## Periodic Reporting for period 1 - INVLOCCY (Invariants of local Calabi-Yau 3-folds)

Reporting period: 2015-06-01 to 2017-05-31

Enumerative geometry on complex algebraic surfaces is a classical topic dating back to the 19th century. Typical problems are: how many lines in the plane go through 2 points (answer 1), or how many rational planar cubics go through 8 general points (answer 12). Modern invariants originating from string theory have provided new tools for attacking such problems as is most strikingly illustrated by M. Kontsevich's determination of all genus zero Severi degrees of the projective plane by using Gromov-Witten invariants. Other invariants, known as Pandharipande-Thomas or stable pair invariants, are relevant for the determination of Severi degrees for sufficiently ample linear systems as was shown in joint work of the author and R.P. Thomas. Together with R.P. Thomas and V. Shende, this led to a proof of Göttsche's conjecture, which loosely states that Severi degrees for sufficiently ample linear systems on surfaces only depend on the topology.

The work of Kool-Thomas and Kool-Shende-Thomas mentioned above, only concerns a special type of stable pairs on the total space of the canonical bundle K_S of a surface S, namely those stable pairs which have the property that they are scheme-theoretically supported in the zero section of K_S. Theme 1 of the project INVLOCCY aims to explore much more general stable pairs on K_S, which need not be constrained to the zero section. Inspired by the applications mentioned above, I expected this has interesting applications as well. Theme 2 of the project INVLOCCY is concerned with refinements of invariants especially with an eye towards the physics literature.

These problems are purely theoretical in nature with no applications and as such their importance is part of the relevance of the development of fundamental mathematics for its own sake. The topic of this proposal has deep links with other branches of mathematics and theoretical physics, namely the theory of modular forms, enumerative geometry, combinatorics, and string theory.

The work of Kool-Thomas and Kool-Shende-Thomas mentioned above, only concerns a special type of stable pairs on the total space of the canonical bundle K_S of a surface S, namely those stable pairs which have the property that they are scheme-theoretically supported in the zero section of K_S. Theme 1 of the project INVLOCCY aims to explore much more general stable pairs on K_S, which need not be constrained to the zero section. Inspired by the applications mentioned above, I expected this has interesting applications as well. Theme 2 of the project INVLOCCY is concerned with refinements of invariants especially with an eye towards the physics literature.

These problems are purely theoretical in nature with no applications and as such their importance is part of the relevance of the development of fundamental mathematics for its own sake. The topic of this proposal has deep links with other branches of mathematics and theoretical physics, namely the theory of modular forms, enumerative geometry, combinatorics, and string theory.

Theme 1 was successfully executed. In joint work with R.P. Thomas, I found an explicit description of the “vertical component” of the moduli space of stable pairs on K_S in the case S has a holomorphic 2-form cutting out a smooth curve. “Vertical” refers to stable pairs with underlying support curve being a thickening “into the fibre direction” of a reduced curve in the zero section. As anticipated, there are interesting applications to enumerative questions; in this case spin Hurwitz numbers. We have written 1 paper on this topic (51 pages, submitted) and I have given 4 talks on this topic at leading conferences/workshops. Theme 1 also led to a side project with A. Gholampour (and parts with B. Young) on generating functions of higher rank sheaves on threefolds. This led to a publication in IMRN (89 pages) and Selecta Math. (19 pages) as well as a very recent further preprint (33 pages, to be submitted). I have given 2 talks on this topic in leading international conferences/workshops. Output:

M. Kool and R. P. Thomas, Stable pairs with descendents on local surfaces I: the vertical component, 51 pages, arXiv:1605.02576

M. Kool, A. Gholampour, and B. Young, Rank 2 sheaves on toric 3-folds: classical and virtual counts, 62 pages, to appear in IMRN, doi.org/10.1093/imrn/rnw302. arXiv:1509.03536

M. Kool and A. Gholampour, Rank 2 wall-crossing and the Serre correspondence, Selecta Math. 23 (2017) 1599-1617. arXiv:1602.03113

M. Kool and A. Gholampour, Higher rank sheaves on threefolds and functional equations, 33 pages, arXiv:1706.05246

Theme 2 was successfully executed. In joint work with L. Göttsche, I found virtual refinements of a formula from the physics literature due to Vafa-Witten. Specifically, we conjecture a formula for the virtual \chi_y genera of moduli spaces of rank 2 stable sheaves on general type surfaces. We have written 1 paper in this topic (36 pages, submitted) and we are currently writing a second paper. I have given 2 talks on this topic in two seminars conferences/workshop.

M. Kool and L. Göttsche, Virtual refinements of the Vafa-Witten formula, 36 pages, arXiv:1703.07196

The above work is openly accessible through arXiv.org. In addition, the above work has been disseminated at 18 invited talks at (mostly international) seminars/conferences/workshops.

M. Kool and R. P. Thomas, Stable pairs with descendents on local surfaces I: the vertical component, 51 pages, arXiv:1605.02576

M. Kool, A. Gholampour, and B. Young, Rank 2 sheaves on toric 3-folds: classical and virtual counts, 62 pages, to appear in IMRN, doi.org/10.1093/imrn/rnw302. arXiv:1509.03536

M. Kool and A. Gholampour, Rank 2 wall-crossing and the Serre correspondence, Selecta Math. 23 (2017) 1599-1617. arXiv:1602.03113

M. Kool and A. Gholampour, Higher rank sheaves on threefolds and functional equations, 33 pages, arXiv:1706.05246

Theme 2 was successfully executed. In joint work with L. Göttsche, I found virtual refinements of a formula from the physics literature due to Vafa-Witten. Specifically, we conjecture a formula for the virtual \chi_y genera of moduli spaces of rank 2 stable sheaves on general type surfaces. We have written 1 paper in this topic (36 pages, submitted) and we are currently writing a second paper. I have given 2 talks on this topic in two seminars conferences/workshop.

M. Kool and L. Göttsche, Virtual refinements of the Vafa-Witten formula, 36 pages, arXiv:1703.07196

The above work is openly accessible through arXiv.org. In addition, the above work has been disseminated at 18 invited talks at (mostly international) seminars/conferences/workshops.

The Marie Curie project have led to several new results, which in turn lead to other projects. Of particular interest is the activity surrounding WP2:

This project has an interesting link with very recent work of Tanaka-Thomas (2017), who define a virtual count of Higgs pairs on surfaces. Their invariants contain our virtual Euler characteristic invariants as well as contributions from connected components with non-zero Higgs field. Our conjectural formula together with their low degree calculations for the contribution of other connected components led Tanaka-Thomas to conjecture that their invariants equal the physics invariants from Vafa-Witten. In this way, our project had an impact on theirs (and vice-versa). In turn, Tanaka-Thomas's invariants are equal to the 2-dimensional Donaldson-Thomas invariants studied by Gholampour-Sheshmani-Yau (2017). This circle of ideas is expected to prompt exciting new activity. I have written a research proposal for an NWO (Dutch funding agency) TOP2 grant based on this research direction.

This research purely theoretical nature with no applications. As such the importance is part of the relevance of the internal development of pure mathematics. Although this is a topic in algebraic geometry, it has connections with other fields of mathematics such as the theory of modular forms, enumerative geometry, combinatorics, and string theory. Concretely, WP1 led to an interesting combinatorial problem solved by Pixton-Zagier. It also involved a collaboration with the combinatorist B. Young. WP2 has links with gauge theory and string theory, and involves modular forms.

I like to give talks about my research, or rather my research field in general, for a non-mathematical audience. During the Marie Curie, I gave two lectures for high school students and a public lecture in the context of the international Imaginary Exhibition.

Krommen tellen: van de Griekse Oudheid tot snaartheorie, public lecture at exhibit Imaginary, Utrecht, 10-2-2017.

Symmetrieën en Du Val singulariteiten, lecture for high school student (4VWO) JCU, 24-3-2015 and 5-4-2016.

I would like to end by mentioning that the Marie Curie project was crucial for my research output over the last two years. It has also played a crucial role in meeting several requirements for tenure at the Mathematical Institute at Utrecht University (such as obtaining my basic teaching qualification).

This project has an interesting link with very recent work of Tanaka-Thomas (2017), who define a virtual count of Higgs pairs on surfaces. Their invariants contain our virtual Euler characteristic invariants as well as contributions from connected components with non-zero Higgs field. Our conjectural formula together with their low degree calculations for the contribution of other connected components led Tanaka-Thomas to conjecture that their invariants equal the physics invariants from Vafa-Witten. In this way, our project had an impact on theirs (and vice-versa). In turn, Tanaka-Thomas's invariants are equal to the 2-dimensional Donaldson-Thomas invariants studied by Gholampour-Sheshmani-Yau (2017). This circle of ideas is expected to prompt exciting new activity. I have written a research proposal for an NWO (Dutch funding agency) TOP2 grant based on this research direction.

This research purely theoretical nature with no applications. As such the importance is part of the relevance of the internal development of pure mathematics. Although this is a topic in algebraic geometry, it has connections with other fields of mathematics such as the theory of modular forms, enumerative geometry, combinatorics, and string theory. Concretely, WP1 led to an interesting combinatorial problem solved by Pixton-Zagier. It also involved a collaboration with the combinatorist B. Young. WP2 has links with gauge theory and string theory, and involves modular forms.

I like to give talks about my research, or rather my research field in general, for a non-mathematical audience. During the Marie Curie, I gave two lectures for high school students and a public lecture in the context of the international Imaginary Exhibition.

Krommen tellen: van de Griekse Oudheid tot snaartheorie, public lecture at exhibit Imaginary, Utrecht, 10-2-2017.

Symmetrieën en Du Val singulariteiten, lecture for high school student (4VWO) JCU, 24-3-2015 and 5-4-2016.

I would like to end by mentioning that the Marie Curie project was crucial for my research output over the last two years. It has also played a crucial role in meeting several requirements for tenure at the Mathematical Institute at Utrecht University (such as obtaining my basic teaching qualification).