arXiv subjects

Benjamin Tighe

Publications and source records attributed to Benjamin Tighe.

4 recordsLinked to original sources

arXiv2025

The LLV Algebra for Primitive Symplectic Varieties with Isolated Singularities

We extend results of Looijenga--Lunts and Verbitsky and show that the total Lie algebra $\mathfrak g$ for the intersection cohomology of a primitive symplectic variety $X$ with isolated singularities is isomorphic to $$\mathfrak g \cong \mathfrak{so}\left(\left(IH^2(X, \mathbb Q), Q_X\right)\oplus \mathfrak h\right),$$ where $Q_X$ is the intersection Beauville--Bogomolov--Fujiki form and $\mathfrak h$ is a hyperbolic plane. This gives a new, algebraic proof for irreducible holomorphic symplectic manifolds which does not rely on the hyperkähler metric. Along the way, we study the structure of $IH^*(X, \mathbb Q)$ as a $\mathfrak{g}$-representation -- with particular emphasis on the Verbitsky component, multidimensional Kuga--Satake constructions, and Mumford--Tate algebras -- and give some immediate applications concerning the $P = W$ conjecture for primitive symplectic varieties.

math.AG↗

arXiv2024

The Holomorphic Extension Property for Higher Du Bois Singularities

Let $X$ be a normal complex variety and $π:\tilde X \to X$ a resolution of singularities. We show that the inclusion morphism $π_*Ω_{\tilde X}^p\hookrightarrow Ω_X^{[p]}$ is an isomorphism for $p < \mathrm{codim}_X(X_{\mathrm{sing}})$ when $X$ has du Bois singularities, giving an improvement on Flenner's criterion for arbitrary singularities. We also study the $k$-du Bois definition from the perspective of holomorphic extension and compare how different restrictions on $\mathscr H^0(\underline Ω_X^p)$ affect the singularities of $X$, where $\underlineΩ_X^p$ is the $p^{th}$-graded piece of the du Bois complex.

math.AG↗

arXiv2024

Symmetries and vanishing theorems for symplectic varieties

We describe the local and Steenbrink vanishing problems for singular symplectic varieties with isolated singularities. We do this by constructing a morphism $$\mathbb D_X(\underline Ω_X^{n+p}) \to \underline Ω_X^{n+p}$$ for a symplectic variety $X$ of dimension $2n$ for $\frac{1}{2}\mathrm{codim}_X(X_{\mathrm{sing}}) < p$, where $\underline Ω_X^k$ is the $k^{th}$-graded piece of the Du Bois complex and $\mathbb D_X$ is the Grothendieck duality functor. We show this morphism is a quasi-isomorphism when $p = n-1$ and that this symmetry descends to the Hodge filtration on the intersection Hodge module. As applications, we describe the higher Du Bois and higher rational properties for symplectic germs and the cohomology of primitive symplectic 4-folds.

math.AG↗

arXiv2024

On an example of Aspinwall, Morrison, and Szendrői

We study the cohomology of a 1-parameter family Y_t of Calabi-Yau 3-folds introduced by Aspinwall and Morrison, related to the mirror quintic family. Szendrői proved that Y_t, Y_{xi t}, ..., Y_{xi^4 t}, where xi is a fifth root of unity, have the same rational Hodge structure but are not isomorphic, and conjectured that they are not birational or even derived equivalent. We confirm this by proving that their integral Hodge structures are different, and discuss how this fits with known Torelli-type theorems and counterexamples.

math.AG↗