对象，研究射影代数簇上的分歧覆盖及群作用在射影代数簇上诱导的等变凝聚层范畴，内容包括：研究加权射影线与射影代数簇上特殊分歧覆盖的对应，刻画加权射影线之间、加权射影线与一些特殊射影代数簇的关系，并将这一思路推广到Geigle-Lenzing射影空间；进一步地，研究可以表示为群作用在空间上的商映射的分歧覆盖及其诱导的等变凝聚层范畴、等变凝聚层导出范畴；考察等变范畴中Serre对偶的存在性；探讨等变化与李代数构造的协调；进行等变范畴的张量运算进而确定加权射影线上向量丛范畴的abelian monoidal性质。以上研究，涉及代数表示论、代数几何、代数数论及李代数等多个热门的代数学研究领域，研究计划的实施与完成将有助于进一步完善射影代数簇的分歧覆盖理论，凝聚层范畴等变过程的不变量理论，有助于建立加权射影线与射影代数簇的联系，有助于曲面奇异的研究。

本项目以加权射影线及其凝聚层范畴为对象，研究有限群作用及等变范畴，具体内容及成果包括：探讨加权射影线及其凝聚层范畴上的群作用，刻画了不同权型加权射影线之间的等变关系；引入范畴上自函子的Frobenius-Perron维数的概念，建立了加权射影线的凝聚层及其导出范畴上的Frobenius-Perron理论；介绍特殊的拟凝聚层及其性质，确定了加权射影线上特殊的拟凝聚层在凝聚层范畴的分类作用；研究加权射影线上向量丛范畴及其稳定范畴的结构，构造了向量丛稳定范畴的倾斜对象。研究成果的取得有助于建立加权射影线与射影代数簇的联系，有助于wild型加权射影线的分类研究，有助于进一步完善加权射影线上向量丛范畴及其稳定范畴的倾斜理论，丰富了群作用的不变量理论。