Time: Tuesday 21:00-22:00 (Beijing Time)
Organizers: Jialong Deng; Akito Futaki
Venue: ZOOM Meeting
Schedule
.Date: 17 Dec., 2024, (Last talk of the Winter Semester)
Tus., 15:30-16:30, Jingzhai 105（静斋 105), Tsinghua University
Speaker: Thomas Schick (Göttingen)
Title: Equivariant T-duality
Abstract: T-duality is a a concept inspired by certain aspects of string theory (the T stand for ``target space''). The physics predicts that one can pass from a torus bundle over a base space, equipped with some background B-field to a ``dual'' torus bundle with dual background B-field without changing the physical content of the theory. Strominger-Yau-Zaslow even propose that mirror symmetry is a special case of T-duality.
To understand the mathematical underpinnings of this duality, the concept of topological T-duality has been introduced. We present the fundamental ideas of this concept: what are the objects we put in duality? what is the precise meaning of ``being dual''? Among the most striking mathematical consequences, we present the T-duality isomorphism between twisted K-theory of the two dual spaces. The last part of the talk will present new developments which concern the investigation of additional symmetries, encoded in the action of a compact group G. We discuss the meaning of ``equivariant T-duality'' and we show that one obtains a T-duality isomorphism in the full Atiyah-Segal equivariant K-theory.
.Date: 17 Dec., 2024 (Canceled and rescheduled to February 25th, 2025)
Speaker: Marco Mazzucchelli (ENS Lyon)
Title: Closed geodesics and the first Betti number
Abstract: In this talk, based on a joint work with Gonzalo Contreras, I will present the following result: on a closed Riemannian manifold of dimension at least two with non-trivial first Betti number, the existence of a minimal closed geodesic, in the sense of Aubry-Mather theory, implies the existence of a transverse homoclinic, and thus of a horseshoe, for the geodesic flow of a suitable -close Riemannian metric. As a corollary of this result and of existing literature, we infer that on any closed manifold of dimension at least two with non-trivial first Betti number, a generic Riemannian metric has infinitely many closed geodesics, and indeed closed geodesics of arbitrarily large length.
Past Talks
.Date: 10 Dec., 2024
Speaker: Ruobing Zhang (UW-Madison)
Title: Geometry of asymptotically hyperbolic Poincaré-Einstein manifolds
Abstract: This talk will summarize recent progress towards understanding some fundamental questions in Poincaré-Einstein manifolds. We will discuss existence, uniqueness, and compactness questions of the Poincaré-Einstein metrics with prescribed conformal infinity.
.Date: 3 Dec., 2024
Speaker: Yifan Chen (Berkeley)
Title: More Complete Calabi-Yau Metrics of Calabi Type
Abstract: We construct more complete Calabi-Yau metrics asymptotic to Calabi ansatz. They are the higher-dimensional analogues of two dimensional ALH* gravitational instantons. Our work builds on and generalizes the results of Tian-Yau and Hein-Sun-Viaclovski-Zhang, creating Calabi-Yau metrics that are only polynomially close to the model space. We also prove the uniqueness of such metrics in a given cohomology class with fixed asymptotic behavior.
.Date: 26 Nov., 2024
Speaker: Lucas Ambrozio (IMPA )
Title: Riemannian manifolds that are determined by their widths
Abstract: A fascinating subject in Geometric Analysis is whether or not a compact Riemannian manifold can be totally reconstructed from the sole knowledge of its Laplace spectrum. Recent developments in the min-max theory of the area functional on the space of codimension one flat cycles made it possible to ask a similar question regarding the area spectrum. We will present some recent work done in collaboration with A. Neves (UChicago) and F. Marques (Princeton) in that direction. The focus will be on the complete characterization of the class of Riemannian metrics on the three dimensional sphere whose four spherical widths are equal. Time permitting, we will also discuss how these ideas can be further developed, yielding the complete characterization of the real projective plane by its length spectrum.
.Date: 19 Nov., 2024
Speaker: Simone Diverio (Sapienza Università di Roma)
Title: A birational version of Gromov's Kähler hyperbolicity
Abstract: In his wonderful book "Shafarevich maps and automorphic forms", J. Kollár asked almost 30 years ago for a "good" birational version of the notion of Kähler hyperbolicity introduced by M. Gromov in the early '90s. Kähler hyperbolic manifolds are compact Kähler manifold admitting a Kähler form whose pull-back to the universal cover becomes d-exact and moreover with a bounded primitive. Such manifolds are of general type (more than this: with ample canonical bundle), Kobayashi hyperbolic, and with large fundamental group.
We shall report on the recently intruduced class of weakly Kähler hyperbolic manifolds which indeed provides such a birational generalization. Among other things, we shall explain that these manifolds are of general type, satisfy a precise quantitative version of the Green-Griffiths conjecture and have generically arbitrarily large fundamental group. Moreover, their properties permit to verify Lang's conjecture for Kähler hyperbolic manifolds. This is a joint work with F. Bei, B. Claudon, P. Eyssidieux, and S. Trapani.
.Date: 12 Nov., 2024
Speaker: Lars Martin Sektnan (University of Gothenburg)
Title: Constant scalar curvature Kähler metrics and semistable vector bundles
Abstract: A central question in Kähler geometry is if a Kähler manifold admits a canonical metric, such as a Kähler-Einstein metric or more generally a constant scalar curvature Kähler (cscK) metric, in a given Kähler class. The Yau-Tian-Donaldson conjecture predicts that this is equivalent to an algebraic notion of stability. In this talk, I will discuss a necessary and sufficient condition for the projectivisation of a slope semistable vector bundle to admit cscK metrics in adiabatic classes, when the base admits a cscK metric. In particular, this shows that the existence of cscK metrics is equivalent to K-stability in this setting. Moreover, our construction reduces K-stability to a finite dimensional criterion in terms of intersection numbers associated to the vector bundle. This is joint work with Annamaria Ortu.
.Date: 5 Nov., 2024
Speaker: Ronan Conlon (University of Texas at Dallas)
Title: A family of Kahler flying wing steady Ricci solitons
Abstract: Steady Kahler-Ricci solitons are eternal solutions of the Kahler-Ricci flow. I will present new examples of such solitons with strictly positive sectional curvature that live on C^n and provide an answer to an open question of H.-D. Cao in complex dimension n>2. This is joint work with Pak-Yeung Chan and Yi Lai.
.Date: 29 Oct., 2024
Speaker: Yusuke Kawamoto (ETH)
Title: Donaldson divisors and quantitative Floer homology
Abstract: I will explain a comparison of quantitative information of Floer homologies of a symplectic manifold and a Donaldson divisor therein, which leads to a solution to a question of Borman posed in 2012.
.Date: 22 Oct., 2024
Speaker: Rohil Prasad (Berkeley)
Title: Low-action pseudoholomorphic curves and invariant sets.
Abstract: Punctured pseudoholomorphic curves have been a key tool in Hamiltonian dynamics since a breakthrough 1993 paper by Hofer. I'll describe a new compactness theorem for sequences of punctured pseudoholomorphic curves that holds without the traditional technical assumption of "bounded Hofer energy". In the limit, one extracts a family of compact invariant subsets that are invariant under the Hamiltonian flow. This has various dynamical applications. I will briefly mention these applications, but I will mostly focus on the geometric analysis behind the compactness theorem. Most of the talk is based on joint work with Dan Cristofaro-Gardiner.
.Date: 15 Oct., 2024
Speaker: Bram Mesland（Universiteit Leiden)
Title: Curvature and Weitzenbock formulae for spectral triples
Abstract: In this talk I will present a new, operator theoretic construction of the Levi-Civita connection on a Riemannian manifold, based on the two projection problem in Hilbert modules. The construction allows us to deduce the mild technical assumptions needed for the existence and uniqueness of the Levi-Civita connection on the module of differential 1-forms over a noncommutative algebra. In particular the construction applies to a large class of noncommutative manifolds (spectral triples). The well-known algebraic theory of curvature for bimodule connections can then be applied to derive a Weitzenbock formula. Examples include toric non commutative manifolds and the Podles quantum sphere. This is joint work with Adam Rennie.
.Date: 8 Oct., 2024
Speaker: Yueqing Feng (Berkeley)
Title: A gluing construction of constant scalar curvature Kähler metrics of Poincaré type
Abstract: In this talk, we construct new examples of constant scalar curvature Kähler(cscK) metrics of Poincaré type from existing cscK ones. The construction is obtained via gluing a cscK metric on a compact Kähler manifold to a complete scalar-flat Kähler metric of Poincaré type on $\mathbb{C}^n$ removing the origin. Assuming the compact Kähler manifold has no non-trivial holomorphic vector field, we prove the existence of cscK metrics of Poincaré type on this compact manifold removing finitely many points.
(Tsinghua's vacation dates include Sept. 17 and Oct. 1, 2024.)
.Date: 24 Sept., 2024
Speaker: Sébastien Picard (UBC)
Title: Non-Kahler Degenerations of Calabi-Yau Threefolds
Abstract: It was proposed in the works of Clemens, Reid, Friedman to connect Calabi-Yau threefolds with different topologies by a process which degenerates 2-cycles and introduces new 3-cycles. This operation may produce a non-Kahler complex manifold with trivial canonical bundle. In this talk, we will discuss the geometrization of this process by special non-Kahler metrics. This talk will survey joint works with T.C. Collins, B. Friedman, C. Suan, and S.-T. Yau.
.Date: 10 Sept., 2024
Speaker: Vamsi Pritham Pingali (Indian Institute of Science)
Title: Abelian and non-abelian gravitating vortices
Abstract: The gravitating vortex (GV) equations are a dimensional reduction of the Kahler-Yang-Mills-Higgs equations to a Riemann surface. The Abelian GV equations contain as a special case, the Einstein-Bogomol'yni equations modelling (the as of now, hypothetical) cosmic strings in the Bogomolo'yni phase in the early universe. The non-abelian GV equations are expected to play a similar role for Electroweak cosmic strings. Moreover, these equations might have interesting moduli spaces and arise out of a moment map interpretation. I shall describe our existence, uniqueness, and obstruction (including a Donaldson-Uhlenbeck-Yau-Kobayashi-Hitchin correspondence) results (joint with L. Alvarez-Consul, M. Garcia-Fernandez, O. Garcia-Prada, and C. Yao) for Abelian GV equations, and my recent existence result for the non-abelian case.
Remark (by Deng): This talk is based on arXiv:2404.18103
.Date: 11 Jun., 2024 (Last talk of the Spring Semester)
Speaker: Matthew Tointon (University of Bristol)
Title: Sharp bounds in a finitary version of Gromov’s polynomial-growth theorem
Abstract: A famous theorem of Gromov states that if a finitely generated group G has polynomial growth (i.e. there is a polynomial p such that the ball of radius n in some Cayley graph of G always contains at most p(n) vertices) then G has a nilpotent subgroup of finite index. Breuillard, Green and Tao proved a finitary refinement of this theorem, stating that if *some* ball of radius n contains at most eps n^d vertices then that ball contains a normal subgroup H such that G/H has a nilpotent subgroup with index at most f(d), where f is some non-explicit function. I will describe joint work with Romain Tessera in which we obtain explicit and even optimal bounds on both the index and the dimension of this nilpotent subgroup. I will also describe an application to random walks on groups.
Remark (by Deng): This talk is based on arXiv:2403.02485
.Date: 4 Jun., 2024
Speaker: Alex Mramor (University of Copenhagen)
Title: The mean curvature flow, singularities, and entropy
Abstract: The mean curvature flow is an example of a geometric flow, where in this case one deforms a submanifold according to its mean curvature vector. Like many such flows though the mean curvature flow will develop singularities, where the flow “pinches.” The entropy, in the sense of Colding and Minicozzi, is an interesting area-like monotone quantity under the flow, for one because it can constrain what sorts of singularity models may arise, and has played an important role in many recent developments. In this talk after introducing the relevant notions we’ll discuss some of these results, including some joint work with S. Wang.
.Date: 28 May, 2024
Speaker: Michael Alexander Hallam (Aarhus University）
Title: Stability of weighted extremal manifolds through blowups
Abstract: The weighted extremal Kähler metrics introduced by Lahdili provide a vast generalisation of Calabi's extremal Kähler metrics, encompassing many examples of canonical metrics in geometry. In this talk, I will give a quick introduction to these metrics, and discuss the proof that weighted extremal manifolds are relatively weighted K-polystable, in a suitable sense. The proof is along the lines of Stoppa--Szekelyhidi's argument that an extremal polarised manifold is relatively K-polystable, which in particular exploits the existence of an extremal metric on the blowup of an extremal manifold at a suitable point.
.Date: 21 May, 2024 (No Talks)
.Date: 14 May, 2024
Speaker: Shaoyun Bai (Columbia University)
Title: Infinitude of Hamiltonian periodic orbits and GLSM
Abstract: Take an irrational rotation of the two-sphere; it only has the north and south poles as its periodic points. However, Franks proved that for any area-preserving diffeomorphism of the two-sphere, if it has more than two fixed points, then it must have infinitely many periodic points. I will discuss a generalization of this result with Guangbo Xu to all compact toric manifolds, for which the gauged linear sigma model (GLSM) plays a surprising role.
.Date: 7 May, 2024
Speaker: Jesús A. Álvarez López (University of Santiago de Compostela (USC))
Title: A trace formula for foliated flows
Abstract: In the lecture, I will try to explain the ideas of a recent paper on the trace formula for foliated flows, written in collaboration with Yuri Kordyukov and Eric Leichtnam. Let $\mathcal{F}$ be a transversely oriented foliation of codimension one on a closed manifold $M$, and let $\phi=\{\phi^t\}$ be a foliated flow on $(M,\mathcal{F})$ (it maps leaves to leaves). Assume the closed orbits of $\phi$ are simple and its preserved leaves are transversely simple. In this case, there are finitely many preserved leaves, which are compact. Let $M^0$ denote their union, and write $M^1=M\setminus M^0$ and $\mathcal{F}^1=\mathcal{F}|_{M^1}$. We consider two locally convex Hausdorff spaces, $I(\mathcal{F})$ and $I'(\mathcal{F})$, consisting of the leafwise currents on $M$ that are conormal and dual-conormal to $M^0$, respectively. They become topological complexes with the differential operator $d_{\mathcal{F}}$ induced by the de~Rham derivative on the leaves, and they have an $\mathbb{R}$-action $\phi^*=\{\phi^{t\,*}\}$ induced by $\phi$. Let $\bar H^\bullet I(\mathcal{F})$ and $\bar H^\bullet I'(\mathcal{F})$ denote the corresponding leafwise reduced cohomologies, with the induced $\mathbb{R}$-action $\phi^*=\{\phi^{t\,*}\}$. The spaces $\bar H^\bullet I(\mathcal{F})$ and $\bar H^\bullet I'(\mathcal{F})$ are shown to be the central terms of short exact sequences in the category of continuous linear maps between locally convex spaces, where the other terms are described using Witten's perturbations of the de~Rham complex on $M^0$ and leafwise Witten's perturbations for $\mathcal{F}^1$. This is used to define some kind of Lefschetz distribution $L_{\rm dis}(\phi)$ of the actions $\phi^*$ on both $\bar H^\bullet I(\mathcal{F})$ and $\bar H^\bullet I'(\mathcal{F})$, whose value is a distribution on $\mathbb{R}$. Its definition involves several renormalization procedures, the main one is the b-trace of some smoothing b-pseudodifferential operator on the compact manifold with boundary obtained by cutting $M$ along $M^0$. We also prove a trace formula describing $L_{\rm dis}(\phi)$ in terms of infinitesimal data from the closed orbits and preserved leaves. This solves a conjecture of C.~Deninger involving two leafwise reduced cohomologies instead of a single one.
Remark (by Deng): This talk is based on arXiv:2402.06671
.Date: 30 Apr., 2024
Speaker: Antonio Trusiani (Chalmers University of Technology）
Title: Singular cscK metrics on smoothable varieties
Abstract: We extend the notion of cscK metrics to singular varieties. We establish the existence of these canonical metrics on Q-Gorenstein smoothable klt varieties when the Mabuchi functional is coercive, these arise as a limit of cscK metrics on close-by fibres. The proof relies on developing a novel strong topology of pluripotential theory in families and establishing uniform estimates for cscK metrics. A key point is the lower semicontinuity of the coercivity threshold of Mabuchi functional along degenerate families of normal compact Kähler varieties with klt singularities. The latter suggests the openness of (uniform) K-stability for general polarized families of normal projective varieties. This is a joint work with Chung-Ming Pan and Tat Dat Tô.
Remark (by Deng): This talk is based on arXiv:2312.13653
.Date: 23 Apr., 2024
Speaker: Giovanni Gentili (Università di Torino )
Title: Special metrics in hypercomplex Geometry
Abstract: The existence and search of special metrics plays a remarkably important role in Differential Geometry. After a brief overview of basic definitions and facts in hypercomplex Geometry, we will discuss certain notions of special metrics in the hypercomplex setting, focusing on the consequences of their existence. We will also introduce an Einstein condition for hyperhermitian metrics and describe the similarities with the Kähler-Einstein case. The talk is based on a joint work with Elia Fusi.
Remark (by Deng): This talk is based on arXiv:2401.13056
.Date: 16 Apr., 2024
Speaker: Simion Filip (University of Chicago)
Title: The volume of a divisor and cusp excursions of geodesics in hyperbolic manifolds
Abstract: The volume of a divisor on an algebraic variety measures the growth rate of the dimension of the space of sections of tensor powers of the associated line bundle. In the case of certain Calabi-Yau N-folds possessing a large group of pseudo-automorphisms, we show that the behavior of the volume can be highly oscillatory as the divisor class approaches the boundary of the pseudo-effective cone. This is explained by relating the volume function to the dynamical behavior of geodesics on certain hyperbolic manifolds.
After providing an introduction and some context for the above notions, I will discuss some of the ideas that go into the proof.
(joint work with John Lesieutre and Valentino Tosatti)
Remark (by Deng): This talk is based on arXiv:2312.01012
.Date: 9 Apr., 2024
Speaker: Charles Cifarelli (Stony Brook)
Title: Non-compact Kähler-Ricci solitons, toric fibrations, and weighted K-stability
Abstract: In recent years, it has been shown that various problems in Kähler geometry can be unified under the framework of the weighted cscK equation, introduced by Lahdili. One feature of this setup is that, if a given manifold Y admits a special kind of fibration structure with toric fiber M and base B, then many interesting equations for the metric on Y can be reduced to the weighted cscK equation on M. This can be thought of as a generalized form of the Calabi Ansatz, which one recovers by taking M = \mathbb{C}. I will present on recent work extending some of this picture to the non-compact setting, with particular attention on (shrinking) Kähler-Ricci solitons. In particular, if a non-compact toric manifold M admits a weighted cscK metric (for suitable choices of weights), then it must be weighted K-stable. I will explain the relationship with Kähler-Ricci solitons, and time permitting I will explain how this leads to a simple proof of a well-known result of Futaki-Wang on the existence of shrinking Kähler-Ricci solitons on the total space of certain line bundles over a compact Kähler-Einstein Fano base.
.Date: 2 Apr., 2024
Speaker: Tomoyuki Hisamoto (Tokyo Metropolitan University)
Title: Quantization of the Kähler-Ricci flow, the entropy, and the optimal degeneration for a Fano manifold
Abstract: In recent years the study of K-unstable Fano manifolds atracts people's attention. In this talk I will introduce the geometric quantization of the Kähler-Ricci flow and the associated entropy functional introduced by Weiyong He. The "quantized entropy" coincides with the terminology in the quantum information theory. We also show some convergence results and discuss about the finite-dimensional analogue of the optimal degeneration.
Remark (by Deng): This talk is based on arXiv:2401.01153
.Date: 26 Mar., 2024
Speaker: Jian Song (Rutgers University)
Title: Minimal slopes for complex Hessian type equations
Abstract: The existence of smooth solutions to a broad class of complex Hessian type equations is related to nonlinear Nakai type criteria on intersection numbers on Kahler manifolds. Such a Nakai criteria can be interpreted as a slope stability condition analogous to the slope stability for Hermitian vector bundles over Kahler manifolds. In the present work, we initiate a program to find canonical solutions to such equations in the unstable case when the Nakai criteria fails. Conjecturally such solutions should arise as limits of natural parabolic flows and should be minimisers of the corresponding moment-map energy functionals. We implement our approach for the J-equation and the deformed Hermitian Yang-Mills equation on surfaces and some examples with symmetry. We further present the bubbling phenomena for the J-equation by constructing minimizing sequences of the moment-map energy functionals.
Remark (by Deng): This talk is based on arXiv:2312.03370
.Date: 19 Mar., 2024
Speaker: Yaxiong Liu (University of Maryland)
Title: The eigenvalue problem of complex Hessian operators
Abstract: In a very recent pair of nice papers of Badiane and Zeriahi, they consider the eigenvalue problem of complex Monge-Ampere and complex Hessian, and show that the C^{1,\bar{1}}-regularity of eigenfunction for MA and C^alpha-regularity for complex Hessian. They posed a question about the C^{1,1}-regularity. We give a positive answer and show the C^{1,1}-regularity and uniqueness of the eigenfunction. We also derive a number of applications, including a bifurcation-type theorem and geometric bounds for the eigenvalue. This is a joint work with Jianchun Chu and Nicholas McCleerey.
Remark (by Deng): This talk is based on arXiv:2402.03098
.Date: 12 Mar., 2024 (Additional talks)
Speaker: Thorsten Hertl (Albert-Ludwigs-Universität Freiburg) 19:30-20:00 (Beijing Time)
Title: Moduli Spaces of Positive Curvature Metrics
Abstract: Besides the space of positive scalar curvature metrics, various moduli spaces have gained a lot of attention. Among those, the observer moduli space arguably has the best behaviour from a homotopy-theoretical perspective because the subgroup of observer diffeomorphisms acts freely on the space of Riemannian metrics if the underlying manifold is connected.
In this talk, I will present how to construct non-trivial elements in the second homotopy of the observer moduli space of positive scalar curvature metrics for a large class for four-manifolds. I will further outline how to adapt this construction to produce the first non-trivial elements in higher homotopy groups of the observer moduli space of positive sectional curvature metrics on complex projective spaces.
Remark (by Deng): This talk is based on arXiv:2310.14115
Speaker: Zhicheng Han (Georg-August-Universität) 20:00-20:30 (Beijing Time)
Title: Spectra of Lie groups and application to L^2-invariants
Abstract: In this talk, I will explore the Laplace operator and Dirac operator on semisimple Lie groups. While the parallel problem on symmetric spaces has been well-studied in the last century, the corresponding problem is much less understood in general homogeneous spaces. We will examine the obstacles in extending existing techniques and discuss how some of them can be resolved in the case of group manifolds. Towards the end, we will see how the spectra data shall aid in computing certain topological L^2-invariants.
Remark (by Deng): This talk is based on Han's thesis.
.Date: 12 Mar., 2024 21:00-22:00 (Beijing Time)
Speaker: Vyacheslav Lysov (London Institute for Mathematical Sciences)
Title: Chern-Gauss-Bonnet theorem via BV localization
Abstract: I will give a brief introduction to supersymmetric localization, BV formalism, and BV localization. I will show that the Euler class integral is a partition function for a zero-dimensional field theory with on-shell supersymmetry. I will describe the partition function as a BV integral, and deform the Lagrangian sub-manifold to evaluate the same integral as a sum over critical points for the Morse function.
Remark (by Deng): This talk is based on arXiv:2402.09162
Date: 5 Mar., 2024
Speaker: Ruadhaí Dervan (University of Glasgow)
Title: The universal structure of moment maps in complex geometry
Abstract: Much of complex geometry is motivated by linking the existence of solutions to geometric PDEs (producing "canonical metrics") to stability conditions in algebraic geometry. I will discuss a more basic question: what is the recipe to actually produce interesting geometric PDEs in complex geometry? The construction will be geometric, using a combination of universal families and tools from equivariant differential geometry. This is joint work with Michael Hallam.
Remark (by Deng): This talk is based on arXiv:2304.01149
Date: 27 Feb., 2024
Speaker: Xin Wang (KIAS)
Title: 5D Wilson Loops and Topological Strings on Fano-threefolds
Abstract: Geometric engineering provides a rich class of 5D supersymmetric gauge theories with eight supercharges, arising from M-theory compactification on non-compact Calabi-Yau threefolds. The counting of BPS states in the low-energy gauge theory is determined by the degeneracies of M2-branes wrapping holomorphic two-cycles in the Calabi-Yau threefold X. These degeneracies can also be calculated from the (refined) topological strings on the same manifold X. In this talk, I will explore the BPS spectrum of the 5D gauge theory with the insertion of a half-BPS Wilson loop operator. Utilizing M-theory realization, we derive the BPS expansion of the expectation value of the Wilson loop operator in terms of BPS sectors. It is found that the BPS sectors can be realized and computed from topological string theories. In the unrefined limit, the BPS sectors act as generating functions for Gromov-Witten invariants on compact (semi)-Fano threefolds constructed from X. We further conjecture a new refined holomorphic anomaly equations for the generating functions of the Wilson loops. These equations can completely solve the refined BPS invariants of Wilson loops for local \mathbb{P}^2 and local \mathbb{P}^1\times\mathbb{P}^1.