We generalize the main result of Demailly \cite{D2} for the bundles $E_{k,m}^{GG}(V^*)$ of jet differentials of order $k$ and weighted degree $m$ to the bundles $E_{k,m}(V^*)$ of the invariant jet differentials of order $k$ and weighted degree $m$. Namely, Theorem 0.5 from \cite{D2} and Theorem 9.3 from \cite{D1} provide a lower bound $\frac{c^k}{k}m^{n+kr-1}$ on the number of the linearly independent holomorphic global sections of $E_{k,m}^{GG} V^* \bigotimes \mathcal{O}(-m δA)$ for some ample divisor $A$. The group $G_k$ of local reparametrizations of $(\mathbb{C},0)$ acts on the $k$-jets by orbits of dimension $k$, so that there is an automatic lower bound $\frac{c^k}{k} m^{n+kr-1}$ on the number of the linearly independent holomorphic global sections of $E_{k,m}V^* \bigotimes \mathcal{O}(-m δA)$. We formulate and prove the existence of an asymptotic duality along the fibers of the Green-Griffiths jet bundles over projective manifolds. We also prove a Serre duality for asymptotic sections of jet bundles. An application is also given for partial application to the Green-Griffiths conjecture.
For an abelian category with a Serre duality and a finite group action, we compute explicitly the Serre duality on the category of equivariant objects. Special cases and examples are discussed. In particular, an abelian category with a periodic Serre duality and its equivariantization are studied.
We formulate and prove the existence of an asymptotic duality along the fibers of the Green-Griffiths jet bundles over projective manifolds. The existence of global sections for these bundles and also for their dual sheaves has been established by the author and J. P. Demailly in a previous article, [3].
Let V and F be holomorphic bundles over a complex manifold M, and s be a holomorphic section of V. We study the cohomology associated to the Koszul complex induced by s, and prove a generalized Serre duality theorem for them.
We discuss the naive duality theory of coherent, torsion free, $S_2$ sheaves on schemes. Version 2: A solution of the original Conjecture 6.2 is added as an appendix by Hailong Dao.
Given Y a non-compact manifold or orbifold, we define a natural subspace of the cohomology of Y called the narrow cohomology. We show that despite Y being non-compact, there is a well-defined and non-degenerate pairing on this subspace. The narrow cohomology proves useful for the study of genus zero Gromov-Witten theory. When Y is a smooth complex variety or Deligne-Mumford stack, one can define a quantum D-module on the narrow cohomology of Y. This yields a new formulation of quantum Serre duality.
Let $X$ be a smooth variety or orbifold and let $Z \subseteq X$ be a complete intersection defined by a section of a vector bundle $E \to X$. Originally proposed by Givental, quantum Serre duality refers to a precise relationship between the Gromov--Witten invariants of $Z$ and those of the dual vector bundle $E^\vee$. In this paper we prove a quantum Serre duality statement for quasimap invariants. In shifting focus to quasimaps, we obtain a comparison which is simpler and which also holds for non-convex complete intersections. By combining our results with the wall-crossing formula developed…
Let A be an abelian hereditary category with Serre duality. We provide a classification of such categories up to derived equivalence under the additional condition that the Grothendieck group modulo the radical of the Euler form is a free abelian group of finite rank. Such categories are called numerically finite, and this condition is satisfied by the category of coherent sheaves on a smooth projective variety.
Let E be a locally free, rank n bimodule over a smooth projective scheme X, and let A be the non-commutative symmetric algebra generated by E. We construct an internal Hom functor on the category of graded right A-modules. When E has rank 2, we prove that A is Gorenstein by computing the right derived functors of the internal Hom functor. When X is a smooth projective variety, we use the Gorensteinness of A to prove a version of Serre duality on Proj A, the non-commutative P^1 bundle defined by A.
Two varieties $Z$ and $\widetilde Z$ are said to be related by extremal transition if there exists a degeneration from $Z$ to a singular variety $\overline Z$ and a crepant resolution $\widetilde Z \to \overline Z$. In this paper we compare the genus-zero Gromov--Witten theory of toric hypersurfaces related by extremal transitions arising from toric blow-up. We show that the quantum $D$-module of $\widetilde Z$, after analytic continuation and restriction of a parameter, recovers the quantum $D$-module of $Z$. The proof provides a geometric explanation for both the analytic continuation and re…
We give an interpretation of quantum Serre of Coates and Givental as a duality of twisted quantum D-modules. This interpretation admits a non-equivariant limit, and we obtain a precise relationship among (1) the quantum D-module of X twisted by a convex vector bundle E and the Euler class, (2) the quantum D-module of the total space of the dual bundle E^\vee over X, and (3) the quantum D-module of a submanifold Z\subset X cut out by a regular section of E. When E is the anticanonical line bundle K_X^{-1}, we identify these twisted quantum D-modules with second structure connections with differ…
According to Taubes, the Gromov invariants of a symplectic four-manifold X with b_+ > 1 satisfy the duality Gr(A) = +/- Gr(K-A), where K is Poincare dual to the canonical class. Extending joint work with Simon Donaldson in math.SG/0012067, we interpret this result in terms of Serre duality on the fibres of a Lefschetz pencil, by proving an analogous symmetry for invariants counting sections of associated bundles of symmetric products. Using similar methods we give a new proof of an existence theorem for symplectic surfaces in four-manifolds with b_+ = 1 and b_1 = 0. This reproves another theor…
In this paper we use recently developed calculus of residue currents together with integral formulas to give a new explicit analytic realization, as well as a new analytic proof of Serre duality on any reduced pure $n$-dimensional paracompact complex space $X$. At the core of the paper is the introduction of concrete fine sheaves $\mathscr{B}_X^{n,q}$ of certain currents on $X$ of bidegree $(n,q)$, such that the Dolbeault complex $(\mathscr{B}_X^{n,\bullet},\,\bar{\partial})$ becomes, in a certain sense, a dualizing complex. In particular, if $X$ is Cohen-Macaulay (e.g., Gorenstein or a comple…
The principal aim of this paper is to extend Abel's theorem to the setting of complex supermanifolds of dimension 1|q over a finite-dimensional local supercommutative C-algebra. The theorem is proved by establishing a compatibility of Serre duality for the supercurve with Poincare duality on the reduced curve. We include an elementary algebraic proof of the requisite form of Serre duality, closely based on the account of the reduced case given by Serre in Algebraic Groups and Class Fields, combined with an invariance result for the topology on the dual of the space of repartitions. Our Abel ma…
In this paper we construct arithmetic analogs of the Riemann-Roch theorem and Serre's duality for line bundles. This improves on the works of Tate and van der Geer - Schoof. We define $H^0(L)$ and $H^1(L)$ as some convolution of measures structures. The $H^1$ is defined by a procedure very similar to the usual Cech cohomology. We get Serre's duality as Pontryagin duality of convolution structures. We get separately Riemann-Roch formula and Serre's duality. Instead of using the Poisson summation formula, we basically reprove it. The whole theory is pretty much parallel to the geometric case.
We formulate and prove Serre's equivalence for $\mathbb{Z}$-graded rings. When restricted to the usual case of $\mathbb{N}$-graded rings, our version of Serre's equivalence also sharpens the usual one by replacing the condition that $A$ be generated by $A_1$ over $A_0$ by a more natural condition, which we call the Cartier condition. For $\mathbb{Z}$-graded rings coming from flips and flops, this Cartier condition relates more naturally to the geometry of the flip/flop in question. We also interpret Grothendieck duality as an instance of Greenlees-May duality for graded rings. These form the b…
Given a holomorphic vector bundle $E:EX X$ over a compact Kähler manifold, one introduces twisted GW-invariants of $X$ replacing virtual fundamental cycles of moduli spaces of stable maps $f: Σ\to X$ by their cap-product with a chosen multiplicative characteristic class of $H^0(Σ, f^* E) - H^1(Σ, f^*E)$. Using the formalism of quantized quadratic hamiltonians, we express the descendent potential for the twisted theory in terms of that for $X$. The result (Theorem 1) is a consequence of Mumford's Riemann -- Roch -- Grothendieck formula applied to the universal stable map. When $E$ is concave, a…
For a left coherent ring A with every left ideal having a countable set of generators, we show that the coderived category of left A-modules is compactly generated by the bounded derived category of finitely presented left A-modules (reproducing a particular case of a recent result of Stovicek with our methods). Furthermore, we present the definition of a dualizing complex of fp-injective modules over a pair of noncommutative coherent rings A and B, and construct an equivalence between the coderived category of A-modules and the contraderived category of B-modules. Finally, we define the notio…
We adapt ideas from Ekedahl [Eke84] to prove a Serre-type duality for Witt-divisorial sheaves of $\mathbb Q$-Cartier divisors on a smooth projective variety over a perfect field of finite characteristic. We also explain its relationship to Tanaka's vanishing theorems [Tan20].
We study Serre duality in the singularity category of an isolated Gorenstein singularity and find an explicit formula for the duality pairing in terms of generalised fractions and residues. For hypersurfaces we recover the residue formula of the string theorists Kapustin and Li. These results are obtained from an explicit construction of complete injective resolutions of maximal Cohen-Macaulay modules.
A formula for the first Chern class of the Verlinde bundle over the moduli space of smooth genus g curves is given. A finite-dimensional argument is presented in rank 2 using geometric symmetries obtained from strange duality, relative Serre duality, and Wirtinger duality together with the projective flatness of the Hitchin connection. A derivation using conformal-block methods is presented in higher rank. An expression for the first Chern class over the compact moduli space of curves is obtained.
We prove non-degeneracy of the cohomological bulk and boundary traces for general open-closed Landau-Ginzburg models associated to a pair $(X,W)$, where $X$ is a non-compact complex manifold with trivial canonical line bundle and $W$ is a complex-valued holomorphic function defined on $X$, assuming only that the critical locus of $W$ is compact (but may not consist of isolated points). These results can be viewed as certain "deformed" versions of Serre duality. The first amounts to a duality property for the hypercohomology of the sheaf Koszul complex of $W$, while the second is equivalent wit…
We give a rigorous mathematical proof for the validity of the toric sheaf cohomology algorithm conjectured in the recent paper by R. Blumenhagen, B. Jurke, T. Rahn, and H. Roschy (arXiv:1003.5217). We actually prove not only the original algorithm but also a speed-up version of it. Our proof is independent from (in fact appeared earlier on the arXiv than) the proof by H. Roschy and T. Rahn (arXiv:1006.2392), and has several advantages such as being shorter and cleaner and can also settle the additional conjecture on "Serre duality for Betti numbers" which was raised but unresolved in arXiv:100…
The notion of higher order dual varieties of a projective variety is a natural generalization of the classical notion of projective duality, introduced by Piene in 1983. In this paper we study higher order dual varieties of projective toric embeddings. We compute the degree of the second dual variety of a smooth toric threefold in geometric and combinatorial terms, and we classify smooth 2-jet spanned projective embeddings of smooth threefolds whose second dual variety has dimension less than expected. We also describe the tropicalization of the k-th dual variety of an equivariantly embedded (…
We show a surprising link between singularity theory and the invariant subspace problem of nilpotent operators as recently studied by C. M. Ringel and M. Schmidmeier, a problem with a longstanding history going back to G. Birkhoff. The link is established via weighted projective lines and (stable) categories of vector bundles on those. The setup yields a new approach to attack the subspace problem. In particular, we deduce the main results of Ringel and Schmidmeier for nilpotency degree p from properties of the category of vector bundles on the weighted projective line of weight type (2,3,p), …
Duality for asymptotic invariants of graded families
University of South Alabama（南阿拉巴马大学）；Tulane University（杜兰大学）；University of Education, Hue University（顺化师范大学）；University of Nebraska-Lincoln（内布拉斯加大学林肯分校）
