arXiv subjects

Ludmil Katzarkov

Publications and source records attributed to Ludmil Katzarkov.

80 records · Page 5Linked to original sources

arXiv2005

Singular Lefschetz pencils

We consider structures analogous to symplectic Lefschetz pencils in the context of a closed 4-manifold equipped with a `near-symplectic' structure (ie, a closed 2-form which is symplectic outside a union of circles where it vanishes transversely). Our main result asserts that, up to blowups, every near-symplectic 4-manifold (X,omega) can be decomposed into (a) two symplectic Lefschetz fibrations over discs, and (b) a fibre bundle over S^1 which relates the boundaries of the Lefschetz fibrations to each other via a sequence of fibrewise handle additions taking place in a neighbourhood of the zero set of the 2-form. Conversely, from such a decomposition one can recover a near-symplectic structure.

math.DG↗

arXiv2004

Mirror symmetry for weighted projective planes and their noncommutative deformations

We study the derived categories of coherent sheaves of weighted projective spaces and their noncommutative deformations, and the derived categories of Lagrangian vanishing cycles of their mirror Landau-Ginzburg models. In particular, we show that the derived category of coherent sheaves (B-branes) on the weighted projective plane $\CP^2(a,b,c)$ is equivalent to the derived category of vanishing cycles (A-branes) on the affine hypersurface $X=\{x^ay^bz^c=1\}\subset (\C^*)^3$ equipped with an exact symplectic form and the superpotential $W=x+y+z$. Hence, the homological mirror symmetry conjecture holds for weighted projective planes. Moreover, we also show that this mirror correspondence between derived categories can be extended to toric noncommutative deformations of $\CP^2(a,b,c)$ where B-branes are concerned, and their mirror counterparts, non-exact deformations of the symplectic structure of $X$ where A-branes are concerned. We also obtain similar results for other examples such as weighted projective lines or Hirzebruch surfaces.

math.AG↗

arXiv2004

On K-Stability of Reductive Varieties

G. Tian and S.K. Donaldson formulated a conjecture relating GIT stability of a polarized algebraic variety to the existence of a Kahler metric of constant scalar curvature. In [Don02] Donaldson partially confirmed it in the case of projective toric varieties. In this paper we extend Donaldson's results and computations to a new case, that of reductive varieties. The changes in the second version are cosmetic.

math.DG↗

arXiv2000

Nonabelian mixed Hodge structures

We propose a definition of ``nonabelian mixed Hodge structure'' together with a construction associating to a smooth projective variety $X$ and to a nonabelian mixed Hodge structure $V$, the ``nonabelian cohomology of $X$ with coefficients in $V$'' which is a (pre-)nonabelian mixed Hodge structure denoted $H=Hom(X_M, V)$. We describe the basic definitions and then give some conjectures saying what is supposed to happen. At the end we compute an example: the case where $V$ has underlying homotopy type the complexified 2-sphere, and mixed Hodge structure coming from its identification with $\pp ^1$. For this example we show that $Hom (X_M,V)$ is a namhs for any smooth projective variety $X$.

math.AG↗

arXiv1997

Complex projective surfaces and infinite groups

The paper contains a general construction which produces new examples of non simply-connected smooth projective surfaces. We analyze the resulting surfaces and their fundamental groups. Many of these fundamental groups are expected to be non-residually finite. Using the construction we also suggest a series of potential counterexamples to the Shafarevich conjecture which claims that the universal covering of smooth projective variety is holomorphically convex. The examples are only potential since they depend on group theoretic questions, which we formulate, but we do not know how to answer. At the end we formulate an arithmetic version of the Shafarevich conjecture.

alg-geom↗

arXiv1997

Families of K3 surfaces

We use automorphic forms to prove that a compact family of Kaehler K3 surfaces with constant Picard number is isotrivial.

alg-geom↗

arXiv1995

Nilpotent groups and universal coverings of smooth projective varieties

In this paper we prove that the universal cover of a smooth projective variety with nilpotent fundamental group is holomorphically convex.

alg-geom↗

arXiv1994

Factorization theorems for the representations of the fundamental groups of quasiprojective varieties and some applications

In this paper, using Gromov-Jost-Korevaar-Schoen technique of harmonic maps to nonpositively curved targets, we study the representations of the fundamental groups of quasiprojective varieties. As an application of the above considerations we give a proof of a weak version of the Shafarevich Conjecture.

alg-geom↗