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In this paper, we verify a part of the Mirror Symmetry Conjecture for Schoen's Calabi-Yau 3-fold, which is a special complete intersection in a toric variety. We calculate a part of the prepotential of the A-model Yukawa couplings of the…
We reconstruct the all-genus Fan-Jarvis-Ruan-Witten invariants of a Fermat cubic Landau-Ginzburg space $(x_1^3+x_2^3+x_3^3: [\mathbb{C}^3/ \mathbold{\mu}_3]\to \mathbb{C})$ from genus-one primary invariants, using tautological relations and…
In this work we prove a bound for the torsion in Mordell-Weil groups of smooth elliptically fibered Calabi-Yau 3- and 4-folds. In particular, we show that the set which can occur on a smooth elliptic Calabi-Yau $n$-fold for ($n\geq 3$) is…
We propose a conjecture extending the classical construction of elliptic units to complex cubic number fields $K$. The conjecture concerns special values of the elliptic gamma function, a holomorphic function of three complex variables…
We explicitly compute family GW invariants of elliptic surfaces for primitive classes. That involves establishing a TRR formula and a symplectic sum formula for elliptic surfaces and then determining the GW invariants using an argument from…
We compute the genus zero family Gromov-Witten invariants for K3 surfaces using the topological recursion formula and the symplectic sum formula for a degeneration of elliptic K3 surfaces. In particular we verify the Yau-Zaslow formula for…
An elliptic orbifold is the quotient of an elliptic curve by a finite group. Eskin and Okounkov proved that generating functions for the number of branched covers of an elliptic curve with specified ramification are quasimodular forms for…
We attempt to define a new invariant I of (almost) Calabi-Yau 3-folds M, by counting special Lagrangian rational homology 3-spheres N in M in each 3-homology class, with a certain weight w(N) depending on the topology of N. This is…
We offer a new construction of Lagrangian submanifolds for the Gopakumar-Vafa conjecture relating the Chern-Simons theory on the 3-sphere and the Gromov-Witten theory on the resolved conifold. Given a knot in the 3-sphere its conormal…
Integrals of the Chern classes of the Hodge bundle in Gromov-Witten theory are studied. We find a universal system of differential equations which determines the generating function of these integrals from the standard descendent potential…
Gromov-Witten theory is used to define an enumerative geometry of curves in Calabi-Yau 4-folds. The main technique is to find exact solutions to moving multiple cover integrals. The resulting invariants are analogous to the BPS counts of…
We study modularity properties of generating series of logarithmic Gromov-Witten invariants of elliptic fibrations relative to singular fibers. Motivated by predictions from Vafa-Witten theory, we conjecture that such generating series are…
An algorithm to determine all the Gromov-Witten invariants of any smooth projective curve was obtained by Okounkov and Pandharipande in 2006. They identified stationary invariants with certain Hurwitz numbers and then presented Virasoro…
This is the first of two papers which construct a purely algebraic counterpart to the theory of Gromov-Witten invariants (at all genera). These Gromov-Witten type invariants depend on a Calabi-Yau A-infinity category, which plays the role…
We develop new techniques to study regularity questions for moduli spaces of pseudoholomorphic curves that are multiply covered. Among the main results, we show that unbranched multiple covers of closed holomorphic curves are generically…
We explore the connection between $K3$ categories and 0-cycles on holomorphic symplectic varieties. In this paper, we focus on Kuznetsov's noncommutative $K3$ category associated to a nonsingular cubic 4-fold. By introducing a filtration on…
In this paper, we compute categorical entropy of spherical twists. In particular, we prove that Gromov-Yomdin type conjecture holds for spherical twists. Moreover, we construct counterexamples of Gromov-Yomdin type conjecture for K3…
In a series of papers \cite{KS,MR}, Krawitz, Milanov, Ruan, and Shen have verified the so-called Landau-Ginzburg/Calabi-Yau (LG/CY) correspondence for simple elliptic singularities $E_N^{(1,1)}$ ($N=6,7,8$). As a byproduct it was also…
Craw and Ishii proved that for a finite abelian group G in SL_3(C) every (projective) relative minimal model of C^3/G is isomorphic to the fine moduli space of \theta-stable G-constellations for some GIT parameter \theta. In this article,…
We prove the standard conjectures for complex projective varieties that are deformations of the Hilbert scheme of points on a K3 surface. The proof involves Verbitsky's theory of hyperholomorphic sheaves and a study of the cohomology…