Searcharxiv⌕ Search

arXiv subjects

Victor Przyjalkowski

Publications and source records attributed to Victor Przyjalkowski.

46 records · Page 3Linked to original sources
arXiv2014

Birational geometry via moduli spaces

In this paper we connect degenerations of Fano threefolds by projections. Using Mirror Symmetry we transfer these connections to the side of Landau--Ginzburg models. Based on that we suggest a generalization of Kawamata's categorical approach to birational geometry enhancing it via geometry of moduli spaces of Landau--Ginzburg models. We suggest a conjectural application to Hasset--Kuznetsov--Tschinkel program based on new nonrationality "invariants" we consider --- gaps and phantom categories. We make several conjectures for these invariants in the case of surfaces of general type and quadric bundles.

math.AG↗
arXiv2012

Toric Degenerations of Fano Threefolds Giving Weak Landau-Ginzburg Models

We show that every rank one smooth Fano threefold has a weak Landau-Ginzburg model coming from a toric degeneration. The fibers of these Landau-Ginzburg models can be compactified to K3 surfaces with Picard lattice of rank 19. We also show that any smooth Fano variety of arbitrary dimension which is a complete intersection of Cartier divisors in weighted projective space has a very weak Landau-Ginzburg model coming from a toric degeneration.

math.AG↗
arXiv2012

Weak Landau-Ginzburg models for smooth Fano threefolds

The paper is joined with arXiv:0911.5428 and improved. We prove that Landau-Ginzburg models for all 17 smooth Fano threefolds with Picard rank 1 can be represented as Laurent polynomials in 3 variables exhibiting them case by case. We check that these Landau-Ginzburg models can be compactified to open Calabi-Yau varieties. In the spirit of L. Katzarkov's program we prove that numbers of irreducible components of the central fibers of compactifications of these pencils are dimensions of intermediate Jacobians of Fano varieties plus 1. In particular these numbers do not depend on compactifications. We state most of known methods of finding Landau-Ginzburg models in terms of Laurent polynomials. We discuss Laurent polynomial representation of Landau-Ginzburg models of Fano varieties and state some problems related to it.

math.AG↗
arXiv2012

Hodge numbers of Fano threefolds via Landau--Ginzburg models

For each smooth Fano threefold $X$ with Picard number 1 we consider a weak Landau--Ginzburg model, that is a fibration over $\mathbb C^1$ given by a certain Laurent polynomial. In the spirit of L. Katzarkov's program we prove that the number of irreducible components of the central fiber of its compactification is $h^{1,2}(X)+1$. In particular, it does not depend on the compactification. The question of dependence on the model is open; however we produce examples of different weak Landau--Ginzburg models for the same variety with the same number of components of the central fiber.

math.AG↗
arXiv2007

Minimal Gromov--Witten ring

We build the abstract theory of Gromov-Witten invariants of genus 0 for quantum minimal Fano varieties (a minimal natural (with respect to Gromov-Witten theory) class of varieties). In particular, we consider ``the minimal Gromov-Witten ring'', i. e. a commutative algebra with generators and relations of the form used in the Gromov-Witten theory of Fano variety (of unspecified dimension). Gromov-Witten theory of any quantum minimal variety is a homomorphism of this ring to $\mathbb C$. We prove the Abstract Reconstruction Theorem which states the particular isomorphism of this ring with a free commutative ring generated by ``prime two-pointed invariants''. We also find the solutions of the differential equations of type DN for a Fano variety of dimension N in terms of generating series of one-pointed Gromov-Witten invariants.

math.AG↗
arXiv2007

Quantum cohomology of smooth complete intersections in weighted projective spaces and singular toric varieties

We generalize Givental's Theorem for complete intersections in smooth toric varieties in the Fano case. In particular, we find Gromov--Witten invariants of Fano varieties of dimension $\geq 3$, which are complete intersections in weighted projective spaces and singular toric varieties. We generalize the Riemann-Roch equations to weighted projective spaces. We compute counting matrices of smooth Fano threefolds with Picard group Z and anticanonical degrees 2, 8, and 16.

math.AG↗
arXiv2007

Gromov-Witten invariants of Fano threefolds of genera 6 and 8

The aim of this paper is to prove Golyshev's conjecture in the cases of Fano threefolds $V_{10}$ and $V_{14}$. This conjecture states modularity of D3 equations for smooth Fano threefolds with Picard group Z. More precisely, we find counting matrices of prime two-pointed Gromov-Witten invariants for them. For this we use the method that lets us find Gromov-Witten invariants of complete intersections in varieties whose invariants are (partially) known.

math.AG↗
arXiv2005

Hyperelliptic and trigonal Fano threefolds

We classify Fano 3-folds with canonical Gorenstein singularities whose anticanonical linear system has no base points but does not give an embedding, and we classify anticanonically embedded Fano 3-folds with canonical Gorenstein singularities which are not intersections of quadrics. We also study the rationality questions for most of these varieties.

math.AG↗