Related papers: On a twisted de Rham complex, II

We study vanishing cycles naturally attached to a meromorphic function with isolated singularities, in both local and global settings.

We give an explicit construction of a deformation quantization of the algebra of functions on a Poisson manifolds, based on Kontsevich's local formula. The deformed algebra of functions is realized as the algebra of horizontal sections of a…

Let f be an isolated plane curve singularity with Milnor fiber of genus at least 5. For all such f, we give (a) an intrinsic description of the geometric monodromy group that does not invoke the notion of the versal deformation space, and…

In this paper we present some linear algebra behind quadratic parts of quadratically flat complex points of codimension two real submanifold in a complex manifold. Assuming some extra nondegenericity and using the result of Hong, complete…

This paper has two parts. We first survey recent efforts on the Bloom conjecture which still remains open in the case of complex dimension at least 4. Bloom's conjecture concerns the equivalence of three regular types. There is a more…

We prove (Theorem 2.4) that the symmetrized deleted join $SymmDelJoin(\mathcal{K})$ of a "balanced family" $\mathcal{K} = \langle K_i\rangle_{i=1}^r$ of collectively $r$-unavoidable subcomplexes of $2^{[m]}$ is $(m-r-1)$-connected. As a…

We define the appropriate homological setting to study deformation theory of complete locally convex (curved) dg-algebras based on Positselski's contraderived categories. We define the corresponding Hochschild complex controlling…

For an associative algebra A we consider the pair "the Hochschild cochain complex C*(A,A) and the algebra A". There is a natural 2-colored operad which acts on this pair. We show that this operad is quasi-isomorphic to the singular chain…

Voevodsky has conjectured that numerical and smash equivalence coincide on a smooth projective variety. We prove the conjecture for one dimensional cycles on an arbitrary product of curves. As a consequence we get that numerically trivial…

A new, simple method to approach enumerative questions about rational curves on rational surfaces is described. Applications include a short proof of Kontsevich's formula for plane curves and a the solution of the analogous problem for the…

This article is part of an ongoing investigation of the two-dimensional Jacobian conjecture. In the first paper of this series, we proved the generalized Magnus' formula. In this paper, inspired by cluster algebras, we introduce a sequence…

We introduce a notion of $n$-commutativity ($0\le n\le \infty$) for cosimplicial monoids in a symmetric monoidal category ${\bf V}$, where $n=0$ corresponds to just cosimplicial monoids in ${\bf V,}$ while $n=\infty$ corresponds to…

We express the Lefschetz number of iterates of the monodromy of a function on a smooth complex algebraic variety in terms of the Euler characteristic of a space of truncated arcs.

We reformulate a conjecture of Beauville on algebraic cycles on an abelian variety in terms of certain compatibility and vanishings of some naturally defined filtrations on the Grothendieck group of the abelian variety.

A quantitative version of the Oppenheim conjecture for inhomogeneous quadratic forms is proved. We also give an application to eigenvalue spacing on flat 2-tori with Aharonov-Bohm flux.

We present an $\ell$-adic trace formula for saturated and admissible dg-categories over a base monoidal dg-category. Moreover, we prove K\"unneth formulas for dg-category of singularities, and for inertia-invariant vanishing cycles. As an…

We conjecture an evaluation of three-partition cyclic Hodge integrals in terms of loop Schur functions. Our formula implies the orbifold Gromov-Witten/Donaldson-Thomas correspondence for toric Calabi-Yau threefolds with transverse type A…

The Kodaira-Nakano Vanishing Theorem has been generalized to the relative setting by A. Sommese. We prove a version of this theorem for non-compact manifolds. As an apllication, we prove that the cohomology of a fiber of a symplectic…

This text surveys cohomological properties of pairs $(U,f)$ consisting of a smooth complex quasi-projective variety $U$ together with a regular function on~it. On the one hand, one tries to mimic the case of a germ of holomorphic function…

A conjecture of Kotschick predicts that a compact K\"ahler manifold $X$ fibres smoothly over the circle if and only if it admits a holomorphic one-form without zeros. In this paper we develop an approach to this conjecture and verify it in…