# Workshop: Curves and K3 surfaces

The goal of the meeting is to bring together experts on K3 surfaces, moduli spaces of curves, and enumerative geometry, all viewed in a broad sense, to discuss the latest in the field in a relaxed atmosphere.

Date: Monday May 9th to Thursday, May 12th

Speakers:

• Younghan Bae (ETH Zürich)
• Luca Battistella (Heidelberg)
• Thomas Blomme (Geneve)
• Tim Bülles (ETH Zürich)
• Yajnaseni Dutta (Bonn)
• Francesca Carocci (EPFL)
• Soheyla Feyzbakhsh (Imperial College)
• Lothar Göttsche (ICTP)
• Frank Gounelas (Göttingen)
• Martijn Kool (Utrecht)
• Nikolas Kuhn (MPI Bonn)
• Denis Nesterov (Bonn)
• Jeongseok Oh (Imperial College)
• Helge Ruddat (Mainz)

Location:

All talks will be held in the Lipschitzsaal at the

Mathematisches Institut
Universität Bonn
Endenicher Allee 60
D-53115 Bonn

### Schedule

Talks will last for one hour. The expected schedule is as follows:

Monday:
09:20-09:30   Welcome
09:30-10:30   Talk 1: Gounelas
10:30-11:00   Coffee break
11:00-12:00   Talk 2: Feyzbakhsh

14:00-15:00   Talk 3: Kuhn
15.15-16:15   Talk 4: Bülles
16.15-16:45   Coffee break and cake
16:45-17:45   Talk 5: Dutta

Tuesday:
09:30-10:30   Talk 6: Blomme
10:30-11:00   Coffee break
11:00-12:00   Talk 7: Battistella

14:00-15:00   Talk 8: Carocci
15.15-16:15   Talk 9: Nesterov
16.15-16:45   Coffee break and cake
16:45-17:45   Talk 10: by Junior participants

Wednesday:
09:00-10:00   Talk 11: Ruddat
10:00-10:30   Coffee break and cake
10.30-11:30   Talk 12: Oh
11:45-12:45   Talk 13: Bojko
Free afternoon

Thursday:
09:30-10:30   Talk 14: Kool
10:30-11:00   Coffee break
11:00-12:00   Talk 15: Bae

14.00-15.00   Talk 16: Göttsche
15.00-15.30   Coffee break with cake and goodbye

Title and abstracts

Younghan Bae: Counting surfaces on Calabi-Yau 4-folds II
Continuing our study of the reduced virtual cycle, we calculate the reduced virtual dimension for local complete intersection surfaces. Subsequently, we consider families of Calabi-Yau 4-folds and prove deformation invariance along the Hodge locus. As the main application, we deduce the variational Hodge conjecture in this setting when the reduced virtual cycle is non-zero. Joint work with M. Kool and H. Park.

Luca Battistella: Logarithmic and orbifold Gromov-Witten invariants
Logarithmic Gromov-Witten theory can be thought of as the study of curves in open manifolds, or, in other words, curves with tangency conditions to a boundary divisor. When the divisor is smooth, several techniques have been developed to compute the invariants, most notably orbifold stable maps. When the divisor is normal crossings, on the other hand, the logarithmic theory remains hardly accessible. The strategy of rank reduction, i.e. looking at the components of the boundary one at a time, is more directly applicable to other theories than the logarithmic one (as shown in Nabijou-Ranganathan and B.-Nabijou-Tseng-You) due to tropical obstructions. Inspired by one of the distinguishing features of the logarithmic theory - namely, birational invariance [Abramovich-Wise] - in joint work with Nabijou and Ranganathan we show that, when the genus is zero, tropical obstructions can be disposed of by blowing up the target sufficiently. The slogan is that the logarithmic theory is the limit orbifold theory under birational modifications along the boundary divisor.

Thomas Blomme: Enumeration of tropical curves in abelian surfaces
Tropical geometry is a powerful tool that allows one to compute enumerative algebraic invariants through the use of some correspondence theorem, transforming an algebraic problem into a combinatorial problem. Moreover, the tropical approach also allows one to twist definitions to introduce mysterious refined invariants, obtained by counting tropical curves with polynomial multiplicities. So far, this correspondence has mainly been implemented in toric varieties. In this talk we will study enumeration of curves in abelian surfaces and use the tropical geometry approach to prove a multiple cover formula that enables an simple and elegant computation of enumerative invariants of abelian surfaces.

Arkadij Bojko: Wall-crossing for non-commutative Calabi--Yau fourfolds
The work of Joyce has set up a conjectural wall-crossing framework for sheaves and complexes on Calabi--Yau fourfolds when changing stability conditions. This has been used by me in my previous work to study the explicit formulae for Hilbert schemes of points and Quot-schemes on Calabi--Yau fourfolds which could be used to derive closed expressions for generating series of all interesting invariants. Unfortunately, until now this machinery has remained conjectural. In the talk, I will not only discuss how to prove the already computed results, but will generalize the statements to be applicable for general non-commutative Calabi--Yau fourfolds. The upshot is going to be a wall-crossing formula for invariants under the change of some generalization of Bayer's standard polynomial stability condition in a heart of a bounded t-structure of any Calabi--Yau 4 triangulated category. With an additional twist added to the construction of vertex algebras which the wall-crossing heavily relies upon, the applications to proving conjectures of Cao--Toda and surface counting conjecture of Bae--Kool--Park become immediate.

Tim-Henrik Bülles: Weyl symmetry for curve counting invariants via spherical twists
The curve counting invariants of Calabi—Yau 3-folds exhibit symmetries induced by the action of derived autoequivalences. I will give a brief overview of the subject and explain some recent development. In joint work with M. Moreira we obtain the Weyl symmetry along a ruled divisor, a new rationality result and functional equation for the generating function of stable pairs invariants. The underlying derived autoequivalence involves spherical twists.

Francesca Carocci: BPS invariant from non Archimedean integrals
We consider moduli spaces M(ß,χ) of one-dimensional semistable sheaves on del Pezzo and K3 surfaces supported on ample curve classes. Working over a non-archimedean local field F, we define a natural measure on the F-points of such moduli spaces. We prove that the integral of a certain naturally defined gerbe on M(ß,χ) with respect to this measure is independent of the Euler characteristic. Analogous statements hold for (meromorphic or not) Higgs bundles. Recent results of Maulik-Shen and Kinjo-Coseki imply that these integrals compute the BPS invariants for the del Pezzo case and for Higgs bundles. This is a joint work with Giulio Orecchia and Dimitri Wyss.

Yajnaseni Dutta: Curves on K3 surfaces and a conjecture of Matsushita.
I will report on a recent joint work with D. Huybrechts where we strudied how wildly the complex structures of smooth (general) curves on a given K3 surface can vary when they deform inside the surface. We prove that if the curve lies in a large enough (>= 3) multiple of an ample and base point free linear system then the complex structures of general curves on that linear system vary maximally. In the genus 2 case we can say more. The original motivation for this problem comes from a conjecture of Matsushita about variation of complex structures of general fibers of Lagrangian fibrations of hyperkaehler manifolds. I will briefly discuss the current state of affairs on this conjecture.

Soheyla Feyzbakhsh: Hyperkaehler varieties as Brill-Noether loci on curves
Consider the moduli space $M_C(r; K_C)$ of stable rank r vector bundles on a curve $C$ with canonical determinant, and let $h$ be the maximum number of linearly independent global sections of these bundles. If $C$ embeds in a K3 surface $X$ as a generator of $Pic(X)$ and the genus of $C$ is sufficiently high, I will show the Brill-Noether locus $\BN_C \subset M_C(r; K_C)$ of bundles with $h$ global sections is a smooth projective Hyperk\"{a}hler manifold, isomorphic to a moduli space of stable vector bundles on $X$. The main technique is to apply wall-crossing with respect to Bridgeland stability conditions on K3 surfaces.

Lothar Göttsche: Generating functions for Segre and Verlinde numbers on surfaces
This is based in part on joint work with Martijn Kool, and is partially joint work with Anton Mellit. Based on conjectural blowup formulas for Segre and Verlinde numbers of moduli spaces of sheaves on projective surfaces with $q=0$ and $p_g>0$ and a virtual version of the strange duality conjecture, we in many cases give conjectural generating functions for these invariants. In the case of the Hilbert scheme of points this gives a complete conjectural determination of the Verlinde and Segre series. We prove this conjecture for surfaces with K^2=0.

Frank Gounelas: Curves of maximal moduli on K3 surfaces
In joint work with Chen, we proved that on any K3 surface one can produce curves of any fixed geometric genus g, each of which deforms maximally in moduli, i.e. in a g-dimensional family of M_g. In this talk I will discuss this and some related results, and various applications. The key inputs in the proof are the existence of infinitely many rational curves on a K3 (obtained jointly with Chen-Liedtke) and the logarithmic Bogomolov-Miyaoka-Yau inequality which provides some (very weak) control of the singularities of these rational curves.

Martijn Kool: Counting surfaces on Calabi-Yau 4-folds I
The Oh-Thomas virtual cycle generally vanishes when applied to surface classes on Calabi-Yau 4-folds. Using methods of Kiem-Li and Kiem-Park, we reduce the Oh-Thomas obstruction theory by H^{3,1}-forms in order to obtain non-trivial invariants. On the semi-regular locus of the moduli space, our reduced virtual cycle coincides with the fundamental class. Joint work with Y. Bae and H. Park.

Nikolas Kuhn: Instanton Vafa-Witten invariants on the blowup of a K3
The virtual Euler characteristics of moduli spaces of sheaves on an algebraic surface X are also known as instanton Vafa-Witten invariants through their connection to S-duality. They have a conjecturally rich structure which has been predicted by Göttsche and Kool. However, these conjectures remain so far unproven beyond the cases where the moduli spaces are unobstructed, so that virtual and topological invariants agree. In this talk, I will describe how to prove the conjectured formula in the case that X is the blowup of a K3 surface. This is the consequence of a blowup formula which holds for arbitrary X.

Denis Nesterov: Gromov-Witten/Hurwtiz wall-crossing
Using a branching divisor, we introduce a one-parameter family of stability conditions, termed \epsilon-admissibility, on maps from nodal curves to X x C for a smooth variety X and a curve C. If X is a point, these stability conditions interpolate between Gromov-Witten and Hurtwiz spaces of C. More generally, $\epsilon$-admissibility provides a wall-crossing between Gromov-Witten theory of an orbifold symmetric product of X and relative Gromov-Witten theory of X x C with a varying C. In conjunction with quasmap wall-crossing we establish the genus-0 3-point Crepant resolution conjecture for Hilbert-Chow morphism of punctorial Hilbert schemes of a toric surface X by reducing it to GW/PT correspondence of X x C.

Jeongseok Oh: Complex Kuranishi spaces
We develop a theory of complex Kuranishi structures on projective schemes. These are sufficiently rigid to be equivalent to weak perfect obstruction theories, but sufficiently flexible to admit global complex Kuranishi charts. We apply the theory to projective moduli spaces M of stable sheaves on Calabi-Yau 4-folds. Borisov-Joyce produced a real virtual homology cycle on M using real derived differential geometry. In the prequel to this work we constructed an algebraic virtual cycle on M. We prove the cycles coincide in homology after inverting 2 in the coefficients. In particular, when Borisov-Joyce’s real virtual dimension is odd, their virtual cycle is torsion. This is a joint work with Richard Thomas.

Helge Ruddat: The proper Landau-Ginzburg potential is the open mirror map
The mirror dual of a smooth toric Fano surface equipped with an anticanonical divisor is a Landau-Ginzburg model with superpotential, W. Carl-Pumperla-Siebert give a definition of the the superpotential in terms of tropical disks using a toric degeneration of the pair (X,E). When E is smooth, the superpotential is proper. We show that this proper superpotential equals the open mirror map for outer Aganagic-Vafa branes in the canonical bundle K_X, in framing zero. As a consequence, the proper Landau-Ginzburg potential is a solution to the Lerche-Mayr Picard-Fuchs equation. Along the way, we prove a generalization of a result about relative Gromov-Witten invariants by Cadman-Chen to arbitrary genus using the multiplication rule of quantum theta functions. In addition, we generalize a theorem of Hu that relates Gromov-Witten invariants of a surface under a blow-up from the absolute to the relative case. One of the two proofs that we give introduces birational modifications of a scattering diagram. We also demonstrate how the Hori-Vafa superpotential is related to the proper superpotential by mutations from a toric chamber to the unbounded chamber of the scattering diagram.

Organization: Georg Oberdieck
Contact: georgo@math.uni-bonn.de
The workshop is funded by the Deutsche Forschungsgemeinschaft (DFG), Grant number OB 512/1-1.
Administrative support is provided by the Hausdorff Center for Mathematics.