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Chengxi Wang

Publications and source records attributed to Chengxi Wang.

23 records · Page 2Linked to original sources
arXiv2019

On strong exceptional collections of line bundles of maximal length on Fano toric Deligne-Mumford stacks

We study strong exceptional collections of line bundles on Fano toric Deligne-Mumford stacks $\mathbb{P}_{\mathbfΣ}$ with rank of Picard group at most two. We prove that any strong exceptional collection of line bundles generates the derived category of $\mathbb{P}_{\mathbfΣ}$, as long as the number of elements in the collection equals the rank of the (Grothendieck) $K$-theory group of $\mathbb{P}_{\mathbfΣ}$.

math.AG↗
arXiv2019

Positivity determines the quantum cohomology of Grassmannians

We prove that if X is a Grassmannian of type A, then the Schubert basis of the (small) quantum cohomology ring QH(X) is the only homogeneous deformation of the Schubert basis of the ordinary cohomology ring of X that multiplies with non-negative structure constants. This implies that the (three point, genus zero) Gromov-Witten invariants of X are uniquely determined by Witten's presentation of QH(X) and the fact that they are non-negative. We conjecture that the same is true for any flag variety X = G/P of simply laced Lie type. For the variety GL(n)/B of complete flags, this conjecture is equivalent to Fomin, Gelfand, and Postnikov's conjecture that the quantum Schubert polynomials of type A are uniquely determined by positivity properties. Our proof for Grassmannians answers a question of Fulton.

math.AG↗
arXiv2018

On stringy Euler characteristics of Clifford non-commutative varieties

It was shown by Kuznetsov that complete intersections of $n$ generic quadrics in ${\mathbb P}^{2n-1}$ are related by Homological Projective Duality to certain non-commutative (Clifford) varieties which are in some sense birational to double covers of ${\mathbb P}^{n-1}$ ramified over symmetric determinantal hypersurfaces. Mirror symmetry predicts that the Hodge numbers of the complete intersections of quadrics must coincide with the appropriately defined Hodge numbers of these double covers. We observe that these numbers must be different from the well-known Batyrev's stringy Hodge numbers, else the equality fails already at the level of Euler characteristics. We define a natural modification of stringy Hodge numbers for the particular class of Clifford varieties, and prove the corresponding equality of Euler characteristics in arbitrary dimension.

math.AG↗
arXiv2015

Finitistic dimension conjecture and radical-power extensions

The finitistic dimension conjecture asserts that any finite-dimensional algebra over a field should have finite finitistic dimension. Recently, this conjecture is reduced to studying finitistic dimensions for extensions of algebras. In this paper, we investigate those extensions of Artin algebras in which some radical-power of smaller algebras is a one-sided ideal in bigger algebras. Our results, however, are formulated more generally for an arbitrary ideal: Let $B\subseteq A$ be an extension of Artin algebras and $I$ an ideal of $B$ such that the full subcategory of $B/I$-modules is $B$-syzygy-finite. Then: (1) If the extension is right-bounded (for example, proj.dim$(A_B)$ is finite), $I A\, rad(B)\subseteq B$ and fin.dim$(A)$ is finite, then fin.dim$(B)$ is finite. (2) If $I\, rad(B)$ is a left ideal of $A$ and $A$ is torsionless-finite, then fin.dim$(B)$ is finite. Particularly, if $I$ is specified to a power of the radical of $B$, then our results not only generalize some ones in the literature (see Corollaries 1.3 and 1.4), but also provide some completely new ways to detect algebras of finite finitistic dimensions.

math.RT↗