In Kontsevich-Soibelman’s theory of motivic Donaldson-Thomas invariants for 3-dimensional noncommutative Calabi-Yau varieties, the integral identity conjecture plays a crucial role as it involves the existence of these ...

A theorem of Weil and Atiyah says that a holomorphic vector bundle $E$ on a compact Riemann surface $X$ admits a holomorphic connection if and only if the degree of every direct summand of $E$ is zero. Fix a finite subset ...

Assume that $M(\mathcal{T})$ is a rational homology sphere plumbed 3--manifold associated with a connected negative definite graph $\mathcal{T}$. We consider the combinatorial multivariable Poincar\'e series associated ...

In this article, we study the Hilbert scheme of effective divisors in smooth hypersurfaces in $\mathbb{P}^3$, a topic not extensively studied. We prove that there exists such effective divisors $D$ satisfying the property: ...

In [Kol13] Kollár defined $i$-th index of a proper scheme over a field. In this note we study how index behaves under specialization, in any characteristic.

Classically, there exists a determinant map from the moduli space of semi-stable sheaves on a smooth, projective variety to the Picard scheme. Unfortunately, if the underlying variety is singular, then such a map does not ...

[TBA]

Let $E_G$ be a holomorphic principal $G$--bundle on a compact connected Riemann surface $X$, where $G$ is a connected reductive complex affine algebraic group. Fix a finite subset $D\, \subset\, X$, and for each $x\,\in\, ...

In Kontsevich-Soibelman’s theory of motivic Donaldson-Thomas invariants for 3-dimensional noncommutative Calabi-Yau varieties, the integral identity conjecture plays a crucial role as it involves the existence of these ...

We explore some equisingularity criteria in one-parameter families of generically reduced curves. We prove the equivalence between Whitney regularity and Zariski’s discriminant criterion. We prove that topological triviality ...