相关论文: Adjacent Singularities, TQFTs, and Zariski's Multi…

We compare the sheaf-theoretic and singular chain versions of Poincare duality for intersection homology, showing that they are isomorphic via naturally defined maps. Similarly, we demonstrate the existence of canonical isomorphisms between…

We study exact orbifold fillings of contact manifolds using Floer theories. Motivated by Chen-Ruan's orbifold Gromov-Witten invariants, we define symplectic cohomology of an exact orbifold filling as a group using classical techniques, i.e.…

If $\bar\rho$ is an automorphic modulo $p$ Galois representation, it is natural to wonder if automorphic points are Zariski dense in the deformation space of $\bar\rho$. We prove new results in this direction in the case of a unitary group…

We compute the Floer homology of mapping classes which do not have any pseudo-Anosov components in the sense of Thurston's theory of surface diffeomorphisms. The formula for the Floer homology is obtained from a topological separation of…

We develop connections between the qualitative dynamics of Hamiltonian isotopies on a surface $\Sigma$ and their chain-level Floer theory using ideas drawn from Hofer-Wysocki-Zehnder's theory of finite energy foliations. We associate to…

In this paper we develope a Morsification Theory for holomorphic functions defining a singularity of finite codimension with respect to an ideal, which recovers most previously known Morsification results for non-isolated singulatities and…

One considers quasihomogeneous isolated singularities of hypersurfaces in arbitrary dimensions through the lenses of three apparently quite apart themes: syzygies, singularity invariants, and foliations. In the first of these, one adds to…

It is argued that singular vectors of the topological conformal (twisted $N=2$) algebra are identical with singular vectors of the $sl(2)$ Kac--Moody algebra. An arbitrary matter theory can be dressed by additional fields to make up a…

Over complex numbers, the Fourier-Mukai partners of abelian varieties are well-understood. A celebrated result is Orlov's derived Torelli theorem. In this note, we study the FM-partners of abelian varieties in positive characteristic. We…

A theorem of Mumford states that, on complex surfaces, any normal isolated singularity whose link is diffeomorphic to a sphere is actually a smooth point. While this property fails in higher dimensions, McLean asks whether the contact…

We provide numerical conditions for a polarized abelian threefold $(A,L)$ to have simple syzygies, in terms of property $(N_p)$ and the vanishing of Koszul cohomology groups $K_{p,1}$. We rely on a reduction method of…

In this note, we give a short proof of the Torelli theorem for cubic fourfolds that relies on the global Torelli theorem for irreducible holomorphic symplectic varieties proved by Verbitsky.

We consider morphisms $\pi: X \to \mathbb{P}^1$ of smooth projective varieties over $\mathbb{C}$. We show that if $\pi$ has at most one singular fibre, then $X$ is uniruled and $\pi$ admits sections. We reach the same conclusions, but with…

The ascent and descent of the Mittag-Leffler property were instrumental in proving Zariski locality of the notion of an (infinite dimensional) vector bundle by Raynaud and Gruson in \cite{RG}. More recently, relative Mittag-Leffler modules…

We prove that the affine closure of the cotangent bundle of the basic affine space of a complex semisimple group has conical symplectic singularities, which confirms a conjecture of Ginzburg and Kazhdan. We also show that this variety is…

In this paper, we prove the Shafarevich conjecture for certain complete intersections of hypersurfaces in abelian varieties defined over a number field $K$ using the Lawrence-Venkatesh method. The main new inputs we need are computation of…

In the early days of the Floer theory, Atiyah asked if there is a Milnor fiber description of the Floer homology of the links of singularities. We answer this question for the Brieskorn-Hamm complete intersection singularities. The…

We investigate the small area limit of the gauged Lagrangian Floer cohomology of Frauenfelder. The resulting cohomology theory, which we call quasimap Floer cohomology, is an obstruction to displaceability of Lagrangians in the symplectic…

We study the interaction between Fourier-Mukai transforms and perverse filtrations for a certain class of dualizable abelian fibrations. Multiplicativity of the perverse filtration and the "Perverse $\supset$ Chern" phenomenon for these…

This is the second of three papers motivated by the author's desire to understand and explain "algebraically" one aspect of Dmitriy Zhuk's proof of the CSP Dichotomy Theorem. In this paper we extend Zhuk's "bridge" construction to arbitrary…