Geometric Construction of Quiver Tensor Products

TL;DR

Beilinson’s quiver presentation of Perf$(\mathbb{P}^n)$ induces a vertexwise monoidal structure, which the authors reinterpret geometrically as an extended convolution with a Fourier–Mukai kernel supported on the torus-graph closure in $(\mathbb{P}^n)^3$. They develop a general EC framework that yields symmetric monoidal structures on Perf of smooth complete toric varieties (Bondal–Ruan type) and on projectivizations of finite-dimensional algebras, via windows, Hitchcock functors, and six-functor formalisms for derived stacks. The central technical achievement is a rigorous $\,\mathbb{E}_n$-monoidal equivalence between QC-EC constructions and quiver tensor products, grounding EC products in explicit kernel descriptions and diagonal resolutions. Under toric mirror symmetry, the EC products correspond to tensor products of constructible sheaves on real tori, providing a bridge between algebraic geometry and constructible–sheaf theory, and suggesting a moduli-theoretic view of monoidal structures on $\mathrm{Perf}(\mathbb{P}^n)$. The framework also yields invariants (Balmer spectrum, Grothendieck ring, Picard group) for EC monoidal structures and demonstrates a reconstruction principle for finite-dimensional algebras from their EC monoidal categories, with implications for categorical compactifications of classical groups. Finally, the paper outlines conjectural Cox-category pictures tying these constructions to birationally glued EC products and the broader Hochschild–Duval–Bondal–Ruán program in toric HMS.”

Abstract

By a classic theorem of Beilinson, the perfect derived category $\operatorname{Perf}(\mathbb{P}^n)$ of projective space is equivalent to the category of derived representations of a certain quiver with relations. The vertex-wise tensor product of quiver representations corresponds to a symmetric monoidal structure $\otimes_{\mathsf{Q}}$ on $\operatorname{Perf}(\mathbb{P}^n)$. We prove that, for a certain choice of equivalence, the symmetric monoidal structure $\otimes_{\mathsf{Q}}$ may be described geometrically as an \emph{extended convolution product} in the sense that the Fourier--Mukai kernel is given by the closure of the torus multiplication map in $(\mathbb{P}^n)^3$. We also set up a general framework for such problems, allowing us to generalize the extended convolution description of quiver tensor products to the case where $\mathbb{P}^n$ is replaced by any smooth complete toric variety of Bondal--Ruan type. Under toric mirror symmetry, this extended convolution product corresponds to the tensor product of constructible sheaves on a real torus. As another generalization of our results for $\mathbb{P}^n$, we show that any finite-dimensional algebra $A$ gives rise to a monoidal structure $\star_A'$ on $\operatorname{Perf}(\mathbb{P}(A))$, providing insights into the moduli of monoidal structures on $\operatorname{Perf}(\mathbb{P}^n)$.

Figures (2)

• Figure 1: The Beilinson quiver for $\mathbb{P}^n$ (with relations $x_i x_j = x_j x_i$ for all $i, j$).
• Figure 2: The infinite Beilinson quiver for $[\mathbb{A}^{n+1} / \mathbb{G}_m]$ (with relations $x_i x_j = x_j x_i$ for all $i, j$). Vertices are indexed by $\mathbb{X}^\bullet(\mathbb{G}_m) = \mathbb{Z}$.

Theorems & Definitions (171)

• Theorem 1.2: \ref{['mainthm_toric']}, \ref{['prop:hms_tensor']}
• Remark 1.3
• Theorem 1.4: \ref{['mainthm_algebra']}, \ref{['prop:algebra_reconstruction']}
• Remark 1.5
• Remark 1.6
• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5