Kyoung-Seog Lee

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 resulting combinatorial formulas lead to an independent proof of the equality of the Casson invariants in the Donaldson and Seiberg–Witten theories. We use similar techniques to express the Heegaard Floer d -invariant of torus knots in terms of their Milnor fibers.

In this paper, we suggest a new approach to study minimal surfaces of general type with p g = 0 p_g=0 via their Cox rings, especially using the notion of combinatorially minimal Mori dream space introduced by Hausen [Mosc. Math. J. 8 (2008), pp. 711–757, 847]. First, we study general properties of combinatorially minimal Mori dream surfaces. Then we discuss how to apply these ideas to the study of minimal surfaces of general type with p g = 0 p_g=0 which are very important but still mysterious objects. In our previous paper [Adv. Math. 347 (2019), pp. 708-738], we provided several examples of Mori dream surfaces of general type with p g = 0 p_g = 0 and computed their effective cones explicitly. In this paper, we study their fibrations, explicit combinatorially minimal models and discuss singularities of the combinatorially minimal models. We also show that many minimal surfaces of general type with p g = 0 p_g=0 arise from the minimal resolutions of combinatorially minimal Mori dream surfaces.

Hodge theory is pivotal in studying algebraic varieties’ intricate geometry and topology: it provides essential insights into their structure. The Hodge decomposition theorem establishes a profound link between the geometry of varieties and their cohomology groups, helping to understand their underlying properties. Moreover, Hodge theory was crucial at the inception of the field of mirror symmetry, revealing deep connections among seemingly disparate algebraic varieties. It also sheds light on studying algebraic cycles and motives, crucial objects in algebraic geometry. This article explores Hodge polynomials and their properties, specifically focusing on non-Kähler complex manifolds. We investigate a diverse range of such manifolds, including (quasi-)Hopf, (quasi-)Calabi–Eckmann, and LVM manifolds, alongside a class of definable complex manifolds encompassing both algebraic varieties and the aforementioned special cases. Our research establishes the preservation of the motivic nature of Hodge polynomials inside this broader context. Through explicit calculations and thorough analyses, this work contributes to a deeper understanding of complex manifold geometry beyond the realm of algebraic varieties. The outcomes of this study have potential applications in various areas of mathematics and physics where complex manifolds play a significant role.

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 resulting combinatorial formulas lead to an independent proof of the equality of the Casson invariants in the Donaldson and Seiberg-Witten theories. We use similar techniques to express the Heegaard Floer d-invariant of torus knots in terms of their Milnor fibers.

We show that the derived categories of symmetric products of a curve are embedded into the derived categories of the moduli spaces of vector bundles of large ranks on the curve. It supports a prediction of the existence of a semiorthogonal decomposition of the derived category of the moduli space, expected by a motivic computation. As an application, we show that all Jacobian varieties, symmetric products of curves and all principally polarized abelian varieties of dimension at most three, are Fano visitors. We also obtain similar results for motives.

We show that the derived category of a curve is embedded into the derived category of the moduli space of vector bundles on the curve of coprime rank and degree. We also generalize the semiorthogonal decomposition constructed by Narasimhan and Belmans-Mukhopadhyay. Finally, we produce a one-dimensional family of ACM bundles over the moduli space.

We show that the derived category of a curve is embedded into the derived category of the moduli space of vector bundles on the curve of coprime rank and degree. We also generalize the semiorthogonal decomposition constructed by Narasimhan and Belmans-Mukhopadhyay. Finally, we produce a one-dimensional family of ACM bundles over the moduli space.