Woonam Lim
Yonsei University · 数学
研究室紹介
Professor Woonam Lim's research lab specializes in algebraic geometry and enumerative geometry, with a focus on moduli spaces of sheaves, Quot schemes, and their virtual invariants. The lab investigates deep connections between enumerative invariants, vertex algebras, and mathematical physics, particularly through the lens of Virasoro constraints, wall-crossing phenomena, and Nekrasov's gauge theory. A central theme is the interplay between cohomological structures, integrality, and geometric relations in moduli stacks, with applications to Gopakumar–Vafa invariants and topological string theory. The lab employs advanced tools from algebraic geometry, including virtual cycles, Seiberg–Witten invariants, and vertex algebraic structures to solve long-standing conjectures in enumerative geometry.
Research Overview
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Selected Papers
12Abstract In enumerative geometry, Virasoro constraints were first conjectured in Gromov-Witten theory with many new recent developments in the sheaf theoretic context. In this paper, we rephrase the sheaf theoretic Virasoro constraints in terms of primary states coming from a natural conformal vector in Joyce’s vertex algebra. This shows that Virasoro constraints are preserved under wall-crossing. As an application, we prove the conjectural Virasoro constraints for moduli spaces of torsion-free
Abstract We prove that the cohomology rings of the moduli space $M_{d,\chi }$ of one-dimensional sheaves on the projective plane are not isomorphic for general different choices of the Euler characteristics. This stands in contrast to the $\chi $ -independence of the Betti numbers of these moduli spaces. As a corollary, we deduce that $M_{d,\chi }$ are topologically different unless they are related by obvious symmetries, strengthening a previous result of Woolf distinguishing them as algebraic
In enumerative geometry, Virasoro constraints were first conjectured in Gromov-Witten theory with many new recent developments in the sheaf theoretic context. In this paper, we rephrase the sheaf-theoretic Virasoro constraints in terms of primary states coming from a natural conformal vector in Joyce's vertex algebra. This shows that Virasoro constraints are preserved under wall-crossing. As an application, we prove the conjectural Virasoro constraints for moduli spaces of torsion-free sheaves o
We prove that the cohomology rings of the moduli space $M_{d,χ}$ of one-dimensional sheaves on the projective plane are not isomorphic for general different choices of the Euler characteristics. This stands in contrast to the $χ$-independence of the Betti numbers of these moduli spaces. As a corollary, we deduce that $M_{d,χ}$ are topologically different unless they are related by obvious symmetries, strengthening a previous result of Woolf distinguishing them as algebraic varieties.
Quot schemes are fundamental objects in the moduli theory of algebraic geometry. Quot schemes of surfaces admit natural perfect obstruction theories if we consider 1-dimensional quotients of trivial vector bundles. We study various virtual invariants of such Quot schemes using the structure of Seiberg-Witten invariants and Hilbert schemes of points. The main result expresses the virtual Quot scheme invariants universally in terms of Seiberg-Witten invariants and certain cohomological data of a s
Nekrasov's gauge origami theory provides a (complex) 4-dimensional generalization of the ADHM quiver and its moduli spaces of representations. We describe the origami moduli space as the zero locus of an isotropic section of a quadratic vector bundle on a smooth space. This allows us to give an algebro-geometric definition of the origami partition function in terms of Oh--Thomas virtual cycles. The key input is the computation of a sign associated to each torus fixed point of the moduli space. F
We study the Virasoro constraints for moduli spaces of representations of quiver with relations by Joyce's vertex algebras. Using the framed Virasoro constraints, we construct a representation of half of the Virasoro algebra on the cohomology of moduli stacks of quiver representations under smoothness assumption. By exploiting the non-commutative nature of the Virasoro operators, we apply our theory for quivers to del Pezzo surfaces using exceptional collections. In particular, the Virasoro cons
We introduce and study the Chern filtration on the cohomology of the moduli of bundles on curves. This can be viewed as a natural cohomological invariant defined via tautological classes that interpolates between additive Betti numbers and the multiplicative ring structure. In the rank two case, we fully compute the Chern filtration for moduli of stable bundles and all intermediate stacks in the Harder--Narasimhan stratification. We observe a curious symmetry of the Chern filtration on the modul
Nekrasov's gauge origami theory provides a (complex) 4-dimensional generalization of the ADHM quiver and its moduli spaces of representations. We describe the origami moduli space as the zero locus of an isotropic section of a quadratic vector bundle on a smooth space. This allows us to give an algebro-geometric definition of the origami partition function in terms of Oh--Thomas virtual cycles. The key input is the computation of a sign associated to each torus fixed point of the moduli space. F
We initiate a systematic study on the cohomology rings of the moduli stack $\mathfrak{M}_{d,χ}$ of semistable one-dimensional sheaves on the projective plane. We introduce a set of tautological relations of geometric origin, including Mumford-type relations, and prove that their ideal is generated by certain primitive relations via the Virasoro operators. Using BPS integrality and the computational efficiency of Virasoro operators, we show that our geometric relations completely determine the co