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Gopakumar–Vafa Invariants in Genus 0 and 1

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Gopakumar–Vafa Invariants in Genus 0 and 1 Imperial College London Department of Mathematics Gopakumar–Vafa invariants in genus 0 and 1 Francesca Carocci September 2018 Submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Pure Mathematics of Imperial College London Declaration of Originality I hereby declare that all material in this dissertation which is not my own work has been properly acknowledged. Copyright Declaration The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work. 3 Acknowledgements I would like to thank first and foremost my supervisor Richard Thomas, for the projects he proposed, for all the mathematical suggestions and discussions, for reading the first draft of this thesis, but mostly for never getting tired of asking me if I had computed an example and for eventually managing to make me love computations, especially when they really do not want to work. I am thankful to Cristina Manolache, for all the discussion about virtual classes, and more in general for all the precious support she offered over four years, mathematical and otherwise. I would also like to thank Tom Coates and Alessio Corti for various useful mathemat- ical discussions and for welcoming me in the Fano group meetings. I wish you thank once again Balazs´ Szendroi¨ and Tom Coates for having carefully examined this thesis, and for their useful comments which helped me improve this manuscript. I have been lucky to have met many nice people I shared this experience with, and I thank them all. Some special thanks go to Luca, who was there for the initial panic, but nonetheless bravely decided to collaborate with me and got through it, and to Andrea, for all the good time spent watching Downtown, and for always agreeing that it is best to buy olive oil separately. Finally I would like to thank Zak, who has been by my side in the best part of this unique adventure. 5 Abstract In this thesis we explore two possible approaches to the study of Gopakumar-Vafa invari- ants in genus 0 and 1, mostly concentrating on their relation with certain moduli spaces of morphisms. The work is divided in two parts. In the first part we study via a new technique inspired by Hitchin’s spectral curve construction, a version of Donaldson-Thomas invari- ants which Katz has proposed as a mathematical definition of genus 0 GV invariants. The spectral curve point of view leads us, in the reduced class case, to express the local invariants of a fixed curve in terms of maps from curves homeomorphic to genus 0 ones. At the end of the first part we look at some examples in the non reduced class case and explore the spectral construction techniques in this set up. In the second part of the thesis I discuss a joint work with Luca Battistella and Cristina Manolache. We first recall Li-Zinger definition of reduced genus 1 invariants, which coincide with genus 1 GV for the quintic 3-folds for the reduced class case, and prove that such invariants coincide with those arising from the moduli space of 1-stable maps. In such moduli space elliptic tails are destabilised and replaced by non degenerate maps from cuspidal curves. We think of this as a first step in a longer project, with the aim is that to define alternate compactifications of the moduli space of maps in higher genus whose associated invariants do not take into account degenerate contributions. 7 Contents 1 Picard groups, Higgs sheaves and Katz invariants 19 1.1 Partial normalisations and simultaneous normalisations . 19 1.1.1 Semi-normalisations and Simultaneous normalisations . 20 1.2 Jacobian of singular curves . 21 1.3 Spectral curve construction . 24 1.4 Moduli of stable sheaves and DT invariants . 28 1.4.1 Behrend function and Local invariants . 30 1.4.2 Reduction to curves of geometric genus 0 . 31 2 Contribution of singular curves: reduced cycle case 33 2.1 Characterisation of fixed point of the action and description of the orbits . 33 2.2 Moduli spaces of partial normalisations . 37 2.2.1 One Step Spectral Construction . 42 2.3 Interpretation of the invariants in terms of generalised tree . 43 3 Towards non reduced class contribution 51 3.1 Invariant higher rank sheaves on rational curves with a single node . 53 3.2 Invariant higher rank sheaves on the cusp . 57 4 Moduli space of stable maps in genus 1 and Li-Zinger reduced invariants 61 4.1 Components and Hu-Li equations . 62 4.1.1 Hu-Li equations . 63 4.2 Genus 1 singularities and smoothability . 67 5 Viscardi Alternate compactification and 1-stabilisation map 75 5.1 Definition of the moduli space and components . 75 5.2 1-stabilisation map . 78 6 Reduced vs Cuspidal invariants 85 6.1 The main idea and Chang-Li solution . 85 6.2 1-stable maps with p fields . 87 6.2.1 Moduli of sections . 88 6.2.2 Obstruction theories . 89 9 10 Contents 6.2.3 Cosection localisation and virtual pullback . 91 6.2.4 A cosection for p-fields . 93 6.2.5 From p-fields to the quintic threefold . 94 6.3 Auxiliary space of p-fields . 98 6.4 Local equations and desingularisation . 101 6.4.1 Equations for Zp relative to XP ..................... 101 6.4.2 Hu-Li blow-up and desingularisation . 103 6.5 Splitting the cone and proof of the Main Theorem . 104 6.5.1 Contribution of the main component . 105 Bibliography 113 Introduction Gromov-Witten theory is the first example of the modern approach to enumerative ge- ometry and, as the first deformation invariant theory constructed, it has been intensively studied for more than 20 years. The theory presents several serious difficulties: it is extremely hard to compute, the invariants are rational due to the presence of finite au- tomorphisms of the maps in the case of multiple cover, and in genus g > 0 the different boundary components of the moduli space give degenerate contributions. When X is a smooth projective Calabi-Yau 3-fold, Gopakumar-Vafa conjectured the existence of an integer valued theory underlying Gromov-Witten counting; via M-theoretic methods they defined the BPS invariants ng,β which should count the embedded curves in geometric genus g underlying the GW and should satisfy: 2g 2 X g X ng,β mλ N qβλ2g 2 = 2sin − qmβ; (0.1) β − m 2 β;g β;g;m g where we denoted by Nβ the Gromov-Witten invariants of X. A direct mathematical definition for these invariants in all genus has been recently proposed in [MT16] via the perverse cohomology of the DT sheaf on the moduli space M1,β(X) of pure 1-dimensional stable sheaves F with support in class β H (X;Z) and holomorphic Euler characteristic 2 2 χ(F ) = 1: In this thesis we explore, via different techniques, Gopakumar-Vafa invariants in ge- nus zero and one and their relations with moduli spaces of maps. Katz invariants and spectral curve construction The numerical invariants arising from M1,β(X) were already proposed as mathematical definition of genus 0 BPS invariants by Katz [Kat08]: Z n0,β = 1 = χ(M1,β(X);ν) vir [M1,β (X)] where ν denotes the Behrend function of the moduli space. It is a priori not clear from the definition why these invariants should capture only the genus 0 curves and our main motivation was to understand whether there exists a technique to reduce Katz invariants to genus 0 curves and thus reveal the relation with maps from genus 0 curves. 11 12 Contents We will actually work with local contribution : for a fixed curve i : C X and γ a ! cycle on C satisfying i γ = β; we can define the contributions of C to Katz’s invariants as ∗ n = χ(M (C);ν ): 0,γ 1,γ jM1,γ (C) where M1,γ (C) is the subspace of M1,β(X) of sheaves set theoretically supported on C whose associated fundamental cycle [F ] (see (1.7)) coincide with γ: In this thesis we develop a new technique to study Katz invariants and their localisa- tion to rational curves; the technique is inspired by Hitchin’s spectral curve construction. Before giving an outline of the main idea, we collect the main results. Reduced class case In the reduced case, i.e. γ = [C] and C reduced, we give an interpretation of the invariants in terms of curves homeomorphic to genus 0 curve. We first define (see Definition 2.13) the moduli functor PN m of partial normalisations of C of fixed genus (which is indeed m in the connected case) and show that: m m Theorem. PN is represented by a closed subscheme PN of the Quot scheme QuotC(S ;m) parametrising length m quotient of the zero dimensional sheaf on S ν OC=OC: We denoted ν ∗ by C C the normalisation. −! Let T m PN m PN m denote the locally closed sub-variety parametrising partial ⊆ conn ⊆ normalisations homeomophic to trees of P1: Theorem. Let C be a connected, reduced projective curve of arithmetic genus g and normal- isation N P1: Let C i X denote its embedding in a smooth Calabi-Yau 3-fold X.
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