Related papers: Generalized complexity of surfaces

In this article, we summarize combinatorial description of complete intersection Calabi-Yau threefolds in Hibi toric varieties. Such Calabi-Yau threefolds have at worst conifold singularities, and are often smoothable to non-singular…

In this paper, we introduce the notion of parabolic stable pairs on Calabi-Yau 3-folds and invariants counting them. By applying the wall-crossing formula developed by Joyce-Song, Kontsevich-Soibelman, we see that they are related to…

For a smooth and proper variety $X$ over an algebraically closed field $k$ of characteristic $p>0$, the group $Br(X)[p^\infty]$ is a direct sum of finitely many copies of $\mathbb{Q}_p/\mathbb{Z}_p$ and an abelian group of finite exponent.…

In arXiv:2011.08830 we established a series of correspondences relating five enumerative theories of log Calabi-Yau surfaces, i.e. pairs $(Y,D)$ with $Y$ a smooth projective complex surface and $D=D_1+\dots +D_l$ an anticanonical divisor on…

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 prove some basic properties of the relative Nakayama-Zariski decomposition. We apply them to the study of lc generalized pairs. We prove the existence of log minimal models or Mori fiber spaces for (relative) lc generalized pairs…

Let $(X,B)$ be a pair, and let $f \colon X \rightarrow S$ be a contraction with $-(K_X + B)$ nef over $S$. A conjecture, known as the Shokurov-Koll\'{a}r connectedness principle, predicts that $f^{-1} (s) \cap \mathrm{Nklt}(X,B)$ has at…

In this article, we study the orbifold fundamental group $\pi_1^{\rm orb}(X,\Delta)$ of a Calabi--Yau pair $(X,\Delta)$ with log canonical singularities. We conjecture that the orbifold fundamental group $\pi_1^{\rm orb}(X,\Delta)$ of a…

We give a criterion for a continuous family of curves on a nodal $K$-trivial threefold $X_0$ to contribute geometrically rigid curves to a general smoothing of $X_0$. As an application, we prove the existence of geometrically rigid curves…

Let X_0 be a compact connected Riemann surface of genus g with D_0\subset X_0 an ordered subset of cardinality n, and let E_G be a holomorphic principal G-bundle on X_0, where G is a complex reductive affine algebraic group, that admits a…

We prove that the mirror map is the SYZ map for every toric Calabi-Yau surface. As a consequence one obtains an enumerative meaning of the mirror map. This involves computing genus-zero open Gromov-Witten invariants, which is done by…

We explain how to form a novel dataset of simply connected Calabi-Yau threefolds via the Gross-Siebert algorithm. We expect these to degenerate to Calabi-Yau toric hypersurfaces with certain Gorenstein (not necessarily isolated)…

The notion of the Yau sequence was introduced by Tomaru, as an attempt to extend Yau's elliptic sequence for (weakly) elliptic singularities to normal surface singularities of higher fundamental genera. In this paper, we obtain the…

The predictions of the Mirror Symmetry are extended in dimensions n>3 and are proven for projective complete intersections Calabi-Yau varieties. Precisely, we prove that the total collection of rational Gromov-Witten invariants of such…

We generalize complex manifolds to manifolds with corners $X$, and to manifolds with generalized corners (g-corners) in the sense of the second author arXiv:1501.00401, using complex structures on the b-tangent bundle (log tangent bundle)…

Recently, Escobar, Harada, and Manon introduced the theory of polyptych lattices. This theory gives a general framework for constructing projective varieties from polytopes in a polyptych lattice. When all the mutations of the polyptych…

In this paper we construct monodromy representing generalized special Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric varieties near the large complex limit.

Assuming the standard framework of mirror symmetry, a conjecture is formulated describing how the diffeomorphism group of a Calabi-Yau manifold Y should act by families of Fourier-Mukai transforms over the complex moduli space of the mirror…

Consider a log Calabi-Yau pair $(X,D)$ consisting of a smooth del Pezzo surface $X$ of degree $\geq 3$ and a smooth anticanonical divisor $D$. We prove a correspondence between genus zero logarithmic Gromov-Witten invariants of $X$…

We study the geometric change of Chow cohomology classes in projective toric varieties under the Weil-McMullen dual of the intersection product with a Lefschetz element. Based on this, we introduce toric chordality, a generalization of…