Higher structures in Enumerative Geometry

June 12-16, 2023 @ Institut Henri Poincaré in Paris

This conference is part of the thematic trimester ‘Higher Structures in Geometry and Mathematical Physics’.


MON 09:00 registration
  10:00 Davesh Maulik
  11:00 coffee break
  11:30 Cristina Manolache
  12:30 lunch break
  14:00 Y.P. Lee
  15:00 coffee break
  15:30 Mauro Porta
  18:00 reception
TUE 10:00 Emily Riehl
  11:00 coffee break
  11:30 Yukinobu Toda
  12:30 lunch break
  14:00 Sarah Scherotzke
  15:00 coffee break
  15:30 Jim Bryan
WED 09:30 Zheng Hua
  10:30 coffee break
  11:00 Wendy Lowen
  12:00 Étienne Mann
THU 10:00 Tobias Dyckerhoff
  11:00 coffee break
  11:30 Ben Davison
  12:30 lunch break
  14:00 Renata Picciotto
  15:00 coffee break
  15:30 Rok Gregoric
FRI 09:30 Penka Georgieva
  10:30 coffee break
  11:00 Andrea Ricolfi
  12:00 Richard Thomas



Apéritif de bienvenue

Refreshments will be provided.

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Jim Bryan (British Columbia)

The geometry and arithmetic of banana nano-manifolds.

The Hodge numbers of a Calabi-Yau threefold X are determined by the two numbers h^{1,1}(X) and h^{1,2}(X) which can be interpreted respectively as the dimensions of the space of Kahler forms and complex deformations respectively. We construct four new examples X_N, where N \in {5,6,8,9}, of rigid Calabi-Yau threefolds (h^{2,1}=0) with Picard number 4 (h^{1,1}=4). These manifolds are of “banana type” and have among the smallest known values for Calabi-Yau Hodge numbers. We (partially) compute the Donaldson-Thomas partition functions of these manifolds and in particular show that the associated genus g Gromov-Witten potential is given by a weight 2g-2 Siegel paramodular form of index N. These manifolds are also modular in the arithmetic sense: there is a weight 4 modular form whose Fourier coefficients are obtained by counting points over finite fields on X_N. We compute this form and observe that it is the unique cusp form of weight 4 and index N. This is joint work with Stephen Pietromonaco.

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Ben Davison (Edinburgh)

Stacks of semistable sheaves on K3 surfaces

I’ll explain some new results on the Borel-Moore homology of stacks of coherent sheaves on K3 surfaces, as well as intersection cohomology of coarse moduli spaces. For nonprimitive Chern classes, these spaces can be highly singular. Nonetheless, considering all multiples of a given class simultaneously, we (in joint work with Lucien Hennecart and Sebastian Schlegel Mejia) have a cohomological upgrade of the integrality theorem from DT theory, connecting the intersection cohomology of the coarse spaces with the Borel-Moore homology of the stacks. Aside from the integrality theorem itself, applications include a new description of Maulik-Toda-style GV invariants of local K3 surfaces, a proof of the Halpern-Leistner purity conjecture for the BM homology of the stack, and wall-crossing invariance results.

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Tobias Dyckerhoff (Hamburg)

Complexes of stable $\infty$-categories

Derived categories have come to play a decisive role in a wide range of topics. Several recent developments, in particular in the context of topological Fukaya categories, arouse the desire to study not just single categories, but rather complexes of categories. In this talk, we will discuss examples that arise in the vicinity of homological mirror symmetry, along with some results and conjectures involving them.

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Penka Georgieva (Sorbonne Université)

Klein TQFT and real Gromov-Witten invariants

TQFT structures appear in relation with Gromov-Witten invariants in at least two separate occasions. In this talk I will explain how the Real Gromov-Witten theory of local 3-folds with base a Real curve gives rise to an extension of a 2d Klein TQFT. This gives strong implications for the structure of the invariants on any compact Calabi-Yau 3-fold with anti-symplectic involution.

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Rok Gregoric (Texas)

Shifting and shearing in spectral algebraic geometry

In derived algebraic geometry, vector bundles can be shifted. This corresponds to the shearing operation on graded objects. However, over the sphere spectrum, the notions of shifting vector bundles and of shearing gradings decouple, in parallel to the existence of two variants of the affine line. In this talk, we will discuss how the picture of shifting and shearing can be reconciled by replacing the integers with more exotic grading groups. We will see how this naturally leads to considering the moduli stack of oriented formal groups, and through it to chromatic homotopy theory and the motivic tau-deformation phenomenon.

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Zheng Hua (Hong Kong)

Modular Poisson structures

Give a Gorenstein Calabi-Yau curve X, the moduli stack of bounded complexes of vector bundles on X admits a 0-shifted Poisson structure. For the component that is a G_m gerbe over a smooth scheme, the 0-shifted Poisson structure descends to an ordinary Poisson structure on its coarse moduli scheme, which we call a “modular Poisson structure”. In this talk we will survey some recent progress on the study of the modular Poisson structures. In particular we will give the example when X is the Kodaira cycle, where we establish a correspondence between certain torus orbits of vector bundle stack and projected open Richardson varieties on grassmannian via modular Poisson structure.

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Yuan-Pin Lee (Utah)

QK = GV for CY3 at g=0

In this talk, I will show that on a Calabi-Yau threefold (CY3) a genus zero quantum K-invariant (QK) can be written as an integral linear combination of a finite number of Gopakumar–Vafa BPS invariants (GV) with coefficients from an explicit multiple cover formula. Conversely, all Gopakumar–Vafa invariants can be determined by a finite number of quantum K-invariants in a similar manner. The technical heart is a proof of a remarkable conjecture by Hans Jockers and Peter Mayr.

This result is consistent with the “virtual Clemens conjecture” for the Calabi–Yau threefolds. A heuristic derivation of the relation between QK and GV via the virtual Clemens conjecture and a “multiple cover formula” will also be explained. This is a joint work with You-Cheng Chou.

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Wendy Lowen (Antwerp)

Box operads and noncommutative schemes

In this talk, we discuss how the concept of algebraic deformation theory, dating back to Gerstenhaber’s deformation theory of algebras, can be applied in geometry leading to ‘noncommutative schemes’ in the sense of Van den Bergh. In the first part of the talk, we focus on models in projective geometry, like graded algebras and Z-algebras, and we describe deformations as ‘noncommutative projective schemes’ under the assumption that H^1(X,O_X) = 0 = H^2(X,O_X). In the second part of the talk, we take a local approach to schemes by deforming the structure sheaf, leading to ‘twisted presheaves’ or ‘prestacks’ in case H^2(X,O_X) is non-zero. Finally, we present an operadic structure on the Gerstenhaber-Schack complex of a prestack recently established in joint work with Hoang Dinh Van and Lander Hermans. This yields an underlying L_{infinity} structure governing deformations of prestacks.

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Etienne Mann (Angers)

Gromov-Witten theory with derived algebraic geometry

Using ideas from Toën-Schuerg-Vezzosi, one can define GW invariants via derived algebraic geometry. In this talk, we will see how some classical statements like Splitting axiom can be seen at the geometric level. An other nice application is to see a derived version of Quantum Lefchetz principle due to David Kern. Part of this work was done with Marco Robalo.

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Cristina Manolache (Sheffield)

Desingularisation of sheaves and reduced Gromov–Witten invariants

Gromov–Witten (GW) invariants of genus g, with g greater than one, do not count curves of genus g in a given space: curves of lower genus also contribute to GW invariants. In genus one this problem was corrected by Vakil and Zinger, who defined more enumerative numbers called “reduced GW invariants.” More recently Hu, Li and Niu gave a construction of reduced GW invariants in genus two. I will define reduced Gromov–Witten invariants in all genera. This is work with A. Cobos-Rabano, E. Mann and R. Picciotto.

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Davesh Maulik (MIT)

P=W conjecture for GL_n

The P=W conjecture, proposed by de Cataldo-Hausel-Migliorini in 2010, gives a link between the topology of the moduli space of Higgs bundles on a curve and the Hodge theory of the corresponding character variety, using non-abelian Hodge theory. In this talk, I will explain this circle of ideas and discuss a recent proof of the conjecture for GL_n (joint with Junliang Shen).

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Renata Picciotto (Angers)

The derived moduli of sections and virtual pushforwards

Derived algebraic geometry provides a powerful set of tools to enumerative geometers, giving geometric spaces which encode the “virtual structures” of the moduli problems . I will discuss a joint work with D. Karn, E. Mann and C. Manolache in which we define a derived enhancement for the moduli space of sections. This enriched space neatly encodes the perfect obstruction theory and virtual structure sheaves of many theories. Special cases include Gromov-Witten and quasimaps theories. To illustrate the potential of this approach, I will explain how we use local derived charts to prove a virtual pushforward formula between stable maps and quasimaps without relying on torus localization.

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Mauro Porta (Strasbourg)

Categorical Hall algebras and their representations

Two-dimensional cohomological Hall algebras have been introduced for the first time by Schiffmann and Vasserot in 2009 and they soon proved to be an exceptional tool for the study of homology and G-theory of several kinds of moduli spaces. More recently they have been revisited with tools from derived geometry, which led to a natural categorification. A current limitation of the subject is that cohomological Hall algebras only yield positive halves of “whole” algebraic objects, such as Yangians or quantum loop groups. With Diaconescu and Sala, I have constructed categorical representations for these algebras, that generalize to a 1-dimensional setting the creation and destruction operators of Nakajima. I will survey this construction and discuss some more recent ongoing work in this direction.

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Andrea Ricolfi (SISSA)

d-critical structure(s) on the Quot scheme of points on a Calabi-Yau 3-fold

D-critical schemes and Artin stacks were introduced by Joyce in 2015, and play a central role in Donaldson-Thomas theory. They typically occur as truncations of (-1)-shifted symplectic derived schemes, but the problem of constructing the d-critical structure on a “DT moduli space” without passing through derived geometry (which is hard) is wide open. We discuss this and related problems, and new results in this direction, when the moduli space is the Quot scheme of points on a Calabi-Yau 3-fold. Joint work with Michail Savvas.

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Emily Riehl (Johns Hopkins)

Do we want a new foundation for “higher structures”?

The fundamental theorem of category theory is the Yoneda lemma, which in its simplest form identifies natural transformations between represented functors with morphisms between the representing objects. The $\infty$-categorical Yoneda lemma is surprisingly hard to prove — at least in the traditional set-based foundations of mathematics. In this talk we’ll describe the experience of developing $\infty$-category theory in an alternate foundation system based on homotopy type theory, in which constructions determined up to a contractible space of choices are genuinely “well-defined” and elementwise mappings are automatically homotopically-coherently functorial. In this setting the proof the $\infty$-categorical Yoneda lemma is arguably easier than the 1-categorical Yoneda lemma. We’ll end by posing the question as to whether similar foundations would be useful for other “higher structures.” This is based on joint work with Mike Shulman and involves computer formalizations written in collaboration with Nikolai Kudasov and Jonathan Weinberger.

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Sarah Scherotzke (Luxembourg)

Cotangent complexes of moduli spaces

We explain how shifted symplectic structures on derived stacks are connected to Calabi-Yau structures on differential graded categories. More concretely, we will show that the cotangent complex to the moduli stack of a differential graded category A is isomorphic to the moduli stack of the Calabi-Yau completion of A, answering a conjecture of Keller-Yeung.

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Richard Thomas (Imperial)

(-2)-shifted symplectic virtual cycles

Behrend-Fantechi and Li-Tian showed how to produce a virtual cycle from a 2-term obstruction theory (or, in higher language, a quasi-smooth derived scheme). I will describe joint work with Jeongseok Oh that produces a virtual cycle from a 3-term symmetric obstruction theory (or, in higher language, a (-2)-shifted symplectic derived scheme). There is also a localisation formula, a K-theoretic refinement, a virtual GRR formula, etc. Earlier Borisov-Joyce used real derived differential geometry to define a real virtual homology class. When their virtual dimension is even we show our class reproduces theirs; when it is odd we show their class is torsion.

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Yukinobu Toda (Kavli IPMU)

Quasi-BPS categories for K3 surfaces

In this talk, I will give semiorthogonal decompositions of derived categories of coherent sheaves on moduli stacks of semistable objects on K3 surfaces. An each summand is given by the categorical Hall product of subcategories called quasi-BPS categories, which approximate the categorification of BPS cohomologies for K3 surfaces. When the weight and the Mukai vector is coprime, the quasi-BPS category is shown to be smooth and proper, with trivial Serre functor etale locally on the good moduli space. So it gives a twisted analogue of categorical crepant resolution of the singular symplectic moduli space, and reminiscents categorical analogue of chi-independence phenomena. This is a joint work in progress with Tudor Padurariu.

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Institut Henri Poincaré