arXiv research

Study on sections of line bundles vanishing along subvarieties in complex spaces.

The study examines spaces of holomorphic sections vanishing along subvarieties in complex spaces.

Introduce generalized Ueda obstruction classes for line bundles and apply them to non-semi-positivity.

For certain compact complex Fano manifolds $M$ with reductive Lie algebras of holomorphic vector fields, we determine the analytic subvariety of the second cohomology group of $M$ consisting of Kähler classes whose Bando-Calabi-Futaki character vanishes. Then a Kähler class contains a Kähler metric of constant scalar c…

I use local differential geometric techniques to prove that the algebraic cycles in certain extremal homology classes in Hermitian symmetric spaces are either rigid (i.e., deformable only by ambient motions) or quasi-rigid (roughly speaking, foliated by rigid subvarieties in a nontrivial way). These rigidity results ha…

A self-dual harmonic 2-form on a 4-dimensional Riemannian manifold is symplectic where it does not vanish. Furthermore, away from the form's zero set, the metric with the 2-form give a compatible almost complex structure and thus pseudo-holomorphic subvarieties. Such a subvariety is said to have finite energy when the …

Upper bound found for dimensions of subspaces where holomorphic sectional curvature vanishes.

Let M be a compact, holomorphically symplectic Kahler manifold, and $η$ a (1,1)-current which is nef (a limit of Kahler forms). Assume that the cohomology class of $η$ is parabolic, that is, its top power vanishes. We prove that all Lelong sets of $η$ are coisotropic. When M is generic, this is used to show that all Le…

We give a simple criterion for slope stability of Fano manifolds $X$ along divisors or smooth subvarieties. As an application, we show that $X$ is slope stable along an ample effective divisor $D\subset X$ unless $X$ is isomorphic to a projective space and $D$ is a hyperplane section. We also give counterexamples to Au…

Cominuscule subvarieties found in flag varieties.

A mapping bends Teichmüller spaces into character varieties, preserving symplectic structure.

New tool helps classify invariant subvarieties in degenerations.

We study the Yang-Mills flow on a holomorphic vector bundle E over a compact Kahler manifold X. We construct a natural barrier function along the flow, and introduce some techniques to study the blow-up of the curvature along the flow. Making some technical assumptions, we show how our techniques can be used to prove t…

Let G be a (real or complex) linear reductive algebraic group acting on an affine variety V. Let W be a subvariety. In this work we study how the G-orbits intersect W. We develop a criterion to determine when the intersection can be described as a finite union of orbits of a reductive subgroup. The conditions of the cr…

New proof for higher rank subvarieties in genus three.

Taubes established fundamental properties of $J-$holomorphic subvarieties in dimension 4 in \cite{T1}. In this paper, we further investigate properties of reducible $J-$holomorphic subvarieties. We offer an upper bound of the total genus of a subvariety when the class of the subvariety is $J-$nef. For a spherical class…

Optimal $L^2$ extension of sections from subvarieties in Kähler manifolds.

Constructs a function to prove meromorphic differential strata don't have complete subvarieties.

Lecture notes on advanced geometry equations and subvarieties.

Let $X$ be a hyperkaehler manifold. Trianalytic subvarieties of $X$ are subvarieties which are complex analytic with respect to all complex structures induced by the hyperkaehler structure. Given a 2-dimensional complex torus $T$, the Hilbert scheme $T^{[n]}$ classifying zero-dimensional subschemes of $T$ admits a hype…

Classifies GL(2,R)-invariant subvarieties with zero Lyapunov exponents.

Study identifies specific subvarieties in translation surfaces with quadratic field.

Constructs subvarieties in translation surface strata using combinatorial input.

Let M be a hyperkaehler manifold, not necessarily compact, and $S\cong CP^1$ the set of complex structures induced by the quaternionic action. Trianalytic subvariety of M is a subvariety which is complex analytic with respect to all $I \in CP^1$. We show that for all $I \in S$ outside of a countable set, all compact co…

The paper explores anomalous subvarieties in hyperbolic 3-manifolds and their geometric implications.

Study identifies subvarieties of projective varieties mapping to models.

We show that Thurston's skinning maps of Teichmuller space have finite fibers. The proof centers around a study of two subvarieties of the SL_2(C) character variety of a surface, one associated to complex projective structures and the other associated to a 3-manifold. Using the Morgan-Shalen compactification of the cha…

Classify projective subvarieties in Bogomolov-Guan manifolds using quasi-diagonals.

The paper studies singularities in a complex flow related to mean curvature.

Classifies GL(2,R)-invariant subvarieties in complex geometry.

A natural oriented (2k+2)-chain in CP^{2k+1} with boundary twice RP^{2k+1}, its complex shade, is constructed. Via intersection numbers with the shade, a new invariant, the shade number of k-dimensional subvarieties with normal vector fields along their real part, is introduced. For an even-dimensional real variety, th…

Paper extends cohomology classes and holomorphic sections on subvarieties.

Totally geodesic subvarieties in moduli space are locally rigid.

We show that under very general assumptions the partial Bergman kernel function of sections vanishing along an analytic hypersurface has exponential decay in a neighborhood of the vanishing locus. Considering an ample line bundle, we obtain a uniform estimate of the Bergman kernel function associated to a singular metr…

We show that the moduli space $M$ of holomorphic vector bundles on $CP^3$ that are trivial along a line is isomorphic (as a complex manifold) to a subvariety in the moduli of rational curves of the twistor space of the moduli space of framed instantons on $\R^4$, called the space of twistor sections. We then use this c…

We compute the algebraic hull of the Kontsevich-Zorich cocycle over any GL^+_2(R) invariant subvariety of the Hodge bundle, and derive from this finiteness results on such subvarieties.

We develop the foundation of the complex symplectic geometry of Lagrangian subvarieties in a hyperkahler manifold. We establish a characterization, a Chern number inequality, topological and geometrical properties of Lagrangian submanifolds. We discuss a category of Lagrangian subvarieties and its relationship with the…

The study finds infinitely many Shimura subvarieties in Jacobian loci for curves of genus 2, 3, and 4.

We show that any totally geodesic submanifold of Teichmuller space of dimension greater than one covers a totally geodesic subvariety, and only finitely many totally geodesic subvarieties of dimension greater than one exist in each moduli space.

In this paper we give examples of closed smooth submanifolds of RP^n which are isotopic to nonsingular projective subvarieties of RP^n but they can not be isotopic to the real parts of nonsingular complex projective subvarieties of CP^n.

Study uses Zilber-Pink conjecture and dynamical methods to solve rigidity problems.

We prove that the non-Kahler locus of a nef and big class on a compact complex manifold bimeromorphic to a Kahler manifold equals its null locus. In particular this gives an analytic proof of a theorem of Nakamaye and Ein-Lazarsfeld-Mustata-Nakamaye-Popa. As an application, we show that finite time non-collapsing singu…

A hypercomplex manifold M is a manifold with a triple I,J,K of complex structure operators satisfying quaternionic relations. For each quaternion L=aI +bJ+cK, L^2=-1, L is also a complex structure operator on M, called an induced complex structure. We are studying compact complex subvarieties of (M,L), when L is a gene…

We prove weak and strong maximum principles, including a Hopf lemma, for smooth subsolutions to equations defined by linear, second-order, partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the smooth subsolutions along this boundary…

We establish a formula for the SL(2,C) Casson invariant of spliced sums of homology spheres along knots. Along the way, we show that the SL(2,C) Casson invariant vanishes for spliced sums along knots in the 3-sphere.

The Oeljeklaus-Toma (OT-) manifolds are compact, complex, non-Kahler manifolds constructed by Oeljeklaus and Toma, and generalizing the Inoue surfaces. Their construction uses the number-theoretic data: a number field $K$ and a torsion-free subgroup $U$ in the group of units of the ring of integers of $K$, with rank of…

This paper gives a complete parametrization of the commensurability classes of totally geodesic subspaces of irreducible arithmetic quotients of $X_{a, b} = (\mathbf{H}^2)^a \times (\mathbf{H}^3)^b$. A special case describes all Shimura subvarieties of type $\mathrm{A}_1$ Shimura varieties. We produce, for any $n\geq 1…

The study proves semisimplicity of totally geodesic subvarieties in moduli spaces of Riemann surfaces.