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An elementary description of nef cone for irreducible holomorphic symplectic manifolds

Anastasia V. Vikulova

Abstract

We describe MBM classes for irreducible holomorphic symplectic manifolds of K3 and Kummer types. These classes are the monodromy images of extremal rational curves which give the faces of the nef cone of some birational model. We study the connection between our results and A. Bayer and E. Macrì's theory. We apply the numerical method of description due to E. Amerik and M. Verbitsky in low dimensions to the K3 type and Kummer type cases.

An elementary description of nef cone for irreducible holomorphic symplectic manifolds

Abstract

We describe MBM classes for irreducible holomorphic symplectic manifolds of K3 and Kummer types. These classes are the monodromy images of extremal rational curves which give the faces of the nef cone of some birational model. We study the connection between our results and A. Bayer and E. Macrì's theory. We apply the numerical method of description due to E. Amerik and M. Verbitsky in low dimensions to the K3 type and Kummer type cases.
Paper Structure (5 sections, 45 theorems, 214 equations, 1 figure)

This paper contains 5 sections, 45 theorems, 214 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lattices of $\mathrm{K3}$ and Kummer types manifolds
  4. Description of nef cone for $\mathrm{K3}$ type manifolds
  5. Description of nef cone for Kummer type manifolds

Key Result

Theorem 1.2

Let $S$ be a $\mathrm{K3}$ surface, $v \in H^*_{\text{alg}}(X,\mathbb{Z})$ be a fixed primitive Mukai vector and $M_{\sigma}(v)$ be the moduli space of $\sigma$-semistable sheaves on $S$ with the Mukai vector $v$ and the stability condition $\sigma \in \mathrm{Stab}(S)$. Consider the set $W$ of elem which is defined in YoshiokaAb. Then the set of $a^{\perp}$ with $a \in W$ defines a chamber decomp

Figures (1)

  • Figure 1: Teichmüller space of IHSM $X.$

Theorems & Definitions (104)

  • Definition 1.1
  • Theorem 1.2: BM
  • Theorem 1.3: Yoshioka
  • Remark 1.4: cf. BeauvilleKnutsen
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Theorem 1.9
  • Corollary 1.10
  • ...and 94 more