We show that Q-Fano varieties of fixed dimension with anti-canonical degrees and alpha-invariants bounded from below form a bounded family. As a corollary, K-semistable Q-Fano varieties of fixed dimension with anti-canonical degrees bounded from below form a bounded family.

Generalizing the continuous rank function of Barja-Pardini-Stoppino, in this paper we consider cohomological rank functions of Q-twisted (complexes of) coherent sheaves on abelian varieties. They satisfy a natural transformation formula with respect to the Fourier-Mukai-Poincar´e transform, which has several consequences. In many concrete geometric contexts these functions provide useful invariants. We illustrate this with two different applications, the first one to GVsubschemes and the second one to multiplication maps of global sections of ample line bundles on abelian varieties.

For each integer we describe the space of stability conditions on the derived category of the n-dimensional Ginzburg algebra associated to the A2 quiver. The form of our results points to a close relationship between these spaces and the Frobenius-Saito structure on the unfolding space of the A2 singularity.

We introduce the cluster exchange groupoid associated to a non-degenerate quiver with potential, as an enhancement of the cluster exchange graph. In the case that arises from an (unpunctured) marked surface, where the exchange graph is modelled on the graph of triangulations of the marked surface, we show that the universal cover of this groupoid can be constructed using the covering graph of triangulations of the surface with extra decorations. This covering graph is a skeleton for a space of suitably framed quadratic differentials on the surface, which in turn models the space of Bridgeland stability conditions for the 3-Calabi–Yau category associated to the marked surface. By showing that the relations in the covering groupoid are homotopically trivial when interpreted as loops in the space of stability conditions, we show that this space is simply connected.

Let $(V, 0)$ be an isolated hypersurface singularity. We introduce a series of new derivation Lie algebras $L_{k}(V)$ associated to $(V,0)$. Its dimension is denoted as $\lambda_{k}(V)$. The $L_{k}(V)$ is a generalization of the Yau algebra $L(V)$ and $L_{0}(V)=L(V)$. These numbers $\lambda_{k}(V)$ are new numerical analytic invariants of an isolated hypersurface singularity. In this article we compute $L_1(V)$ for fewnomial isolated singularities (Binomial, Trinomial) and obtain the formulas of $L_{1}(V)$. We also formulate a sharp upper estimate conjecture for the $L_k(V)$ of weighted homogeneous isolated hypersurface singularities and we prove this conjecture for large class of singularities. Furthermore, we formulate another conjecture: $\cdots >\lambda^{(i+1)}(V) > \lambda^i(V) \cdots >\lambda^0(V) \geq \lambda_l (V),\; l=1,2,\cdots,n$. We partially prove it for binomial and trinomial singularities.

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No abstract uploaded!

We construct a smooth algebraic stack of tuples consisting of genus two nodal curves, line bundles, and twisted fields. It leads to a desingularization of the moduli of genus two stable maps to projective spaces. The construction is based on systematical application of the theory of stacks with twisted fields (STF), which has its prototype appeared in arXiv:1906.10527 and arXiv:1201.2427 and is fully developed in this article. The results of this article are the second step of a series of works toward the resolutions of the moduli of stable maps of higher genera.

We construct a smooth Artin stack parameterizing the stable weighted curves of genus one with twisted fields and prove that it is isomorphic to the blowup stack of the moduli of genus one weighted curves studied by Hu and Li. This leads to a blowup-free construction of Vakil-Zinger's desingularization of the moduli of genus one stable maps to projective spaces. This construction provides the cornerstone of the theory of stacks with twisted fields, which is thoroughly studied in arXiv:2005.03384 and leads to a blowup-free resolution of the stable map moduli of genus two.

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Let $(V, 0)$ be an isolated hypersurface singularity defined by the holomorphic function $f: (\mathbb{C}^n, 0)\rightarrow (\mathbb{C}, 0)$. In our previous work, we introduced a series of novel Lie algebras associated to $(V,0)$, i.e., $k$-th Yau algebra $L^k(V), k\geq 0$. It was defined to be the Lie algebra of derivations of the $k$-th moduli algebras $A^k(V)= \mathcal{O}_n/(f, m^k J(f)), k\geq 0$, where $m$ is the maximal ideal of $\mathcal{O}_n$. I.e., $L^k(V):=\text{Der}(A^k(V), A^k(V))$. The dimension of $L^k(V)$ was denoted by $\lambda^k(V)$. The number $\lambda^k(V)$, which was called $k$-th Yau number, is a subtle numerical analytic invariant of $(V, 0)$. Furthermore, we formulated two conjectures for these $k$-th Yau number invariants: a sharp upper estimate conjecture of $\lambda^k(V)$ for weighted homogeneous isolated hypersurface singularities (see Conjecture \ref{conj2}) and an inequality conjecture $\lambda^{(k+1)}(V) > \lambda^k(V), k\geq0$ (see Conjecture \ref{conj1}). In this article, we verify these two conjectures when $k$ is small for large class of singularities.

We analyze the asymptotic behavior of certain twisted orbital integrals arising from the study of affine Deligne-Lusztig varieties. The main tools include the Base Change Fundamental Lemma and q-analogues of the Kostant partition functions. As an application we prove a conjecture of Miaofen Chen and Xinwen Zhu, relating the set of irreducible components of an affine Deligne-Lusztig variety modulo the action of the σ-centralizer group to the Mirkovic-Vilonen basis of a certain weight space of a representation of the Langlands dual group.

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No abstract uploaded!

Let Y be the zero loci of a regular section of a convex vector bundle Y over Y . We provide a proof of a conjecture of Cox, Katz and Lee for the virtual class of the genus zero moduli of stable maps to Y . This in turn yields the expected relationship between Gromov-Witten theories of Y and Y which together with Mirror Theorems allows for the calculation of enumerative invariants of Y inside of Y .

We show that the analogue of Hamilton's Ricci flow in the combinatorial setting produces solutions which converge exponentially fast to Thurston's circle packing on surfaces. As a consequence, a new proof of Thurston's existence of circle packing theorem is obtained. As another consequence, Ricci flow suggests a new algorithm to find circle packings.

As a continuation of\lianyaufour, we study modular properties of the periods, the mirror maps and Yukawa couplings for multi-moduli Calabi-Yau varieties. In Part A of this paper, motivated by the recent work of Kachru-Vafa, we degenerate a three-moduli family of Calabi-Yau toric varieties along a codimension one subfamily which can be described by the vanishing of certain Mori coordinate, corresponding to going to the``large volume limit''in a certain direction. Then we see that the deformation space of the subfamily is the same as a certain family of K3 toric surfaces. This family can in turn be studied by further degeneration along a subfamily which in the end is described by a family of elliptic curves. The periods of the K3 family (and hence the original Calabi-Yau family) can be described by the squares of the periods of the elliptic curves. The consequences include:(1) proofs of various conjectural formulas of physicists\vk\lkm involving mirror maps and modular functions;(2) new identities involving multi-variable hypergeometric series and modular functions--generalizing\lianyaufour. In Part B, we study for two-moduli families the perturbation series of the mirror map and the type A Yukawa couplings near certain large volume limits. Our main tool is a new class of polynomial PDEs associated with Fuchsian PDE systems. We derive the first few terms in the perturbation series. For the case of degree 12 hypersurfaces in\P^ 4 [6, 2, 2, 1, 1], in one limit the series of the couplings are expressed in terms of the j function. In another limit, they are expressed in terms of rational functions. The latter give explicit formulas for infinite sequences of``instanton

1. INDEX THEORY, ELLIPTIC CURVES AND LOOP GROUPS One can look at elliptic genus from several different points of view; from index theory, from representation theory of Kac-Moody affine Lie algebras or from the theory of elliptic functions and modular forms. Each of them shows us some quite different interesting features of ellitic genus. On the other hand we can also combine the forces of these three different mathematical fields to derive many interesting results in topology such as rigidity, divisibility and vanishing of topological invariants..

Spin polynomial invariants, as denned by Pidstrigatch and Tyurin, are used to define differentiable invariants for Dolgachev surfaces. In order to define and compute these invariants, some machinery is developed to study the wall structure for Spin polynomials, to deal with the fact that the moduli spaces have the wrong dimension and to handle nonreduced structures. The invariants are then computed.

We survey the theory of complex manifolds that are related to Riemann surface, Hodge theory, Chern class, Kodaira embedding and HirzebruchRiemannRoch, and some modern development of uniformization theorems, KhlerEinstein metric and the theory of DonaldsonUhlenbeckYau on Hermitian YangMills connections. We emphasize mathematical ideas related to physics. At the end, we identify possible future research directions and raise some important open questions.

Let X be a compact Khler manifold and X a subvariety of X with higher co-dimension. The aim is to study complete constant scalar curvature Khler metrics on non-compact Khler manifold X with Poincar--Mok--Yau asymptotic property (see Definition\ref {def}). In this paper, the methods of Calabi's ansatz and the moment construction are used to provide some special examples of such metrics.

In this paper, we study the zero loci of local systems of the form \delta\Pi , where \delta\Pi is the period sheaf of the universal family of CY hypersurfaces in a suitable ambient space \delta\Pi , and \delta\Pi is a given differential operator on the space of sections \delta\Pi . Using earlier results of three of the authors and their collaborators, we give several different descriptions of the zero locus of \delta\Pi . As applications, we prove that the locus is algebraic and in some cases, non-empty. We also give an explicit way to compute the polynomial defining equations of the locus in some cases. This description gives rise to a natural stratification to the zero locus.

From the work of Lian, Liu, and Yau on" Mirror Principle", in the explicit computation of the Euler data Q=\{Q_0, Q_1,...\} for an equivariant concavex bundle Q=\{Q_0, Q_1,...\} over a toric manifold, there are two places the structure of the bundle comes into play:(1) the multiplicative characteric class Q=\{Q_0, Q_1,...\} of Q=\{Q_0, Q_1,...\} one starts with, and (2) the splitting type of Q=\{Q_0, Q_1,...\} . Equivariant bundles over a toric manifold has been classified by Kaneyama, using data related to the linearization of the toric action on the base toric manifold. In this article, we relate the splitting type of Q=\{Q_0, Q_1,...\} to the classifying data of Kaneyama. From these relations, we compute the splitting type of a couple of nonsplittable equivariant vector bundles over toric manifolds that may be of interest to string theory and mirror symmetry. A code in Mathematica that carries out the computation of some of these examples is attached.

Let <i>M</i> be a compact Khler manifold and <i>N</i> be a subvariety with codimension greater than or equal to 2. We show that there are no complete KhlerEinstein metrics on M - N . As an application, let <i>E</i> be an exceptional divisor of <i>M</i>. Then M - N cannot admit any complete KhlerEinstein metric if blow-down of <i>E</i> is a complex variety with only canonical or terminal singularities. A similar result is shown for pairs.

We study various geometrical quantities for CalabiYau varieties realized as cones over Gorenstein Fano varieties, obtained as toric varieties from reflexive polytopes in various dimensions. Focus is made on reflexive polytopes up to dimension 4 and the minimized volumes of the SasakiEinstein base of the corresponding CalabiYau cone are calculated. By doing so, we conjecture new bounds for the SasakiEinstein volume with respect to various topological quantities of the corresponding toric varieties. We give interpretations about these volume bounds in the context of associated field theories via the AdS/CFT correspondence.