Proof of the Rokhlin's Conjecture on Arnold's surfaces

量子指标、阿诺德 - 罗赫林曲面与实枚举几何

Quantum index, Arnold-Rokhlin surfaces, and real enumerative geometry

研究关联用于定义复曲面实枚举不变量的两种韦尔申格型符号规则，其关系涉及量子指标与实曲线点集几何，还提出量子指标新定义用于非复曲面实代数曲线精细不变量枚举。

Topological properties of real algebraic varieties: du côté de chez Rokhlin

The survey gives an overview of the achievements in topology of real algebraic varieties in the direction initiated in the early 70th by V.I.Arnold and V.A.Rokhlin. We make an attempt to systematize the principal results in the subject. After an exposition of general tools and results, special attention is paid to surfaces and curves on surfaces.

Rokhlin Conjecture and Topology of Quotients of Complex Surfaces by Complex Conjugation

Quotients $Y=X/conj$ of complex surfaces by anti-holomorphic involutions $conj\: X\to X$ tend to be completely decomposable when they are simply connected, i.e., split into connected sums, $n CP^2\#m\barCP2$, if $w_2(Y)\ne0$, or into $n(S^2\times S^2)$ if $w_2(Y)=0$. If $X$ is a double branched covering over $CP^2$, this phenomenon is related to unknottedness of Arnold surfaces in $S^4=CP^2/conj$, which was conjectured by V.Rokhlin. The paper contains proof of Rokhlin Conjecture and of decomposability of quotients for plenty of double planes and in certain other cases. This results give, in pa…

On a conjecture of Teissier: the case of log canonical thresholds

For a smooth germ of algebraic variety $(X,0)$ and a hypersurface $(f=0)$ in $X$, with an isolated singularity at $0$, Teissier conjectured a lower bound for the Arnold exponent of $f$ in terms of the Arnold exponent of a hyperplane section $f\vert_H$ and the invariant $θ_0(f)$ of the hypersurface. By building on an approach due to Loeser, we prove the conjecture in the case of log canonical thresholds.

Arnold-Thom gradient conjecture for the arrival time

We prove conjectures of Rene Thom and Vladimir Arnold for C^2 solutions to the degenerate elliptic equation that is the level set equation for motion by mean curvature. We believe these results are the first instances of a general principle: Solutions of many degenerate equations behave as if they are analytic, even when they are not. If so, this would explain various conjectured phenomena.

On the growth of local intersection multiplicities in holomorphic dynamics: a conjecture of Arnold

We show by explicit example that local intersection multiplicities in holomorphic dynamical systems can grow arbitrarily fast, answering a question of V. I. Arnold. On the other hand, we provide results showing that such behavior is exceptional, and that typically local intersection multiplicities grow subexponentially.