In this article we explicitly compute the number of maximal subbundles of rank $k$ of a generically stable bundle of rank $r$ and degree $d$ over a smooth projective curve $C$ of genus $g\ge 2$ over $\C$, when the dimension of the quot scheme of maximal subbundles is zero. Our method is to describe the this number purely in terms of the Gromov invariants of the Grassmannian and then use the formula of Vafa and Intriligator to compute them.
Let $C$ be a smooth projective curve and $W$ a symplectic bundle over $C$. Let $LQ_e (W)$ be the Lagrangian Quot scheme parametrizing Lagrangian subsheaves $E \subset W$ of degree $e$. We give a closed formula for intersection numbers on $LQ_e (W)$. As a special case, for $g \ge 2$, we compute the number of Lagrangian subbundles of maximal degree of a general stable symplectic bundle, when this is finite. This is a symplectic analogue of Holla's enumeration of maximal subbundles in [13].
We define the Gromov-Witten invariants for the parabolic bundles over an orbifold $C$ in various situation. Those bring us to refine this notion to get an accurate computation of the number of maximal subbundles of a sufficiently general parabolic bundle by means of the Intriligator-Varfa formula.
Let $E$ be a rank 2, degree $d$ vector bundle over a genus $g$ curve $C$. The loci of stable pairs on $E$ in class $2[C]$ fixed by the scaling action are expressed as products of $\Quot$ schemes. Using virtual localization, the stable pairs invariants of $E$ are related to the virtual intersection theory of $\Quot E$. The latter theory is extensively discussed for an $E$ of arbitrary rank; the tautological ring of $\Quot E$ is defined and is computed on the locus parameterizing rank one subsheaves. In case $E$ has rank 2, $d$ and $g$ have opposite parity, and $E$ is sufficiently generic, it is…
A symplectic bundle over an algebraic curve has a natural invariant $\sLag$ determined by the maximal degree of its Lagrangian subbundles. This can be viewed as a generalization of the classical Segre invariants of a vector bundle. We give a sharp upper bound on $\sLag$ which is analogous to the Hirschowitz bound on the classical Segre invariants. Furthermore, we study the stratifications induced by $\sLag$ on moduli spaces of symplectic bundles, and get a full picture for the case of rank four.
An orthogonal bundle over a curve has an isotropic Segre invariant determined by the maximal degree of a Lagrangian subbundle. This invariant, and the induced stratifications on moduli spaces of orthogonal bundles, were studied for bundles of even rank in [4]. In this paper, we obtain analogous results for bundles of odd rank. We obtain a sharp upper bound on the isotropic Segre invariant. We show the irreducibility of the induced strata on the moduli spaces of orthogonal bundles of odd rank, and compute their dimensions. As a key ingredient of the proofs, we study the correspondence between L…
Let Gr(2, E) be the Grassmann bundle of two-planes associated to a general bundle E over a curve X. We prove that an embedding of Gr(2, E) by a certain twist of the relative Plücker map is not secant defective. This yields a new and more geometric proof of the Hirschowitz-type bound on the Lagrangian Segre invariant for orthogonal bundles over X, analogous to those given for vector bundles and symplectic bundles in [2, 3]. From the non-defectivity we also deduce an interesting feature of a general orthogonal bundle over X, contrasting with the classical and symplectic cases: Any maximal Lagran…
We propose an intersection-theoretic method to reduce questions in genus zero logarithmic Gromov-Witten theory to questions in the Gromov-Witten theory of smooth pairs, in the presence of positivity. The method is applied to the enumerative geometry of rational curves with maximal contact orders along a simple normal crossings divisor and to recent questions about its relationship to local curve counting. Three results are established. We produce counterexamples to the local/logarithmic conjectures of van Garrel-Graber-Ruddat and Tseng-You. We prove that a weak form of the conjecture holds for…
Suppose N is an affine SL(2,R)-invariant submanfold of the moduli space of pairs (M,w) where M is a curve, and w is a holomorphic 1-form on M. We show that the Forni bundle of N (i.e. the maximal SL(2,R)-invariant isometric subbundle of the Hodge bundle of N) is always flat and is always orthogonal to the tangent space of N. As a corollary, it follows that the Hodge bundle of N is semisimple.
We derive a recursive formula for certain relative Gromov-Witten invariants with maximal tangency condition via the Witten-Dijkgraaf-Verlinde-Verlinde equation. For certain relative pairs, we get explicit formulae of invariants using the recursive formula.
This contribution to the 2015 AMS Summer Institute in Algebraic Geometry (Salt Lake City) announces a general mirror construction. This construction applies to log Calabi-Yau pairs (X,D) with maximal boundary D or to maximally unipotent degenerations of Calabi-Yau manifolds. The new ingredient is a notion of "punctured Gromov-Witten invariant", currently in progress with Abramovich and Chen. The mirror to a pair (X,D) is constructed as the spectrum of a ring defined using the punctured invariants of (X,D). An analogous construction leads to mirrors of Calabi-Yau manifolds. This can be viewed a…
We prove a simple equivalence between the virtual count of rational curves in the total space of an anti-nef line bundle and the virtual count of rational curves maximally tangent to a smooth section of the dual line bundle. We conjecture a generalization to direct sums of line bundles.
We establish a correspondence between the disk invariants of a smooth toric Calabi-Yau 3-fold $X$ with boundary condition specified by a framed Aganagic-Vafa outer brane $(L, f)$ and the genus-zero closed Gromov-Witten invariants of a smooth toric Calabi-Yau 4-fold $\widetilde{X}$, proving the open/closed correspondence proposed by Mayr and developed by Lerche-Mayr. Our correspondence is the composition of two intermediate steps: $\circ$ First, a correspondence between the disk invariants of $(X,L,f)$ and the genus-zero maximally-tangent relative Gromov-Witten invariants of a relative Calabi-Y…
We prove a higher genus version of the genus $0$ local-relative correspondence of van Garrel-Graber-Ruddat: for $(X,D)$ a pair with $X$ a smooth projective variety and $D$ a nef smooth divisor, maximal contact Gromov-Witten theory of $(X,D)$ with $λ_g$-insertion is related to Gromov-Witten theory of the total space of $\mathcal{O}_X(-D)$ and local Gromov-Witten theory of $D$. Specializing to $(X,D)=(S,E)$ for $S$ a del Pezzo surface or a rational elliptic surface and $E$ a smooth anticanonical divisor, we show that maximal contact Gromov-Witten theory of $(S,E)$ is determined by the Gromov-Wit…
We study the isotropic Quot schemes $IQ_e (V)$ parameterizing degree $e$ isotropic subsheaves of maximal rank of an orthogonal bundle $V$ over a curve. The scheme $IQ_e (V)$ contains a compactification of the space $IQ^o_e (V)$ of degree $e$ maximal isotropic subbundles, but behaves quite differently from the classical Quot scheme, and the Lagrangian Quot scheme in [6]. We observe that for certain topological types of $V$, the scheme $IQ_e (V)$ is empty for all $e$. In the remaining cases, for infinitely many $e$ there are irreducible components of $IQ_e (V)$ consisting entirely of nonsaturate…
We use techniques from Gromov-Witten theory to construct new invariants of matroids taking value in the Chow groups of spaces of rational curves in the permutohedral toric variety. When the matroid is realizable by a complex hyperplane arrangement, our invariants coincide with virtual fundamental classes used to define the logarithmic Gromov-Witten theory of wonderful models of arrangement complements, for any logarithmic structure supported on the wonderful boundary. When the boundary is empty, this implies that the quantum cohomology ring of a hyperplane arrangement's wonderful model is a co…
A great number of theoretical results are known about log Gromov-Witten invariants, but few calculations are worked out. In this paper we restrict to surfaces and to genus 0 stable log maps of maximal tangency. We ask how various natural components of the moduli space contribute to the log Gromov-Witten invariants. The first such calculation by Gross-Pandharipande-Siebert deals with multiple covers over rigid curves in the log Calabi-Yau setting. As a natural continuation, in this paper we compute the contributions of non-rigid irreducible curves in the log Calabi-Yau setting and that of the u…
For a certain maximal unipotent family of Abelian varieties over the punctured disc, we show that after a base change, one can complete the family over a disc such that the whole degeneration can be simultaneously balanced embedded into a projective space by the theta functions. Then we study the relationship between the balanced filling-in and the Gromov-Hausdorff limit of flat Kähler metrics on the nearby fibers.
We prove a conjecture of Bousseau, van Garrel and the first-named author relating, under suitable positivity conditions, the higher genus maximal contact log Gromov-Witten invariants of Looijenga pairs to other curve counting invariants of Gromov-Witten/Gopakumar-Vafa type. The proof consists of a closed-form $q$-hypergeometric resummation of the quantum tropical vertex calculation of the log invariants in presence of infinite scattering. The resulting identity of $q$-series appears to be new and of independent combinatorial interest.
We show that the elliptic genus of the higher rank E-strings can be computed based solely on the genus 0 Gromov-Witten invariants of the corresponding elliptic geometry. To set up our computation, we study the structure of the topological string free energy on elliptically fibered Calabi-Yau manifolds both in the unrefined and the refined case, determining the maximal amount of the modular structure of the partition function that can be salvaged. In the case of fibrations exhibiting only isolated fibral curves, we show that the principal parts of the topological string partition function at gi…
We confirm a conjecture of Guth concerning the maximal number of $δ$-tubes, with $δ$-separated directions, contained in the $δ$-neighborhood of a real algebraic variety. Modulo a factor of $δ^{-\varepsilon}$, we also prove Guth and Zahl's generalized version for semialgebraic sets. Although the applications are to be found in harmonic analysis, the proof will employ deep results from algebraic and differential geometry, including Tarski's projection theorem and Gromov's algebraic lemma.
Floor diagrams are a class of weighted oriented graphs introduced by E. Brugalle and the second author. Tropical geometry arguments lead to combinatorial descriptions of (ordinary and relative) Gromov-Witten invariants of projective spaces in terms of floor diagrams and their generalizations. In a number of cases, these descriptions can be used to obtain explicit (direct or recursive) formulas for the corresponding enumerative invariants. In particular, we use this approach to enumerate rational curves of given degree passing through a collection of points on the complex plane and having maxim…
