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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 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
$1$
.
We propose two systems of “intrinsic” weights for counting such curves. In both cases the result acquires an exceptionally strong invariance property: it does not depend on the choice of a surface. One of our counts includes all divisor classes of canonical degree 2 and gives in total 30. The other one excludes the class
$-2K$
, but adds up the results of counting for a pair of real structures that differ by Bertini involution. This count gives 96.
Given a quiver with potential
$(Q,W)$
, Kontsevich–Soibelman constructed a cohomological Hall algebra (CoHA) on the critical cohomology of the stack of representations of
$(Q,W)$
. Special cases of this construction are related to work of Nakajima, Varagnolo, Schiffmann–Vasserot, Maulik–Okounkov, Yang–Zhao, etc. about geometric constructions of Yangians and their representations; indeed, given a quiver Q, there exists an associated pair
$(\widetilde{Q}, \widetilde{W})$
whose CoHA is conjecturally the positive half of the Maulik–Okounkov Yangian
$Y_{\text {MO}}(\mathfrak {g}_{Q})$
.
For a quiver with potential
$(Q,W)$
, we follow a suggestion of Kontsevich–Soibelman and study a categorification of the above algebra constructed using categories of singularities. Its Grothendieck group is a K-theoretic Hall algebra (KHA) for quivers with potential. We construct representations using framed quivers, and we prove a wall-crossing theorem for KHAs. We expect the KHA for
$(\widetilde{Q}, \widetilde{W})$
to recover the positive part of quantum affine algebra
$U_{q}(\widehat {\mathfrak {g}_{Q}})$
defined by Okounkov–Smirnov.
We compute, via motivic wall-crossing, the generating function of virtual motives of the Quot scheme of points on
${\mathbb{A}}^3$
, generalising to higher rank a result of Behrend–Bryan–Szendrői. We show that this motivic partition function converges to a Gaussian distribution, extending a result of Morrison.
We introduce and study a fermionisation procedure for the cohomological Hall algebra
$\mathcal{H}_{\Pi_Q}$
of representations of a preprojective algebra, that selectively switches the cohomological parity of the BPS Lie algebra from even to odd. We do so by determining the cohomological Donaldson–Thomas invariants of central extensions of preprojective algebras studied in the work of Etingof and Rains, via deformed dimensional reduction. Via the same techniques, we determine the Borel–Moore homology of the stack of representations of the
$\unicode{x03BC}$
-deformed preprojective algebra introduced by Crawley–Boevey and Holland, for all dimension vectors. This provides a common generalisation of the results of Crawley-Boevey and Van den Bergh on the cohomology of smooth moduli schemes of representations of deformed preprojective algebras and my earlier results on the Borel–Moore homology of the stack of representations of the undeformed preprojective algebra.
For X a smooth projective variety and
$D=D_1+\dotsb +D_n$
a simple normal crossing divisor, we establish a precise cycle-level correspondence between the genus
$0$
local Gromov–Witten theory of the bundle
$\oplus _{i=1}^n \mathcal {O}_X(-D_i)$
and the maximal contact Gromov–Witten theory of the multiroot stack
$X_{D,\vec r}$
. The proof is an implementation of the rank-reduction strategy. We use this point of view to clarify the relationship between logarithmic and orbifold invariants.
For oriented $-1$-shifted symplectic derived Artin stacks, Ben-Bassat, Brav, Bussi and Joyce introduced certain perverse sheaves on them which can be regarded as sheaf-theoretic categorifications of the Donaldson–Thomas invariants. In this paper, we prove that the hypercohomology of the above perverse sheaf on the $-1$-shifted cotangent stack over a quasi-smooth derived Artin stack is isomorphic to the Borel–Moore homology of the base stack up to a certain shift of degree. This is a global version of the dimensional reduction theorem due to Davison. We give two applications of our main theorem. Firstly, we apply it to the study of the cohomological Donaldson–Thomas invariants for local surfaces. Secondly, regarding our main theorem as a version of the Thom isomorphism theorem for dual obstruction cones, we propose a sheaf-theoretic construction of the virtual fundamental classes for quasi-smooth derived Artin stacks.
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.
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.
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.
We propose an intersection-theoretic method to reduce questions in genus 0 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 product geometries. Finally, we explicitly determine the difference between local and logarithmic theories, in terms of relative invariants for which efficient algorithms are known. The polyhedral geometry of the tropical moduli of maps plays an essential and intricate role in the analysis.
We prove that if X is a complex projective K3 surface and
$g>0$
, then there exist infinitely many families of curves of geometric genus g on X with maximal, i.e., g-dimensional, variation in moduli. In particular, every K3 surface contains a curve of geometric genus 1 which moves in a nonisotrivial family. This implies a conjecture of Huybrechts on constant cycle curves and gives an algebro-geometric proof of a theorem of Kobayashi that a K3 surface has no global symmetric differential forms.
We use Noether–Lefschetz theory to study the reduced Gromov–Witten invariants of a holomorphic-symplectic variety of
$K3^{[n]}$
-type. This yields strong evidence for a new conjectural formula that expresses Gromov–Witten invariants of this geometry for arbitrary classes in terms of primitive classes. The formula generalizes an earlier conjecture by Pandharipande and the author for K3 surfaces. Using Gromov–Witten techniques, we also determine the generating series of Noether–Lefschetz numbers of a general pencil of Debarre–Voisin varieties. This reproves and extends a result of Debarre, Han, O’Grady and Voisin on Hassett–Looijenga–Shah (HLS) divisors on the moduli space of Debarre–Voisin fourfolds.
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. 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. 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. 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. 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 prove that the maximal number of conics in a smooth sextic
$K3$
-surface
$X\subset \mathbb {P}^4$
is 285, whereas the maximal number of real conics in a real sextic is 261. In both extremal configurations, all conics are irreducible.
Let $L$ be a very ample line bundle on a projective scheme $X$ defined over an algebraically closed field $\Bbbk$ with ${\rm char}\,\Bbbk \neq 2$. We say that $(X,L)$ satisfies property $\mathsf {QR}(k)$ if the homogeneous ideal of the linearly normal embedding $X \subset {\mathbb {P}} H^{0} (X,L)$ can be generated by quadrics of rank less than or equal to $k$. Many classical varieties, such as Segre–Veronese embeddings, rational normal scrolls and curves of high degree, satisfy property $\mathsf {QR}(4)$. In this paper, we first prove that if ${\rm char}\,\Bbbk \neq 3$ then $({\mathbb {P}}^{n} , \mathcal {O}_{{\mathbb {P}}^{n}} (d))$ satisfies property $\mathsf {QR}(3)$ for all $n \geqslant 1$ and $d \geqslant 2$. We also investigate the asymptotic behavior of property $\mathsf {QR}(3)$ for any projective scheme. Specifically, we prove that (i) if $X \subset {\mathbb {P}} H^{0} (X,L)$ is $m$-regular then $(X,L^{d} )$ satisfies property $\mathsf {QR}(3)$ for all $d \geqslant m$, and (ii) if $A$ is an ample line bundle on $X$ then $(X,A^{d} )$ satisfies property $\mathsf {QR}(3)$ for all sufficiently large even numbers $d$. These results provide affirmative evidence for the expectation that property $\mathsf {QR}(3)$ holds for all sufficiently ample line bundles on $X$, as in the cases of Green and Lazarsfeld's condition $\mathrm {N}_p$ and the Eisenbud–Koh–Stillman determininantal presentation in Eisenbud et al. [Determinantal equations for curves of high degree, Amer. J. Math. 110 (1988), 513–539]. Finally, when ${\rm char}\,\Bbbk = 3$ we prove that $({\mathbb {P}}^{n} , \mathcal {O}_{{\mathbb {P}}^{n}} (2))$ fails to satisfy property $\mathsf {QR}(3)$ for all $n \geqslant 3$.
In our previous paper we suggested a conjecture relating the structure of the small quantum cohomology ring of a smooth Fano variety of Picard number 1 to the structure of its derived category of coherent sheaves. Here we generalize this conjecture, make it more precise, and support it by the examples of (co)adjoint homogeneous varieties of simple algebraic groups of Dynkin types $\mathrm {A}_n$ and $\mathrm {D}_n$, that is, flag varieties $\operatorname {Fl}(1,n;n+1)$ and isotropic orthogonal Grassmannians $\operatorname {OG}(2,2n)$; in particular, we construct on each of those an exceptional collection invariant with respect to the entire automorphism group. For $\operatorname {OG}(2,2n)$ this is the first exceptional collection proved to be full.