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Let X⊂Pr be an integral and non-degenerate variety. Set n:=dim(X). We prove that if the (k+n−1)-secant variety of X has (the expected) dimension (k+n−1)(n+1)−1<r and X is not uniruled by lines, then X is not k-weakly defective and hence the k-secant variety satisfies identifiability, i.e. a general element of it is in the linear span of a unique S⊂X with ♯(S)=k. We apply this result to many Segre-Veronese varieties and to the identifiability of Gaussian mixtures G1,d. If X is the Segre embedding of a multiprojective space we prove identifiability for the k-secant variety (assuming that the (k+n−1)-secant variety has dimension (k+n−1)(n+1)−1<r, this is a known result in many cases), beating several bounds on the identifiability of tensors.

We study the moduli stack of degree 0 semistable G-bundles on an irreducible curve E of arithmetic genus 1, where G is a connected reductive group. Our main result describes a partition of this stack indexed by a certain family of connected reductive subgroups H of G (the E-pseudo-Levi subgroups), where each stratum is computed in terms of bundles on H together with the action of the relative Weyl group. We show that this result is equivalent to a Jordan–Chevalley theorem for such bundles equipped with a framingat a fixed basepoint. In the case where E has a single cusp (respectively, node), this gives a new proof of the Jordan–Chevalley theorem for the Lie algebra g (respectively, group G). We also provide a Tannakian description of these moduli stacks and use it to show that if E is an ordinary elliptic curve, the collection of framed unipotent bundles on E is equivariantly isomorphic to the unipotent cone in G. Finally, we classify the E-pseudo-Levi subgroups using the Borel–de Siebenthal algorithm, and compute some explicit examples.

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This paper studies a notion of enumerative invariants for stable A-branes, and discusses its relation to invariants defined by spectral and exponential networks. A natural definition of stable A-branes and their counts is provided by the string theoretic origin of the topological A-model. This is the Witten index of the supersymmetric quantum mechanics of a single D3 brane supported on a special Lagrangian in a Calabi-Yau threefold. Geometrically, this is closely related to the Euler characteristic of the A-brane moduli space. Using the natural torus action on this moduli space, we reduce the computation of its Euler characteristic to a count of fixed points via equivariant localization. Studying the A-branes that correspond to fixed points, we make contact with definitions of spectral and exponential networks. We find agreement between the counts defined via the Witten index, and the BPS invariants defined by networks. By extension, our definition also matches with Donaldson-Thomas invariants of B-branes related by homological mirror symmetry.

Finite Quot schemes were used by Bertram, Johnson, and the first author to study Le Potier’s strange duality conjecture on del Pezzo surfaces when one of the moduli spaces is the Hilbert scheme of points. In order to rigorously enumerate the finite Quot scheme, we study the moduli space of limit stable pairs in which the target has rank one on a smooth complex projective surface. We obtain an embedding of this moduli space into a smooth space that induces a perfect obstruction theory. This obstruction theory yields a virtual fundamental class that can be computed explicitly. Because the moduli space coincides with the Quot scheme when they have dimension 0, this makes the desired count of the finite Quot scheme in Bertram et al. rigorous. As another application, we obtain a universality result for tautological integrals on the moduli space of stable pairs.

In this paper, we prove that the degree of regularity of square systems, a subfamily of the HFE systems, over a prime finite field of odd characteristic q is exactly q and, therefore, prove that inverting square systems algebraically using Gröbner basis algorithm is exponential, when q = \Omega(n), where n is the number of variables of the system.

Let M and N be two compact complex manifolds. We show that if the tautological line bundle O_{T^∗}M(1) is not pseudo-effective and O_{T^∗}N(1) is nef, then there is no non-constant holomorphic map from M to N. In particular, we prove that any holomorphic map from a compact complex manifold M with RC-positive tangent bundle to a compact complex manifold N with nef cotangent bundle must be a constant map. As an application, we obtain that there is no non-constant holomorphic map from a compact Hermitian manifold with positive holomorphic sectional curvature to a Hermitian manifold with non-positive holomorphic bisectional curvature.

In this paper, we prove that if a compact Kähler manifold X has a smooth Hermitian metric ω such that (T_X,ω) is uniformly RC-positive, then X is projective and rationally connected. Conversely, we show that, if a projective manifold X is rationally connected, then there exists a uniformly RC-positive complex Finsler metric on T_X .

The authors describe the relationships between categories of B-branes in different phases of the non-Abelian gauged linear sigma model. The relationship is described explicitly for the model proposed by Hori and Tong with non-Abelian gauge group that connects two non-birational Calabi-Yau varieties studied by Rødland. A grade restriction rule for this model is derived using the hemisphere partition function and it is used to map B-type D-branes between the two Calabi-Yau varieties.

We provide a uniform construction of “mixed versions” or “graded lifts” in the sense of Beilinson-Ginzburg–Soergel which works for arbitrary Artin stacks. In particular, we obtain a general construction of graded lifts of many categories arising in geometric representation theory and categorified knot invariants. Our new theory associates to each Artin stack of finite type Y over F_q a symmetric monoidal DG-category Shv_{gr,c}(Y) of constructible graded sheaves on Y along with the six-functor formalism, a perverse t-structure, and a weight (or co-t-)structure in the sense of Bondarko and Pauksztello. This category sits in between the category Shv_{m,c}(Y_n) of constructible mixed ℓ-adic sheaves in the sense of Beilinson–Bernstein–Deligne--Gabber for any F_{q^n} -form Y_n of Y and the category Shv_c(Y) of constructible ℓ-adic sheaves on Y, compatible with the six-functor formalism, perverse t-structures, and Frobenius weights. Classically, mixed versions were only constructed in very special cases. However, the category Shv_{gr,c}(Y) agrees with those previously constructed when they are available. For example, for any reductive group G with a fixed pair T ⊂ B of a maximal torus and a Borel subgroup, we have an equivalence of monoidal DG weight categories Shv_{gr,c}(B\G/B) ≃ Ch^b(SBim_W), where Ch^b(SBim_W) is the monoidal DG-category of bounded chain complexes of Soergel bimodules and W is the Weyl group of G.

The theory of virtual fundamental class defines important invariants such as the Gromov–Witten and the Donaldson– Thomas invariants. It has been generalized to the cosection localized virtual cycle which has applications in Seiberg– Witten, Fan–Jarvis–Ruan–Witten and other invariants. In this paper, we prove the formulas of virtual pullback, torus localization and wall crossing for cosection localized virtual cycles.

Given a smooth projective variety X and a smooth divisor D\subset X. We study relative Gromov-Witten invariants of (X,D) and the corresponding orbifold Gromov-Witten invariants of the r-th root stack X_{D,r}. For sufficiently large r, we prove that orbifold Gromov- Witten invariants of X_{D,r} are polynomials in r. Moreover, higher genus relative Gromov-Witten invariants of (X,D) are exactly the constant terms of the corresponding higher genus orbifold Gromov-Witten invari- ants of X_{D,r}. We also provide a new proof for the equality between genus zero relative and orbifold Gromov-Witten invariants, originally proved by Abramovich-Cadman-Wise. When r is sufficiently large and X=C is a curve, we prove that stationary relative invariants of C are equal to the stationary orbifold invariants in all genera.

Given a gauged linear sigma model (GLSM) $\mathcal{T}_{X}$ realizing a projective variety $X$ in one of its phases, i.e. its quantum K\"ahler moduli has a geometric point, we propose an \emph{extended} GLSM $\mathcal{T}_{\mathcal{X}}$ realizing the homological projective dual category $\mathcal{C}$ to $D^{b}Coh(X)$ as the category of B-branes of the Higgs branch of one of its phases. In most of the cases, the models $\mathcal{T}_{X}$ and $\mathcal{T}_{\mathcal{X}}$ are anomalous and the analysis of their Coulomb and mixed Coulomb-Higgs branches gives information on the semiorthogonal/Lefschetz decompositions of $\mathcal{C}$ and $D^{b}Coh(X)$. We also study the models $\mathcal{T}_{X_{L}}$ and $\mathcal{T}_{\mathcal{X}_{L}}$ that correspond to homological projective duality of linear sections $X_{L}$ of $X$. This explains why, in many cases, two phases of a GLSM are related by homological projective duality. We study mostly abelian examples: linear and Veronese embeddings of $\mathbb{P}^{n}$ and Fano complete intersections in $\mathbb{P}^{n}$. In such cases, we are able to reproduce known results as well as produce some new conjectures. In addition, we comment on the construction of the HPD to a nonabelian GLSM for the Pl\"ucker embedding of the Grassmannian $G(k,N)$.

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We introduce the notions of strong local Torelli and T-class for polarized manifolds, and prove that strong local Torelli implies global Torelli theorem on the Torelli spaces for polarized manifolds in the T-class. We discuss many new examples of projective manifolds for which such global Torelli theorem holds. As applications we prove that, in these cases, a canonical completion of the Torelli space is a bounded pseudoconvex domain in complex Euclidean space, and show that generic Torelli implies global Torelli on moduli space with certain level structure.

We give a new construction of (φ,Ĝ )-modules using the theory of prisms developed by Bhatt and Scholze. As an application, we give a new proof about the equivalence between the category of prismatic F-crystals in finite locally free Δ-modules over (K)Δ and the category of lattices in crystalline representations of GK, where K is a complete discretely valued field of mixed characteristic with perfect residue field. We also generalize this result to semi-stable representations using the absolute logarithmic prismatic site defined by Koshikawa.

We prove the representability theorem in derived analytic geometry. The theorem as- serts that an analytic moduli functor is a derived analytic stack if and only if it is compatible with Postnikov towers, has a global analytic cotangent complex, and its truncation is an analytic stack. Our result applies to both derived complex analytic geometry and derived nonarchimedean analytic geometry (rigid analytic geometry). The representability theorem is of both philosophical and prac- tical importance in derived geometry. The conditions of representability are natural expectations for a moduli functor. So the theorem confirms that the notion of derived analytic space is natural and sufficiently general. On the other hand, the conditions are easy to verify in practice. So the theo- rem enables us to enhance various classical moduli spaces with derived structures, thus providing plenty of down-to-earth examples of derived analytic spaces. For the purpose of proof, we study analytification, square-zero extensions, analytic modules and cotangent complexes in the context of derived analytic geometry. We will explore applications of the representability theorem in our subsequent works. In particular, we will establish the existence of derived mapping stacks via the representability theorem.

Using the mirror theorem [CCIT15], we give a Landau-Ginzburg mirror description for the big equivariant quantum cohomology of toric Deligne-Mumford stacks. More precisely, we prove that the big equivariant quantum D-module of a toric Deligne-Mumford stack is isomorphic to the Saito structure associated to the mirror Landau-Ginzburg potential. We give a GKZ-style presentation of the quantum D-module, and a combinatorial description of quantum cohomology as a quantum Stanley-Reisner ring. We establish the convergence of the mirror isomorphism and of quantum cohomology in the big and equivariant setting.

In this article we provide a proof of the Iitaka Cnm conjecture for algebraic fiber spaces over tori.

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.

We construct the Frobenius structure on a rigid connection Be_{\hat G} on G_m for a split reductive group \hat G introduced by Frenkel-Gross. These data form a \hat G-valued overconvergent F-isocrystal Be^†_{\hat G} on G_{m,F_p}, which is the p-adic companion of the Kloosterman \hat G-local system Kl_{\hat G} constructed by Heinloth-Ngô-Yun. By studying the structure of the underlying differential equation, we calculate the monodromy group of Be^†_{\hat G} when \hat G is almost simple (which recovers the calculation of monodromy group of Kl_{\hat G} due to Katz and Heinloth–Ngô–Yun), and prove a conjecture of Heinloth-Ngô-Yun on the functoriality between different Kloosterman \hat G-local systems. We show that the Frobenius Newton polygons of Kl_{\hat G} are generically ordinary for every \hat G and are everywhere ordinary on |G_{m,F_p}| when \hat G is classical or G_2.

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.

Let (X, ) be a pair. We study how the condition (X, ) causes surjectivity or birationality of the Albanese map and the Albanese morphism of (X, ) in both characteristic (X, ) and characteristic (X, ) . In particular in characteristic (X, ) we generalize Kawamata's result to the cases of log canonial pairs, and in characteristic (X, ) we generalize a result of Hacon-Patakfalvi to the cases of log pairs. Moreover we show that if (X, ) is a normal projective threefold in characteristic (X, ) , the coefficients of the components of (X, ) are (X, ) and (X, ) is semiample, then the Albanese morphism of (X, ) is surjective under reasonable assumptions on (X, ) and the singularities of the general fibers of the Albanese morphism. This is a positive characteristic analog in dimension 3 of a result of Zhang on a conjecture of Demailly-Peternell-Schneider.

We use the mirror theorem for toric Deligne-Mumford stacks, proved recently by the authors and by Cheong-Ciocan-Fontanine-Kim, to compute genus-zero Gromov-Witten invariants of a number of toric orbifolds and gerbes. We prove a mirror theorem for a class of complete intersections in toric Deligne-Mumford stacks, and use this to compute genus-zero Gromov-Witten invariants of an orbifold hypersurface.