==== Symplectic versus algebraic geom Sat 28th May to Wed 1st Jul Some abstracts ==== Confirmed speakers: Tom Ducat (Durham), Jonny Evans (Lancaster), Jeffrey Hicks (Edinburgh) Dominic Joyce (Oxford), Baris Kartal (Princeton, online), Jonathan Lai (Imperial) Alex Pieloch (Columbia), Laura Starkston (UC Davis, online), Sara Tukachinsky (Tel Aviv, online), Abigail Ward (MIT, online), Rachel Webb (Berkeley), TBA I and II Andreas Hoering ==== Ilaria Di Dedda (KCL) Title: A symplectic interpretation of the Auslander correspondence ==== Tom Ducat (Durham) Title: Log Calabi--Yau 3-folds with tropicalisation RR^3/G Abstract: The tropicalisation of a log Calabi--Yau variety U is an integral affine manifold with singularities (IAMS) which is a generalisation of the cocharacter space when U is a torus. The quotient spaces RR^2/G, for G one of the four nontrivial finite subgroups of SL(2,ZZ), are all 2-dimensional IAMS which appear naturally in the classification of tropicalisations of log Calabi--Yau surfaces due to Mandel. For the 23 nontrivial finite subgroups G of SL(3,ZZ) I will explain how to construct a family of log Calabi--Yau 3-folds whose tropicalisation is RR^3/G. ==== Jonny Evans (Lancaster) Title: Open problems in symplectic and algebraic geometry Abstract: For several years now, I've been thinking about problems on the interface between symplectic and algebraic geometry. I will give an overview of some of the problems I still can't solve. These range from combinatorial questions about mutations, to topological questions about surfaces of general type, to quantitative questions about symplectic embeddings, to classification problems for degenerations of varieties. ==== Jeffrey Hicks (Edinburgh) Title: Realization in tropical geometry and unobstructedness of Lagrangian submanifolds ==== Dominic Joyce Title: A universal theory of enumerative invariants and wall-crossing formulae Abstract: I outline a (very long and complicated, sorry) programme which gives a common universal structure to many theories of enumerative invariants counting semistable objects in abelian or derived categories in Algebraic Geometry, for example, counting coherent sheaves on curves, surfaces, Fano 3-folds, Calabi-Yau 3- or 4-folds, or representations of quivers (with relations). Write A for your abelian or derived category, K(A) for its numerical Grothendieck group, t for the stability condition, M for the usual moduli stack of objects in A, and M^{pl} for the 'projective linear’ moduli stack of objects modulo "projective linear” isomorphisms (quotient by multiples of identity morphisms). Then (oversimplifying a bit): (i) the homology H_*(M,Q) has the structure of a graded vertex algebra (or a graded vertex Lie algebra in the 3-Calabi-Yau case). (ii) We have H_*(M^{pl},Q) = H_*(M,Q) / D(H_*(M,Q)), where D is the translation operator in the vertex algebra. Therefore H_*(M^{pl},Q) has the structure of a graded Lie algebra. It seems very difficult to understand this Lie bracket without going via the vertex algebra. (iii) For each class a in K(A) we have a moduli stack M_a^{ss}(t) of t-semistable objects in A in class K(A). We can define invariants [M_a^{ss}(t)]_{inv} in H_*(M^{pl}_a,Q). If there are no semistables in class a, this is just the virtual class of M_a^{ss}(t), and is defined over Z. If there are semistables, it is defined over Q, and has a complicated definition involving auxiliary pair invariants. (iv) If t, t* are two stability conditions, there is a universal wall-crossing formula which writes [M_a^{ss}(t*)]_{inv} as a Q-linear combination of repeated Lie brackets of invariants [M_b^{ss}(t)]_{inv}, using the Lie bracket on H_*(M^{pl},Q) from (ii). The programme above is proved for invariants defined using Behrend-Fantechi perfect obstruction theories and virtual classes. I expect to extend it to Calabi-Yau 4-fold obstruction theories and virtual classes, a la Borisov-Joyce / Oh-Thomas. The programme includes “reduced” invariants (for example, counting coherent sheaves on surfaces with p_g > 0), for which the wall-crossing formula is modified. The wall-crossing formulae are effective computational tools in examples. I am currently using them to compute invariants counting semistable sheaves on projective surfaces (algebraic Donaldson invariants) from Seiberg-Witten invariants. Based on: arXiv:2005.05637 (joint with Jacob Gross and Yuuji Tanaka), arXiv:2111.04694, and work in progress. ==== Rachel Webb (Berkeley) Title: Abelianization and quantum Lefschetz for orbifold I-functions Abstract: Let G be a connected reductive group with maximal torus T, and let V and E be two representations of G. Then E defines a vector bundle on the orbifold V//G; let X//G be the zero locus of a regular section. The quasimap I-function of X//G encodes the geometry of maps from PP^1 to X//G and is related to Gromov-Witten invariants of X//G. By directly analyzing these maps from PP^1, we explain how to relate the I-function of X//G to that of V//T. Our formulas validify a mirror symmetry computation of Oneto-Petracci that relates the quantum period of X//G to a certain Laurent polynomial defined by a Fano polytope. Finally, we describe a large class of examples to which our formulas apply, examples that are an orbifold analog of quiver flag varieties. ==== Alex Pieloch (Columbia) Title: Sections and unirulings of families over the projective line Abstract: We will discuss the existence of rational (multi)sections and unirulings for projective families f: X -> CP^1 with at most two singular fibres. In particular, we will discuss two ingredients that are used to construct the above algebraic curves. The first is local symplectic cohomology groups associated to compact subsets of convex symplectic domains. The second is a degeneration to the normal cone argument that allows one to produce closed curves in X from open curves (which are produced using local symplectic cohomology) in the complement of X by a singular fibre. ==== Laura Starkston (UC Davis, online) Title: Comparing complex curves with symplectic surfaces (joint work with Marco Golla) Abstract: I will explain a bit in the talk how we use birational transformations and pseudoholomorphic curves to understand singular symplectic isotopy classifications. We have somewhat ad-hoc birational transformations tailored for explicit examples, but there is some resemblance to the log minimal model program, and I would be enthusiastic to learn more from algebraic geometers about how we may be able to find a more systematic and general procedure. ==== Sara Tukachinsky (Tel Aviv, online) Title: Relative quantum cohomology for complete intersections ==== Irit Huq-Kuruvilla (Berkeley) Title: Euler-theoretic Gromov-Witten invariants Abstract: We introduce a version of Gromov-Witten theory modeled on the ordinary topological Euler characteristic, discuss under what circumstances it can be interpreted geometrically, and show how it can be computed as a limit of a particular modification of quantum K-theory. ==== Warwick Alg Geom seminar, Wed 1st Jun 3:00 pm Andreas Hoering (Nice) Title: Fano manifolds with big tangent bundle Abstract: Let X be a Fano manifold, that is a smooth projective manifold such that the anticanonical bundle det T_X is ample. On the one hand it is well known that the positivity of the anticanonical bundle rarely implies a positivity property for the tangent bundle T_X (for example Mori's theorem tells us that T_X is ample only when X is the projective space). On the other hand, manifolds with a "positive" tangent have a very rich geometry and are therefore particularly interesting. In this talk I will discuss the case of manifolds with big tangent bundle: we will see that there are many pathologies, but that the study of the rational curves makes it possible to show some first classification results. This talk is based on a joint work with Jie Liu. ==== Baris Kartal (Princeton) Title: Algebraic Floer homology sheaves via torus actions on the Fukaya category Abstract: One can deform a compact Lagrangian L in a symplectic manifold M by symplectomorphisms. This lets one to obtain a subset of the moduli of Lagrangians, and analytic coherent sheaves of Floer homology groups. In this talk, we show in the monotone/exact setting that this analytic sheaf is actually algebraic. The main tool for this result is algebraic torus actions on the Fukaya category. We use this result to deduce tameness results for the change of Floer homology groups, and discuss applications to mirror symmetry.