About me

I work in enumerative algebraic geometry with a focus on sheaf-counting theories. My main interests lie in advancing the understanding of wall-crossing theories on the sheaf side and their application to studying invariants and structures of moduli spaces. Most of my work has focused on sheaf-counting on Calabi–Yau fourfolds and Virasoro constraints where I used vertex algebras and their formal families to prove multiple existing conjectures.

I am (co)organizing the Algebraic Geometry Seminar at Academia Sinica. If you think I missed your work on the ArXiv, you can contact me by sending me a url to the article you would like to give a talk on.

Ongoing Projects:

Proving wall-crossing for Calabi–Yau fourfolds : I am proving that for some class of polynomial Bridgeland stabilities on the derived category of coherent sheaves, there are wall-crossing formulae relating the virtual fundamental classes counting semistable objects. One of the interesting outcomes is that this can be applied to Calabi–Yau dg-categories as long as assumptions are satisfied and to wall-crossing with certain nice insertions. I use this theory to prove multiple existing conjectures in the literature related to point and curve counting together with the results in 3. This is based on the work of Joyce in lower dimensions.
Virasoro constraints for abelian categories : Following our success in pdf where I, Woonam Lim, and Miguel Moreira proved Virasoro constraints for moduli schemes of sheaves on curves and surfaces, I continue studying further settings where they may be present. The idea is to compare the constraints to the virtual fundamental class being a physical state of a geometrically constructed vertex algebra. This vertex algebra was introduced by Joyce with the goal of describing wall-crossing of sheaf-counting invariants. Consequently, Virasoro constraints are preserved by wall-crossing under changing stability conditions which can be also used to prove them for (framed) representations of quivers as I do in my more recent work.
Equivariant Segre and Verlinde invariants for Quot schemes joint with J. Huang pdf : Together with Jiahui (who is a master’s student!) we use my previous results about Segre and Verlinde invariants for Quot schemes in the compact case to study their equivariant versions. We do so both for $\mathbb{C}^2$ and $\mathbb{C}^4$ by using equivariant localization. In the two-dimensional case, we find a new Segre-Verlinde correspondence between the higher degree terms which have no compact analogues. For $\mathbb{C}^4$, we simultaneously prove the integrality of the Verlinde series defined in my previous work and formulate multiple conjectures about the structure of the Segre and Verlinde series together with their correspondence.
Proving wall-crossing in special cases In some of the cases, the wall-crossing formulae have not been shown to hold. This is because one needs to prove the properness of all moduli spaces that come into play. An example of this is the wall-crossing leading to Virasoro constraints for M_{\beta,\chi} moduli spaces of 1-dimensional sheaves or the wall-crossing in 4. As a consequence, the approach to proving Virasoro constraints for M_{\beta,\chi} described in 2. will be made into a self-contained theorem.
Wall-crossing for surface counting theories on Calabi–Yau fourfolds : I am proving multiple conjectures of Bae–Kool–Park about (reduced) invariants counting surfaces on Calabi–Yau fourfolds. The main tool I rely on is the wall-crossing which I prove in project 2.
A comprehensive guide to derived algebraic geometry for enumerative geometers pdf : As the name suggests, these lecture notes are meant to give a gentle introduction to derived algebraic geometry and local models of derived stacks. It contains a new result about relative -2-shifted cotangent bundles which I was hoping to use to prove 2.
K-theoretic wall-crossing for Calabi–Yau fourfolds (joint with Henry Liu) : We extend to K-theory my work in 2. and 3. in collaboration with Henry Liu. The goal is to address wall-crossing conjectures for K-theoretic invariants relying on the techniques applied by Liu in the case of Vafa–Witten theory.

Master’s students:

Yifan Zhao: Thesis - ``Stable pair invariants for Calabi–Yau fourfold local surfaces”, Yifan is now doing his PhD at the London School of Geometry and Number Theory.

Huang Jiahui: Thesis - ``Equivariant Segre and Verlinde invariants for Quot schemes”, Jiahui is moving on to do his PhD at the University of Waterloo after successfully turning his thesis into a paper.

Selected Talks:

Wall-Crossing for Holomorphic Donaldson Invariants and Its Applications, Workshop on moduli spaces, virtual invariants and shifted symplectic structures, KIAS, June 2023. slides

Wall-Crossing for Calabi-Yau Fourfolds and Applications, Skye 2023: Workshop on Hall algebras and vertex algebras in enumerative geometry, Isle of Skye, April 2023 notes.

Equivariant Segre and Verlinde series for Quot-schemes, Algebra and geometry seminar, Stockholm University, March 2023 notes

An Intuitive Approach to Sheaf-theoretic Virasoro Constraints, Algebraic geometry seminar, Academia Sinica in Taipei, February 2023. slides video

Equivariant Segre and Verlinde Series for Quot-Schemes, Algebraic geometry and moduli seminar- ETH Zurich, January 2023. notes, Jiahui Huang gave the second part of the talk notes

Wall-Crossing for Calabi–Yau Fourfolds and Applications, Simons Collaboration on Special Holonomy Meeting, University of Oxford, January 2023. slides video

Wall-Crossing for Virasoro Constraints, Utrecht Geometry Centre Seminar, University of Utrecht, November 2022. slides

Wall-Crossing for Calabi-Yau Fourfolds and Applications, M-Seminar, Kansas State University, November 2022. slides video

Virasoro for Moduli of Sheaves: The Wall-Crossing, Diablerets, Helvetic Algebraic Geometry Seminar, June 2022. The first version of the proof appears in these notes notes Miguel gave the introductory talk. notes

Wall-Crossing in Non-Commutative Calabi-Yau Four Categories, May 2022, slides

Wall-Crossing for Punctual Quot-schemes, University of Oslo, Algebra Seminar, January 2022.

Wall-Crossing for Hilbert Schemes of Fourfolds and Quot-Schemes of Surfaces, The London geometry and topology seminar, June 2021

Hilbert Schemes for Fourfolds and Quot-Schemes for Surfaces, Geometry & Analysis seminar - Oxford, April 2021

Wall-Crossing for Hilbert Schemes on Fourfolds and Quot-Schemes on Surfaces, UCSD Algebraic Geometry Seminar, April 2021

Wall-Crossing for Hilbert Schemes on CY 4-Folds II, Algebraic geometry and moduli seminar- ETH Zurich, March 2021.

Wall-Crossing for Hilbert Schemes on CY 4-Folds I, Algebraic geometry and moduli seminar- ETH Zurich, March 2021.

Orientation on the Moduli Stack of Compactly Supported Perfect Complexes over a Calabi-Yau 4-Fold, online, Oxford–London Gauge Assembly, August 2020. slides video

Orientations for DT Invariants on Quasi-Projective Calabi–Yau 4-Folds, Algebraic geometry seminar - Nottingham, November 2020. slides video

Non-Commutative Counting and Stability, University of Warwick, Summer-school on Bridgeland stability conditions, December 2019.

Teaching experience:

Derived algebraic geometry at ETH Zürich 2023 Spring-term (Lecturer) recordings of the lectures

Derived algebraic geometry at ETH Zürich 2022 Spring-term (Lecturer) notes

Scattering diagrams at ETH Zürich Autumn-term (Organizer)

String theory at Oxford 2021 Hilary (Tutor)

Projective geometry at Merton College in Oxford 2020 Trinity (Tutor)

Algebraic topology at Oxford 2018 Michaelmas (Teaching assistent)

Representation theory at Oxford 2018 Michaelmas (Teaching assistent)

Arkadij Bojko, DPhil