We use the tropical geometry approach to compute absolute and relative enumerative invariants of complex surfaces which are $\mathbb {C} P^1$-bundles over an elliptic curve. We also show that the tropical multiplicity used to count curves can be refined by the standard Block–Göttsche refined multiplicity to give tropical refined invariants. We then give a concrete algorithm using floor diagrams to compute these invariants along with the associated interpretation as operators acting on some Fock space. The floor diagram algorithm allows one to prove the piecewise polynomiality of the relative invariants, and the quasi-modularity of their generating series.

We study plane curves over finite fields whose tangent lines at smooth $\mathbb {F}_q$-points together cover all the points of $\mathbb {P}^2(\mathbb {F}_q)$.

Let $\sigma $ be a stability condition on the bounded derived category $D^b({\mathop{\mathrm {Coh}}\nolimits } W)$ of a Calabi–Yau threefold W and $\mathcal {M}$ a moduli stack parametrizing $\sigma $-semistable objects of fixed topological type. We define generalized Donaldson–Thomas invariants which act as virtual counts of objects in $\mathcal {M}$, fully generalizing the approach introduced by Kiem, Li and the author in the case of semistable sheaves. We construct an associated proper Deligne–Mumford stack $\widetilde {\mathcal {M}}^{\mathbb {C}^{\ast }}$, called the $\mathbb {C}^{\ast }$-rigidified intrinsic stabilizer reduction of $\mathcal {M}$, with an induced semiperfect obstruction theory of virtual dimension zero, and define the generalized Donaldson–Thomas invariant via Kirwan blowups to be the degree of the associated virtual cycle $[\widetilde {\mathcal {M}}]^{\mathrm {vir}} \in A_0(\widetilde {\mathcal {M}})$. This stays invariant under deformations of the complex structure of W. Applications include Bridgeland stability, polynomial stability, Gieseker and slope stability.

Gross and Siebert developed a program for constructing in arbitrary dimension a mirror family to a log Calabi–Yau pair (X, D), consisting of a smooth projective variety X with a normal-crossing anti-canonical divisor D in X. In this paper, we provide an algorithm to practically compute explicit equations of the mirror family in the case when X is obtained as a blow-up of a toric variety along hypersurfaces in its toric boundary, and D is the strict transform of the toric boundary. The main ingredient is the heart of the canonical wall structure associated to such pairs (X, D), which is constructed purely combinatorially, following our previous work with Mark Gross. In the case when we blow up a single hypersurface we show that our results agree with previous results computed symplectically by Aroux–Abouzaid–Katzarkov. In the situation when the locus of blow-up is formed by more than a single hypersurface, due to infinitely many walls interacting, writing the equations becomes significantly more challenging. We provide the first examples of explicit equations for mirror families in such situations.

We prove a formula expressing the K-theoretic log Gromov-Witten invariants of a product of log smooth varieties $V \times W$ in terms of the invariants of V and W. The proof requires introducing log virtual fundamental classes in K-theory and verifying their various functorial properties. We introduce a log version of K-theory and prove the formula there as well.

We prove that the moduli space of rational curves with cyclic action, constructed in our previous work, is realizable as a wonderful compactification of the complement of a hyperplane arrangement in a product of projective spaces. By proving a general result on such wonderful compactifications, we conclude that this moduli space is Chow-equivalent to an explicit toric variety (whose fan can be understood as a tropical version of the moduli space), from which a computation of its Chow ring follows.

We determine all the extremal Gromov-Witten invariants of the Hilbert scheme of $3$ points on a smooth projective complex surface. Our result for the genus-$1$ case verifies a conjecture that we propose for the genus-$1$ extremal Gromov-Witten invariant of the Hilbert scheme of n points with n being arbitrary. The main ideas in the proofs are to use geometric arguments involving the cosection localization theory of Kiem and J. Li [17, 23], algebraic manipulations related to the Heisenberg operators of Grojnowski [13] and Nakajima [34], and the virtual localization formulas of Gromov-Witten theory [12, 20, 30].

For each central essential hyperplane arrangement $\mathcal{A}$ over an algebraically closed field, let $Z_\mathcal{A}^{\hat\mu}(T)$ denote the Denef–Loeser motivic zeta function of $\mathcal{A}$. We prove a formula expressing $Z_\mathcal{A}^{\hat\mu}(T)$ in terms of the Milnor fibers of related hyperplane arrangements. This formula shows that, in a precise sense, the degree to which $Z_{\mathcal{A}}^{\hat\mu}(T)$ fails to be a combinatorial invariant is completely controlled by these Milnor fibers. As one application, we use this formula to show that the map taking each complex arrangement $\mathcal{A}$ to the Hodge–Deligne specialization of $Z_{\mathcal{A}}^{\hat\mu}(T)$ is locally constant on the realization space of any loop-free matroid. We also prove a combinatorial formula expressing the motivic Igusa zeta function of $\mathcal{A}$ in terms of the characteristic polynomials of related arrangements.

We address Hodge integrals over the hyperelliptic locus. Recently Afandi computed, via localisation techniques, such one-descendant integrals and showed that they are Stirling numbers. We give another proof of the same statement by a very short argument, exploiting Chern classes of spin structures and relations arising from Topological Recursion in the sense of Eynard and Orantin.

These techniques seem also suitable to deal with three orthogonal generalisations: (1) the extension to the r-hyperelliptic locus; (2) the extension to an arbitrary number of non-Weierstrass pairs of points; (3) the extension to multiple descendants.

We introduce a conjecture on Virasoro constraints for the moduli space of stable sheaves on a smooth projective surface. These generalise the Virasoro constraints on the Hilbert scheme of a surface found by Moreira and Moreira, Oblomkov, Okounkov and Pandharipande. We verify the conjecture in many nontrivial cases by using a combinatorial description of equivariant sheaves found by Klyachko.

In this paper, we study the Koszul property of the homogeneous coordinate ring of a generic collection of lines in $\mathbb {P}^n$ and the homogeneous coordinate ring of a collection of lines in general linear position in $\mathbb {P}^n.$ We show that if $\mathcal {M}$ is a collection of m lines in general linear position in $\mathbb {P}^n$ with $2m \leq n+1$ and R is the coordinate ring of $\mathcal {M},$ then R is Koszul. Furthermore, if $\mathcal {M}$ is a generic collection of m lines in $\mathbb {P}^n$ and R is the coordinate ring of $\mathcal {M}$ with m even and $m +1\leq n$ or m is odd and $m +2\leq n,$ then R is Koszul. Lastly, we show that if $\mathcal {M}$ is a generic collection of m lines such that

$$ \begin{align*} m> \frac{1}{72}\left(3(n^2+10n+13)+\sqrt{3(n-1)^3(3n+5)}\right),\end{align*} $$

$n \leq 6$

$m \leq 6$

We prove the flow tree formula conjectured by Alexandrov and Pioline, which computes Donaldson–Thomas invariants of quivers with potentials in terms of a smaller set of attractor invariants. This result is obtained as a particular case of a more general flow tree formula reconstructing a consistent scattering diagram from its initial walls.

Using new explicit formulas for the stationary Gromov–Witten/Pandharipande–Thomas ($\mathrm {GW}/{\mathrm {PT}}$) descendent correspondence for nonsingular projective toric threefolds, we show that the correspondence intertwines the Virasoro constraints in Gromov–Witten theory for stable maps with the Virasoro constraints for stable pairs proposed in [18]. Since the Virasoro constraints in Gromov–Witten theory are known to hold in the toric case, we establish the stationary Virasoro constraints for the theory of stable pairs on toric threefolds. As a consequence, new Virasoro constraints for tautological integrals over Hilbert schemes of points on surfaces are also obtained.

We establish adjunction and inversion of adjunction for log canonical centers of arbitrary codimension in full generality.

We propose a variation of the classical Hilbert scheme of points, the double nested Hilbert scheme of points, which parametrizes flags of zero-dimensional subschemes whose nesting is dictated by a Young diagram. Over a smooth quasi-projective curve, we compute the generating series of topological Euler characteristic of these spaces, by exploiting the combinatorics of reversed plane partitions. Moreover, we realize this moduli space as the zero locus of a section of a vector bundle over a smooth ambient space, which therefore admits a virtual fundamental class. We apply this construction to the stable pair theory of a local curve, that is the total space of the direct sum of two line bundles over a curve. We show that the invariants localize to virtual intersection numbers on double nested Hilbert scheme of points on the curve, and that the localized contributions to the invariants are controlled by three universal series for every Young diagram, which can be explicitly determined after the anti-diagonal restriction of the equivariant parameters. Under the anti-diagonal restriction, the invariants are matched with the Gromov–Witten invariants of local curves of Bryan–Pandharipande, as predicted by the Maulik–Nekrasov–Okounkov–Pandharipande (MNOP) correspondence. Finally, we discuss $K$-theoretic refinements à la Nekrasov–Okounkov.

We study pencils of hypersurfaces over finite fields $\mathbb {F}_q$ such that each of the $q+1$ members defined over $\mathbb {F}_q$ is smooth.

We prove the abelian/nonabelian correspondence with bundles for target spaces that are partial flag bundles, combining and generalising results by Ciocan-Fontanine–Kim–Sabbah, Brown, and Oh. From this, we deduce how genus-zero Gromov–Witten invariants change when a smooth projective variety X is blown up in a complete intersection defined by convex line bundles. In the case where the blow-up is Fano, our result gives closed-form expressions for certain genus-zero invariants of the blow-up in terms of invariants of X. We also give a reformulation of the abelian/nonabelian Correspondence in terms of Givental’s formalism, which may be of independent interest.

We study open-closed orbifold Gromov-Witten invariants of 3-dimensional Calabi-Yau smooth toric Deligne-Mumford stacks (with possibly nontrivial generic stabilisers K and semi-projective coarse moduli spaces) relative to Lagrangian branes of Aganagic-Vafa type. An Aganagic-Vafa brane in this paper is a possibly ineffective $C^\infty $ orbifold that admits a presentation $[(S^1\times \mathbb {R} ^2)/G_\tau ]$, where $G_\tau $ is a finite abelian group containing K and $G_\tau /K \cong \boldsymbol {\mu }_{\mathfrak {m}}$ is cyclic of some order $\mathfrak {m}\in \mathbb {Z} _{>0}$.

1. 1. We present foundational materials of enumerative geometry of stable holomorphic maps from bordered orbifold Riemann surfaces to a 3-dimensional Calabi-Yau smooth toric DM stack $\mathcal {X}$ with boundaries mapped into an Aganagic-Vafa brane $\mathcal {L}$. All genus open-closed Gromov-Witten invariants of $\mathcal {X}$ relative to $\mathcal {L}$ are defined by torus localisation and depend on the choice of a framing $f\in \mathbb {Z} $ of $\mathcal {L}$.

2. 2. We provide another definition of all genus open-closed Gromov-Witten invariants in (1) based on algebraic relative orbifold Gromov-Witten theory, which agrees with the definition in (1) up to a sign depending on the choice of orientation on moduli of maps in (1). This generalises the definition in [57] for smooth toric Calabi-Yau 3-folds and specifies an orientation on moduli of maps in (1) compatible with the canonical orientation on moduli of relative stable maps determined by the complex structure.

3. 3. When $\mathcal {X}$ is a toric Calabi-Yau 3-orbifold (i.e., when the generic stabiliser K is trivial), so that $G_\tau =\boldsymbol {\mu }_{\mathfrak {m}}$, we define generating functions $F_{g,h}^{\mathcal {X},(\mathcal {L},f)}$ of open-closed Gromov-Witten invariants of arbitrary genus g and number h of boundary circles; it takes values in $H^*_{ {\mathrm {CR}} }(\mathcal {B} \boldsymbol {\mu }_{\mathfrak {m}}; \mathbb {C} )^{\otimes h}$, where $H^*_{ {\mathrm {CR}} }(\mathcal {B} \boldsymbol {\mu }_{\mathfrak {m}}; \mathbb {C} )\cong \mathbb {C} ^{\mathfrak {m}}$ is the Chen-Ruan orbifold cohomology of the classifying space $\mathcal {B} \boldsymbol {\mu }_{\mathfrak {m}}$ of $\boldsymbol {\mu }_{\mathfrak {m}}$.

4. 4. We prove an open mirror theorem that relates the generating function $F_{0,1}^{\mathcal {X},(\mathcal {L},f)}$ of orbifold disk invariants to Abel-Jacobi maps of the mirror curve of $\mathcal {X}$. This generalises a conjecture by Aganagic-Vafa [6] and Aganagic-Klemm-Vafa [5] (proved in full generality by the first and the second authors in [33]) on the disk potential of a smooth semi-projective toric Calabi-Yau 3-fold.

We propose a conjectural list of Fano manifolds of Picard number $1$ with pseudoeffective normalised tangent bundles, which we prove in various situations by relating it to the complete divisibility conjecture of Francesco Russo and Fyodor L. Zak on varieties with small codegree. Furthermore, the pseudoeffective thresholds and, hence, the pseudoeffective cones of the projectivised tangent bundles of rational homogeneous spaces of Picard number $1$ are explicitly determined by studying the total dual variety of minimal rational tangents (VMRTs) and the geometry of stratified Mukai flops. As a by-product, we obtain sharp vanishing theorems on the global twisted symmetric holomorphic vector fields on rational homogeneous spaces of Picard number $1$.

In this paper, we prove a series of identities of the quasi-map K-theoretical I-functions with level structure between the Grassmannian and its dual Grassmannian. Those identities prove the quantum K-theory version mutation conjecture stated in [13]. Here we find an interval of levels within which two I-functions are the same, and on the boundary of that interval, two I-functions intertwine. We call this phenomenon the level correspondence in Grassmann duality.