Related papers: The generalized Frankel conjecture in Sasaki geome…

We introduce and study a special class of almost contact metric manifolds, which we call anti-quasi-Sasakian (aqS). Among the class of transversely K\"ahler almost contact metric manifolds $(M,\varphi, \xi,\eta,g)$, quasi-Sasakian and…

We address a conjecture that $\pi_1$-surjective maps between closed aspherical 3-manifolds having the same rank on $\pi_1$ must be of non-zero degree. The conjecture is proved for Seifert manifolds, which is used in constructing the first…

We introduce a holomorphic sheaf E on a Sasaki manifold and study two new notions of stability for E along the Sasaki-Ricci flow related to the `jumping up' of the number of global holomorphic sections of E at infinity. First, we show that…

We discuss the Sasakian geometry of odd dimensional homotopy spheres. In particular, we give a completely new proof of the existence of metrics of positive Ricci curvature on exotic spheres that can be realized as the boundary of a…

Based on the compactness of the moduli of non-collapsed Calabi-Yau spaces with mild singularities, we set up a structure theory for polarized K\"ahler Ricci flows with proper geometric bounds. Our theory is a generalization of the structure…

For all complex dimensions n>=2, we construct complete Kaehler manifolds of bounded curvature and non-negative Ricci curvature whose Kaehler--Ricci evolutions immediately acquire Ricci curvature of mixed sign.

For the Bach-flat closed manifold with positive scalar curvature, we prove a rigidity result under a given inequality involving the Weyl curvature and the traceless Ricci curvature. Moveover, under an inequality involving…

Let X be a smooth, connected, projective variety over an algebraically closed field of positive characteristic. In "Flat vector bundles and the fundamental group in non-zero characteristics" (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2…

In this paper, for any compact Lie group $G$, we show that the space of $G$-invariant Riemannian metrics with positive scalar curvature (PSC) on any closed three-manifold is either empty or contractible. In particular, we prove the…

We define reduction of locally conformal Kaehler manifolds, considered as conformal Hermitian manifolds, and we show its equivalence with an unpublished construction given by Biquard and Gauduchon. We show the compatibility between this…

We prove that any non-Sasakian contact metric (\kappa,\mu)-space admits a canonical \eta-Einstein Sasakian or \eta-Einstein paraSasakian metric. An explicit expression for the curvature tensor fields of those metrics is given and we find…

In this short paper, we will give a simple and transcendental proof for Mok's theorem of the generalized Frankel conjecture. This work is based on the maximum principle in \cite{BS2} proposed by Brendle and Schoen.

In this paper we introduce the concept of $(\varepsilon)$-almost paracontact manifolds, and in particular, of $(\varepsilon)$-para Sasakian manifolds. Several examples are presented. Some typical identities for curvature tensor and Ricci…

We extend to manifolds of arbitrary dimension the Castelnuovo-de Franchis inequality for surfaces. The proof is based on the theory of Generic Vanishing and higher regularity, and on the Evans-Griffith Syzygy Theorem in commutative algebra.…

We show that any closed biquotient with finite fundamental group admits metrics of positive Ricci curvature. Also, let M be a closed manifold on which a compact Lie group G acts with cohomogeneity one, and let L be a closed subgroup of G…

We give several generalizations of the Kodaira vanishing and embedding theorems for K\"ahler manifolds to the case where the relevent line bundle has a small region of negative curvature. To prove the vanishing theorems we adapt techniques…

We study Gushel-Mukai (GM) varieties of dimension 4 or 6 in characteristic $p$. Our main result is the Tate conjecture for all such varieties over finitely generated fields of characteristic $p\geq 5$. In the case of GM sixfolds, we follow…

We consider a generalized Ricci flow with a given (not necessarily closed) three-form and establish the higher derivatives estimates for compact manifolds. As an application, we prove the compactness theorem for this generalized Ricci flow.…

We prove that a connected mean convex region in $\mathbb{R}^{n+1}$ with at least two components cannot have strictly positive mean curvature. This answers a question of Gromov. We also obtain estimates for how quickly the mean curvature…

The Gromov-Lawson-Rosenberg conjecture for a group G states that a compact spin manifold with fundamental group G admits a metric of positive scalar curvature if and only if a certain topological obstruction vanishes. It is known to be true…