相关论文: On Conjecture $\mathcal O$ for projective complete…
Gamma conjecture I and the underlying Conjecture $\mathcal{O}$ for Fano manifolds were proposed by Galkin, Golyshev and Iritani recently. We show that both conjectures hold for all two-dimensional Fano manifolds. We prove Conjecture…
In this paper, we show that general homogeneous manifolds $G/P$ satisfy Conjecture $\mathcal{O}$ of Galkin, Golyshev and Iritani which `underlies' Gamma conjectures I and II of them. Our main tools are the quantum Chevalley formula for…
The Debarre-de Jong conjecture predicts that the Fano variety of lines on a smooth Fano hypersurface in $\mathbb{P}^n$ is always of the expected dimension. We generalize this conjecture to the case of Fano complete intersections and prove…
We compute the quantum Euler class of Fano complete intersections X in a projective space. In particular, we prove a recent conjecture of A. Buch and R. Pandharipande. Finally we apply our result to obtain formulas for the virtual Tevelev…
It is known that a necessary condition for the existence of K\"ahler-Ricci solitons is the vanishing of the modified Futaki invariant introduced by Tian-Zhu. In a recent work of Berman-Nystr\"om, it was generalized for (singular) Fano…
The purpose of this paper is to show that for a complete intersection curve $C$ in projective space (other than a few stated exceptions), any morphism $f: C \to \mathbb{P}^r$ satisfying $\text{deg}\, f^*\mathcal{O}_{\mathbb{P}^r}(1)…
Consider the Fano manifold $X$ formed by blowing up $\mathbb{P}^n$ along its linear subspace $\mathbb{P}^r$, we check the conifold conditions [3, 1] for its mirror Laurent polynomial $f$, which can imply that $X$ satisfies both Conjecture…
We prove that the derived category of a smooth complete intersection variety is equivalent to a full subcategory of the derived category of a smooth projective Fano variety. This enables us to define some new invariants of smooth projective…
We prove that a general Fano hypersurface in a projective space over an algebraically closed field of arbitrary characteristic is separably rationally connected.
We generalize the toric residue mirror conjecture of Batyrev and Materov to not necessarily reflexive polytopes. Using this generalization we prove the toric residue mirror conjecture for Calabi-Yau complete intersections in Gorenstein…
We obtain criteria for detecting complete intersections in projective varieties. Motivated by a conjecture of Hartshorne concerning subvarieties of projective spaces, we investigate situations when two-codimensional smooth subvarieties of…
Small codimensional embedded manifolds defined byequations of small degree are Fano and covered by lines. They are complete intersections exactly when the variety of lines through a general point is so and has the right codimension. This…
We prove birational superrigidity of generic Fano complete intersections $V$ of type $2^{k_1}\cdot 3^{k_2}$ in the projective space ${\mathbb P}^{2k_1+3k_2}$, under the condition that $k_2\geq 2$ and $k_1+2k_2=\mathop{\rm dim} V\geq 12$,…
We consider the Fano scheme $F_k(X)$ of $k$--dimensional linear subspaces contained in a complete intersection $X \subset \mathbb{P}^n$ of multi--degree $\underline{d} = (d_1, \ldots, d_s)$. Our main result is an extension of a result of…
We verify Katzarkov-Kontsevich-Pantev conjecture for Landau-Ginzburg models of smooth Fano threefolds.
We prove two conjectures on weighted complete intersections and give the complete classification of threefold weighted complete intersections in weighted projective space that are canonically or anticanonically embedded.
We provide enumerative formulas for the degrees of varieties parameterizing hypersurfaces and complete intersections which contain pro-jective subspaces and conics. Besides, we find all cases where the Fano scheme of the general complete…
We show that a refined version of Golyshev's canonical strip hypothesis does hold for the Hilbert polynomials of complete intersections in rational homogeneous spaces.
We study Fano schemes $F_k(X)$ for complete intersections $X$ in a projective toric variety $Y\subset \mathbb{P}^n$. Our strategy is to decompose $F_k(X)$ into closed subschemes based on the irreducible decomposition of $F_k(Y)$ as studied…
We get sharp degree bound for generic smoothness and connectedness of the space of conics in low degree complete intersections which generalizes the old work about Fano scheme of lines on Hypersurfaces.